Thesis yu chieh cheng 2015

Flat PhD Thesis

Y. C. Cheng

Focusing

Mirrors

K. Staliunas & J. Trull & D. Wiersma

Yu-Chieh Cheng is a joint PhD student at the Universitat Politecnica de Catalunya (UPC), Barcelona, Spain and European Laboratory for Non-Linear Spectroscopy (LENS), Florence, Italy. Since 2011, she has started her PhD in Europe, sponsored by the EuroPhotonics program of Erasmus Mundus Joint Doctorate (EMJD) for three years. Her supervisors are K. Staliunas (UPC), J. Trull (UPC) and D. Wiersma (LENS). Her PhD study is mainly on beam modulations in optics or in acoustics, photonic crystals.

Presently she has published 10 journal papers.

Flat Focusing Mirrors

A thesis submitted for the degree of Doctor of Philosophy in Physical Science of

Yu-Chieh Cheng

Directors Dr. Kestutis Staliunas Dr. Jose Trull Dr. Diederik Wiersma

Departament de FĂ­sica i Enginyeria Nuclear Terrassa, 2015

CONTENTS Preface List of publications 1

Chapter 1 Introduction

4

Chapter 2 Near field focusing

5 6 6

2.1 Light diffraction 2.2 Near and far field lens 2.3 Anomalous diffraction

10 Chapter 3 Flat focusing mirrors with multilayer thin film structure 12 14 15 17 18 20

3.1 Anomalous temporal dispersion of a chirped mirror 3.2 Anomalous angular dispersion of a chirped mirror 3.3 Lateral shift and focal length 3.4 Numerical aperture of flat focusing mirrors 3.5 Optimized structure with longer focal length 3.6 Experiment and conclusion

23 Chapter 4 Flat focusing mirrors from subwavelength gratings 24 25 27 29

4.1 Near field lens with periodic subwavelength gratings 4.2 The comparisons between near field and far field lens 4.3 Fabrication and measurement results 4.4 Near field lens with two parallel waveguide-like subwavelength gratings

31 Chapter 5 Conclusion Appendix collections of publications I II III IV V

“Beam focusing in reflection from flat chirped mirrors.”

“Flat Focusing Mirror.”

“Beam focusing in reflections from flat subwavelength diffraction gratings.“ “Beam focalization in reflection from flat dielectric subwavelength gratings.”

“Negative Goos-Hänchen shift in reflection from subwavelength gratings.”

Flat focusing mirror - a fancy focusing micro-derive

Flat focusing mirrors with two different configurations : a multilayer structure (upper left corner) and a subwavelength grating (upper right corner). The principle of flat focusing mirror is similar with the one of a flat lens (lower left corner) by a negative refraction or an anomalous diffraction. One of applications for flat focusing mirror is the non-diffractive waveguide, composed of two flat focusing mirrors (lower right corner).

Preface The evolution of lensing systems seems a challenging knowledge all the time. It was started from the conventional lensing systems such as geometric optics which treat light as particles which propagate as a straight ray. Reflection and refraction occur as light ray passes objects without considering the superposition of phase of light. As the phase coherent is relevant, the light properties such as interference and diffraction are not able to be explained by geometric optics. Fourier optics considers light as waves so light diffraction and interference occur. Both of geometric optics and Fourier optics could simplify and explain most of lensing systems until the advanced technology brings optical systems to micro- or nano-scale. The light in such a small scale has to be considered as an electromagnetic wave and an exact model is required such as Maxwell’s equations which describe how electric and magnetic fields are generated and altered by each other. Suddenly, a new research appeared after the observation of forbidden band from a periodic structure: photonic crystals. It opened a new field in the research of wave propagation. The new field of science of photonic crystals, combines electromagnetic wave, solid state physics and nanotechnology. It is a very challenging and popular topic and I am happy to be one of the researcher in such field.

In my master study, I have devoted to developing a design of light source for photonic integrated circuits by photonic crystals cavities under directing from my master supervisor, Chi-Chang Chen, at National Central University. During my PhD, I continued working on other photonic crystals devices, flat focusing mirrors. The principle of the flat focusing mirror is based on the flat lens which can break the diffraction limit and allow for subwavelength imaging (but without magnification). There is still a problem that the flat lens cannot be used to image into objects thicker than one wavelength, as what it was initially proposed. However, such flat focusing micro-device still has a promising property: the flat lenses/mirrors do not have any optical axis which benefits to photonic integrated circuits for a faster and better quality of light communication. Therefore, the development of flat focusing mirrors with different configurations, is the main objective of my PhD work.

Finally, I would like to extend my sincere gratitude to my three supervisors, Dr. Kestutis Staliunas, Dr. Jose Trull and Dr. Diederik Wiersma, for their instructive advice and useful suggestions on my PhD thesis. I am deeply grateful of their help in the completion of this thesis. High contribute shall be paid to Shubham Kumar who has put considerable time and effort into his comments on the draft. I am also deeply indebted to all the other partners: Lina Maigyte, Ricard Menchon, Alejandro Cebrecos, Martynas Peckus, Simonas Kicas, Mangirdas Malinauskas, Darius Gailevicius, Hao Zeng, Chi-Hua Ho. Much thanks for professors: Ramon Herrero, Muriel Botey, Crina Cojocaru, Ramon Vilaseca, Rubén Picó, Vicent Romero-Garcia and Victor Sanchez Morcillo for their direct and indirect help to me.

Special thanks should go to my family, Nuria Parra Garcia, Jordi Girona Farres, Oriol Gorina Parra and Milu Gorina Cheng. Finally, I am indebted to my parents, 鄭銘爍 and 連麗文 and for their continuous support and encouragement.

People around me, friends, family, parents, always ask me: “what do you study in your PhD? ” It is always difficult to explain my research to someone who doesn’t study physics, even to someone who does, like colleagues sharing my office. My general and ambiguous answer is “ I’m a hardware designer of “a magic mirror” which can focus light in a micro scale with a flat surface. It can be applied for a new generation of light computers which use light or photons to compute in a faster way.” “para un super, super rapido ordenador…” Their responses are unsurprisingly similar.

“How…interesting…”

“Que…interesante…”

The conversation about my research always stops there and never continues again. Now, I can bring them this magazine, with my signature, and show directly what I study, “Look, this is what I do.” “Mira, mi libro…” I’m sure they will understand everything, well perhaps not everything, but at least read the first chapter, the introduction of flat focusing mirrors. Perhaps, except of my daughter, she will take it as her drawing book.

List of Publications 1.

2.

3.

4.

5.

6.

7.

8.

9.

M. Botey, Y.C. Cheng, V. Romero-Garcia, R. Picó, R. Herrero, V. Sánchez-Morcillo, and K. Staliunas, “Unlocked evanescent waves in periodic structures,” Opt. Lett. 38, 1890 (2013).

Y. C. Cheng, M. Peckus, S. Kicas, J. Trull, C. Cojocaru, R. Vilaseca, R. Drazdys, and K. Staliunas, “Beam focusing in reflection from flat chirped mirrors,” Phys. Rev. A 87, 045802 (2013).

Y. C. Cheng, S. Kicas, J. Trull, M. Peckus, C. Cojocaru, R. Vilaseca, R. Drazdys, and K. Staliunas, “Flat Focusing Mirror,” Sci. Rep. 4:6326 (2014).

Y. C. Cheng, J. Redondo and K. Staliunas, “Beam focusing in reflections from flat subwavelength diffraction gratings,” Phys. Rev. A 89, 33814 (2014).

Y. C. Cheng, H. Zeng, J. Trull, C. Cojocaru, M. Malinauskas, T. Jukna, D. S. Wiersma, and K. Staliunas, “Beam focalization in reflection from flat dielectric subwavelength gratings,” Opt. Lett. 39, 6086 (2014).

Y. C. Cheng and K. Staliunas, “Negative Goos-Hänchen shift in reflection from subwavelength gratings,” J. Nanophotonics, 8, 084093 (2014).

C. H. Ho, Y. C. Cheng, L. Maigyte, H. Zeng, J. Trull, C. Cojocaru, D. S. Wiersma and K. Staliunas, “Controllable light diffraction in woodpile photonic crystals filled with liquid crystal,” Appl. Phys. Lett. 106, 021113 (2015) A. Cebrecos, V. Romero-García, R. Picó, V. J. Sánchez-Morcillo, M. Botey, R. Herrero, Y. C. Cheng, K. Staliunas,“ Acoustically penetrable sonic crystals based on fluid-like scatterers,” J. Phys. D: Appl. Phys. 48 025501 (2015).

Y. C. Cheng, S. Kicas and K. Staliunas, “Flat focusing in reflection from a chirped dielectric mirror with a defect layer,” J. Nanophotonics 9, 093084 (2015).

CHAPTER 1

Chapter 1 Introduction

Image: welcomia/Shutterstock

The photonics technology is a new approach to use light (photons) rather than use electrical signals over a copper cable. Bringing the optical technology into the computing industry will transform the way computers work. A novel computing system by applying optical system could be the smaller and faster. Many active and passive optical components have been in the process of development. One of the important optical elements is the focusing microdevice. The traditional focusing devices can be miniaturized to mirco-scale devices such as micro-lenses 1,2 or micro-mirrors 3 thanks to recent developments in advanced technologies. However, many technical problems appear after shrinking the size of optical devices to a microscale. For example, the unavoidable effects of the optical diffraction strongly increase at a micro-scale. In addition, high performances of such focusing devices depend on critical conditions such as an ideal parabolic curvature and a smooth surface.

Recently carefully engineered subwavelength gratings (SWGs) 4 and optical antennas 5 with flat interfaces have been proposed as flat, tiny and compact focusing lenses/mirrors. These flat engineered SWGs lenses/mirrors 6 with customized properties can substitute traditional lenses, and seem to be more suitable for future potential applications for photonic integrated circuits, vertical-cavity surface-emitting lasers (VCSELs) 7-8 , light-emitting diodes (LEDs) 9 and many others. This noticeable feature opens a new field of ultra-compact focusing micro-devices. All

these devices, however, present optical axes, which brings many difficulties in especially perfect alignments of optical components. “Is there any focusing or imaging device without optical axis?� Yes, the flat lens or superlens proposed by J. Pendry 10 can achieve that. The fancy and famous flat focusing lens can manipulate light / sound, using artificially designed media such as photonic 11-12 / sonic crystals 13 and optical/acoustic metamaterials 14,15. The principle is based on the unprecedented properties such as negative or anomalous refraction 16-18 which can provides distortionfree imaging, potentially with arbitrarily-large apertures.

The diffractive broadening of a beam can also be manipulated to obtain collimated or even focused beams 1922. It is noted that the principle of flat lens works also outside optics and acoustics such as in atomic condensates 23.

Such a slab lens does not possess any optical axis so focusing can be obtained anywhere along a transverse space which brings a wide range of applications. The implementations of these focusing features had been realized only in transmission over the past decades. The idea of flat lens can also be extended to reflection and it has been numerical and experimentally demonstrated firstly by us since 2013 24. Therefore, the development of flat focusing mirrors with a transversal or lateral invariance i.e. lacking any optical axis is the main subject of this PhD work.

Page 1 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 1 Fig. 1.1 Two different concepts of flat focusing mirrors are considered in my PhD: (a) the structure with a flat surface which keeps the transversal invariance and (b) the subwavelength periodic structure which also has the transversal invariance in a larger-than-l scale.

As a comparison, (c) the engineered structure with laterally varying periods breaks the transversal invariance.

(a)

(b)

(c)

x z

The implementations of flat focusing mirrors with the transversal invariance can be conceived from two different geometrical approaches:

1)

Multilayer thin film structures The first idea of flat focusing mirror is based on the modulation in the normal-to-surface direction (z-axis) to keep the transversal invariance (x-axis) simultaneously as shown in Fig. 1.1a. The physical process of light focusing can be understood under the idea of “bringing into correct phase�.

One of the examples of correcting phase is the technique for solving a chromatic dispersion of optical pulses. The common solution is using one-dimensional (1D) chirped mirrors (CMs), consisting in multiple layered structures with two alternating materials of high and low refractive indices 25. The interesting perspective of such 1D CM is that its longitudinal periods dz(z) vary along the propagation distance z so that the different frequencies of the optical pulse reflect at different depths inside the CM. The phase shift of different frequency components can be manipulated in reflections. Therefore, a pulse compression can be obtained from CMs 26.

The same general principle can be applied for a monochromatic beam. The monochromatic beam has only one frequency component but many angular components.

The dephasing between the angular components makes a monochromatic beam spreads or diverges. To obtain a change of the shape of the beam upon reflection, the spatial components of the beam should be modulated. In this case, a CM is used to compensate the phase delays among different angular components for a monochromatic beam. It is a similar way as a CM which fix the problem of the chromatic dispersion for an optical pulse. If the phase delays of the angular components of beam could be also compensated or even over-compensated in the mirror, the original beam is restored or imaged at the focal plane after reflecting from the flat focusing mirror.

One of the limitations is that the flat focusing mirror can only perform a compensation of phase delay so that the focal beam waist is the same as the source one. In addition, the flat focusing mirror works only for a beam and never for a plane wave which, for example, only can be focused by spherical lens or curved mirrors.

Page 2 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

2) Subwavelength periodic structures Another possibility is to modulate the incoming light using a surface grating with a transverse period dx(x) as shown in Fig.1.1b. The modulation in the x direction should be periodic in order to keep the transversal invariance in a large space scale (larger than a modulation period). For the operation of the focusing gratings, only the zero-order reflection mode is needed and this can be achieved by working in the subwavelength range. In this regime, the diffractive angle of the high-order modes becomes very large so the high-order modes decay exponentially as evanescent waves. These high-order modes can still interact with a small number of grating periods. The phase shifts of angular components of a beam can be manipulated between the first and adjacently second scatters. As a result, the beam can be diffused (diverging) or anti-diffused (converging) depending on the ratio between the transverse period and the wavelength 27.

It is noted that the principles of both flat focusing devices with the transversal invariance are similar with the near field lensing effects of flat lenses. They are very different from the far field lenses/mirrors such as conventionally spherical lenses/mirrors, Fresnel lenses, gradient-index material (GRIN) 28-30 lenses and engineered gratings [Fig.1.1c]. The difference is that the far field lenses/mirrors never achieve transversal invariance because their spatial modulations break the transversal invariance.

These non-invariant focusing mirrors are not considered in the thesis for three reasons. First, the compact devices can be applied generally in photonic integrated circuits.

However, the fabrication tolerances are more stringent because the desired phase response is sensitive to the local periods which must be accurately patterned in nanoscale range. Second, although these mirrors are flat, the existence of optical axes also limits their applications. Third, these specially designed SWGs are hard to be designed for narrow beams because large phase variation is required in the latter case which results in much more complicated structures 4.

CHAPTER 1 THIS THESIS focuses on the flat focusing mirrors with two different geometric approaches which keep the transversal invariance . Any modulation of a lateral surface of a mirror should be homogeneous so that flat focusing mirrors can keep transversal invariance.

Such flat focusing mirror is a near field lensing effect similar to the flat lens. It is different from those far field focusing devices such as conversional curved mirrors and lenses. The different physical focusing mechanisms between far field lens/mirrors and near field lens/mirrors are discussed and compared in the next chapter 2.

More details of flat focusing mirrors with two different structures: the Bragg-like multilayer structure and periodic subwavelength gratings are discussed in chapter 3 and chapter 4, respectively. In addition, chapter 3 also demonstrated differently advanced multiplayer structures for a better focusing performance. In chapter 4, in addition to periodic subwavelength gratings, a waveguide-like subwavelength structure consisting of two row gratings are also proposed to observe the near field focusing effect which accompanies with negative Goos-Hänchen effects 31.

Finally, the chapter 5 summarizes and concludes our research in the field of flat focusing mirrors with the transversal invariance. Moreover, some proposals for the implementations of the flat focusing mirrors adapting to photonic integrated circuits are discussed with a future outlook.

In the appendix, the collections of my scientific publications are listed with the brief highlights.

Reference 1. Pitchumani, M., Hockel, H., Mohammed, W. & Johnson, E. G. Additive lithography for fabrication of diffractive optics. Appl.

Opt. 41, 6176-6181 (2002).

2. Popovic, Z. D., Sprague, R. A. & Neville Connell, G. A. Technique for monolithic fabrication of microlens arrays. Appl.

Opt. 27, 1281-1284 (1988).

3. Sabry, Y. M., Saadany, B., Khalil, D. & Bourouina, T. Silicon micromirrors with three-dimensional curvature enabling lensless efficient coupling of free-space light. Light Sci. Appl. 2, e94 (2013).

4. Fattal, D., Li, J., Peng, Z., Fiorentino, M. & Beausoleil, R. G. Flat dielectric grating reflectors with focusing abilities. Nat. Photonics 4, 466-470 (2010).

5. Aieta, F. et al. Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces. Nano Lett. 12(9), 4932-4936 (2012).

6. Chrostowski, L. Optical gratings: Nano-engineered lenses. Nat. Photonics 4(7), 413-415 (2010).

7. Huang, M. C. Y., Zhou, Y. & Chang-Hasnain, C. J. A surfaceemitting laser incorporating a high-index-contrast subwavelength grating. Nat. Photonics 1, 119-122 (2007).

8. Huang, M. C. Y., Zhou, Y. & Chang-Hasnain, C. J. Single mode high-contrast subwavelength grating vertical cavity surface emitting lasers. Appl. Phys. Lett. 92, 171108 (2008).

9. Xie, R.-J., Hirosaki, N., Mitomo, M., Takahashi, K. & Sakuma, K. Highly efficient white-light-emitting diodes fabricated with short-wavelength yellow oxynitride phosphors. Appl. Phys. Lett.

88, 101104 (2006).

10. Pendry, J. B. Negative refraction makes a perfect lens. Phys.

Rev. Lett. 85, 3966-3969 (2000).

11. Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58(20), 2059-2062 (1987).

12. John, S. Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 2486-2489 (1987).

13. Miyashita, T. Sonic crystals and sonic wave-guides. Meas. Sci. Technol. 16, R47-R63 (2005).

14. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative refractive index. Science 305, 788-792 (2004).

15. Ding, Y., Liu, Z., Qiu, C., & Shi, J. Metamaterial with simultaneously negative bulk modulus and mass density. Phys.Rev. Lett. 99, 093904 (2007) 16. Veselago, V. G. The Electrodynamics of substances with simultaneously negative values of ε and μ. Sov. Phys. Usp. 10, 509-514 (1968).

17. Cubukcu, E., Aydin, K., Ozbay, E., Foteinopoulou, S. & Soukoulis, C. M. Electromagnetic waves: Negative refraction by photonic crystals. Nature 423, 604-605 (2003).

18. Yao, J. et al. Optical Negative Refraction in Bulk Metamaterials of Nanowires. Science 321, 930 (2008).

19. Parimi, P. V., Lu, W. T., Vodo, P. & Sridhar, S. Photonic crystals: imaging by flat lens using negative refraction. Nature 426, 404 (2003).

20. Lu, Z. et al. Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies. Phys. Rev. Lett. 95, 153901 (2005).

21. Maigyte, L. et al. Flat lensing in the visible frequency range by woodpile photonic crystals. Opt. Lett. 38, 2376-2378 (2013).

22. Cebrecos, A. et al. Formation of collimated sound beams by three-dimensional sonic crystals. J. Appl. Phys. 111, 104910 (2012).

23. Staliunas, K. & Longhi, S. Subdiffractive solitons of BoseEinstein condensates in time-dependent optical lattices. Phys. Rev. A 78, 033606 (2008).

24. Cheng, Y. C. et al. Beam focusing in reflection from flat chirped mirrors. Phys. Rev. A 87, 045802 (2013).

25. De Silvestri, S., Laporta, P. & Svelto, O. Analysis of quarterwave dielectric-mirror dispersion in femtosecond dye-laser cavities. Opt. Lett. 9(8), 335-337 (1984).

26. Szipocs, R., Ferencz, K., Spielmann, C. & Krausz, F. Chirped multilayer coatings for broadband dispersion control in femtosecond lasers. Opt. Lett. 19(3), 201 (1994).

27. Cheng, Y. C., Redondo, J. & Staliunas, K. Beam focusing in reflections from flat subwavelength diffraction gratings. Phys. Rev. A 89, 33814 (2014).

28. Gomez-Reino, C., Perez, M. V. & Bao, C. in Gradient-Index Optics: Fundamentals and Applications (Springer, Berlin, 2002).

29. Wilkinson, P. B., Fromhold, T. M., Taylor, R. P. & Micolich, A. P. Electromagnetic wave chaos in gradient refractive index optical cavities. Phys. Rev. Lett. 86, 5466-5469 (2001).

30. Smith, D. R., Mock, J. J., Starr, A. F. & Schurig, D. Gradient index metamaterials. Phys. Rev. E 71, 036609 (2005).

31. Cheng, Y.-C. & Staliunas, K. Negative Goos-Hänchen shift in reflection from subwavelength gratings. J. Nanophotonics 8, 084093 (2014).

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CHAPTER 2

Chapter 2 A frequently asked question is: why our focusing effects can be treated as a “near-field focusing”?

Especially since the focal distance of our flat focusing mirrors can be more than a hundred microns.

The confusion comes from one of the fancies focusing devices: a flat lens with negative index of refraction. It is known that the flat lens, with effective negative index of refraction, can enhance evanescent waves to achieve a perfect image. This could be the reason why people often confuse the near field with the evanescent field which exists only at a few wavelengths. The long focal length of our focusing mirrors is definitely not in the evanescent area. It should be noted that our flat focusing mirrors can focus in the near as well as the far field zone. The near field focusing is a name of the physical mechanism of focusing. All these misunderstandings will be clarified in this chapter.

Near Field Focusing

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ocusing and spatial manipulation of radiation in optical systems is conventionally performed by curved mirrors and spherical lenses. The physical principles of flat focusing mirrors in this thesis are intrinsically different from those of curved mirrors. While the focusing by our flat focusing mirrors with a transversal invariance is a near-field focusing (NFF), the focusing by conventional mirrors is a far-field focusing (FFF). The difference between NFF and FFF is not very trivial and often requires some clarification. The well-known ‘thinlens’ formula is 1/lobject+1/limage = 1/F for conventional far field focusing devices. This relation becomes lobject+limage = F for flat lensing, which is a crucial difference. This chapter is devoted to the clarification of these concepts of NFF and the FFF.

CHAPTER 2 2.1

Light diffraction Fig. 2.1b. As the finite plane wave with many angular components propagate along z direction, these angular components obtain different phase shifts f = (k0-kcos(q))∙z.

It means that an infinite plane wave never diffracts and that the beam diffracts due to a broadened angular spectrum as shown in Fig. 2.1c. Therefore, the formalism of diffraction can also describe how an optical beam, such as a Gaussian beam, propagates in free space. Similarly, the wave front is also flat at the beam waist but it starts to curve due to the relative phase shift among the different angular components as shown in Fig. 2.1d.

Before starting the discussion of NFF and FFF, let us define diffraction in the near field and the far. Diffraction refers to the phenomena which occurs when a wave encounters an obstacle or a slit, as illustrated in Fig. 2.1a.

The geometry of the slit acts as a source of secondary emission of radiation, which gives new wave fronts and new far field distribution. A Far field is essentially a composition of many plane waves traveling at different angles. Any arbitrary wave envelope can be decomposed into many plane wave components at different angles (angular components) in the angular spectrum 1 as shown in

Fig. 2.1 The illustration of diffraction. (a) The diffraction pattern from a single slit. (b) A wave vector at a given angle q, kx = k0sin(q) and kz = k0cos(q). (c) A Gaussian beam decomposed into many plane waves and (d) its angular spectrum.

How to distinguish between the near field and the far field regions? One can see the difference between these two regimes in the following two classical examples corresponding to a plane wave passing through a slit and a Gaussian beam propagating in free space.

1. In the study of wave diffraction from an obstacle or slit, the region called the near field (where Fresnel diffraction occurs) is the distance below df = 2D2/ from the obstacle or aperture, where D is the width of the aperture and  is the wavelength of the diffracting wave. The region beyond this plane is called the far field (where Fraunhoffer diffraction occurs). 2 2. For a Gaussian beam, the boundary between near field and far field is defined as the Rayleigh length Zr = 2/ where 0 is the radius of the beam waist. The traditional ray optics description can be applied only in the far field where the radius of curvature R(z) = z(1+(Zr/z)2) is approximated R(z)  z for z >> Zr. 3

Fig. 2.2 The near and far field diffractions. (a) A plane wave passing through an slit and its far field is changed to the distribution as a sinc function. (b) A Gaussian beam propagating in free space and its far field keep the same as a Gaussian distribution.

(a) Near field

Far field

Far field

Near field

Far field

Far field

D

d f  2D

2



(b)

flat wave Near field front

Near field

Far field

Far field

q (angle) Far field

Far field

20

Zr 

02



flat wave front

q (angle)

Page 5 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 2 2.2

Near and far field lens

It is observed that the phase and intensity in the near field dramatically changes during propagation while its far field (Fourier image) is a fixed pattern which does not change during the propagation in free space. The propagation affects the phase and distribution in the near field due to the phase shift among different plane wave components (the far field). However, the distribution in the far field is unchanged. Note that the far field is only changed after passing through an object such as a lens or prism. For example, a prism has a linearly varying optical thickness in transverse direction given by d(x) ~ x which introduces in the transmitted beam a transversely varying phase with a linear slope A(x)′ = A(x)eix . The far field becomes A(kx)′ = A(kx-) and this effect corresponds to the well-known refraction phenomena. As the surface of the optical element is spherical, the angle of refraction at any positions depends on the local curvature which is how a conventional lens focuses. Therefore, the far field is changed from one to many different plane wave components with varying orientation.

It should be emphasized that the flat lens 4 composed of two parallel interfaces cannot change angles of refraction or, equivalently, cannot change the far field distribution. In order to gain insight into the purpose of this work, the comparison between flat lens (near-field lens) and a conventional thin lens (far field lens) is illustrated as shown in Fig. 2.3.

The far-field lens means that it modulates the distribution of the far field. It is noted that only modulating the far-field distribution is possible to obtain different size of the focal spots. The field distributions at the focal plane in fact are far fields. For example, the far field is broader or narrower immediately after conventional lenses which results in narrower or broader focal beam waist as shown in Fig. 2.3a and 2.3b respectively.

It is known that the conventional phase arrays (similar to Fresnel lens) which manipulate near-field interference are used to produce a desired far-field pattern. Many specialized subwavelength gratings 5 or metasurfaces 6 which modulate the phase of near field can result in the focal distances lying in the near field. It is noted that the principle of focusing of those engineered arrays should be far field lensing like Fresnel lens rather than near field lensing like flat lens.

On the contrary, the near-field lens such as flat lenses means that it modulates the distribution of the near field.

The transmitted beam behind the flat lens can be broader or narrower as shown in Fig. 2.3c and 2.3d. However, the far field of the beam remains unchanged due to its two parallel flat surfaces, so the focal beam waist is the same with the source beam waist.

Remember that, the formula of flat lensing is lobject + limage = F. The difference is that the distance from the flat lens to an image plane is limited by the focal length F of the lens and the image can never be infinitely far away. For example, the limitation of the flat lenses or superlenses is that the projected image only exists near the surface of the

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lens, with no magnification. This fact limits the practical applications for the flat lenses or superlens, since any camera needs to be within the lens’s “near field”. To make a lens useful for far-field imaging below the diffraction limit, the evanescent waves have to be converted into propagating waves, which is what the hyperlens 7 does. The Hyperlens can project the high-resolution image from near field to the far field. Developing a near-field lens with a long focal distance for a observable and measurable phenomenon is always challenging and it is one of the main objectives of this thesis.

The ABCD matrix method in optics can also show the difference between far-field lens and near-field lens. A light wave is characterized by the distance x from the axis of the lens and an angle θ with respect to the propagation direction z. These quantities in front and behind the element are related by the ABCD matrix for a far-field lens with a focal length f

 x2   1    q 2   1 f

0   x1    1   q1 

(2.2.1)

For a near-field lens, the angle of incidence and refraction are the same due to the flat interface. However, the distance x is modified. Therefore, the simple ABCD matrix for a near field lens can be obtained as

 x2   1  F   x1       q 2   0 1   q1 

(2.2.2)

and the focal length F of the near-field lens follows lobject+limage = F.

(2.2.3)

These two ABCD matrix also show that the angle (far field) is changed/unchanged and the transmitted field (near field) is unchanged/changed for the far/near field lens .

2.3 Anomalous diffraction The idea of near field lens in the previous section is discussed by geometric optics or ray optics. In mathematics, the focal length can also be calculated from the Maxwell’s equations in free space which has the form called the Helmholtz equation   2  k 2  E (r )  0 , k = n0/c.

For the paraxial approximation of the Helmholtz equation :

 2 A  2ik

A 0, z

(2.3.1)

2 2 2 2 2 where      x    y is the transverse part of the Laplace operator.

In the paraxial approximation, the complex amplitude of the electric field is E r  A(r )e ikz and A(r ) describes transverse profile with its variation in propagation.

In the Fourier domain, Eq. (2.3.1) can be written as



CHAPTER 2

(a)

(c)

(b)

(d)

Fig. 2.3 The comparison of far field lens and near field lens. A beam passes through convectional thin lens with (a) a smaller f1and (b) larger f2 focal length.(b) The beam passes through flat lens with (c) smaller F1 and (d) larger F2 focal length.

A(k ) ik x2  A(k x ) z 2k

(2.3.2)

The solution of Eq. (2.3.2) is: A(kx , z )  A(k ,0)ei ( kx

2

2k ) z

(2.3.3)

which means that during propagation in homogeneous space, the angular components acquire a phase shift:

 (k x )  z k x2 2k0

(2.3.4)

proportional to the propagation distance z and to kx2; thus each transverse component acquires a different phase.

Eq. (2.3.4) shows that if a beam is incident with a distance 2 of lobject, the phase becomes   k x 2k0  lobject . In order to obtain flat lensing, such a phase spread of angular components is to be compensated. For example, it is supposed that L is the effective propagation distance of the flat lens in free space so the phase delay 2 becomes   k x 2k0  (lobject  L) . As beam is in-phase again in transmission at a distance limage behind the structure, it follows a relation L + lobject + limage = 0. To achieve focusing, the effective propagation distance provided by the flat lens

should be negative (L < 0), which means the focal length F = -L. Therefore, the relation of focal length for near-field lens lobject + limage = -L = F can be obtained. Such dephasing of the angular components (far field) does not alter the intensity distribution of the far field but instead results in a change of the near field profile.

It is important to find a modulated material with such anomalous diffraction. The profile of dispersion in free 2 space  (kx , z)   kx 2k  z has negative curvature –z/k0, so the curvature of the anomalous diffraction should be positive, as marked by the red line in Fig 2.4b.

Summarizing, as the second derivative of the particular phase shift (curvature) is positive, the focusing can be expected with a focal length F

F   2 (k x ) k x2  k0

(2.3.5)

The Eq. (2.3.5) can also be written in terms of the first derivative of phase shift s    (k x ) k x

F  s kx  k0

(2.3.6)

Note that the lateral shift must decrease linearly with increasing angles to obtain a positive F and an aberrationfree focusing.

Page 7 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

kz

CHAPTER 2

x (mm)

(c)

lobject z

(b)

kx

limage

d kx

kz

0

20

40

kz

60

80

100

Material

(d)

kx kz

Material

0

-20 -20 20

x (mm)

x

kz

20

lobject  limage  d  F

(a)

kz

0

focusing

source -20 -20

0

20

40

60

80

100

z (mm)

x (mm)

20

(c) 2.4 The principle of a flat lens.

Material Fig. (a) The illustration of focusing by a flat lens with thickness d. The anomalous diffraction is also d as if the parabolic curvature of normal dispersion (blue) and anomalous dispersion (red) is the same but of an opposite sign. (b) The spatial 0 spectrum of a Gaussian beam in free space (blue) and its angular range in the presence of anomalous diffraction of a flat lens. (c) The field and (d) intensity distributions of a Gaussian beam passing through a flat lens.

x (mm)

-20 -20 0 20 40 60 80 100 In this section, the flat lensing is discussed based on the 20

concept of anomalous diffraction.

The anomalous Material (d) diffraction can be obtained in the modulated structure, 0photonic crystals (PhCs). However, the angular range of the anomalous diffraction is usually not broad enough to focus focusing source a point source but is available for a narrow beam. This is -20very similar to the super-prism effect where there is a -20 0 20 40 60 80 100 strong bending of the angular dispersion curves with some z (mm) regions of sharp corners. This effect can separate plane waves at slightly different incidence angles with a large variation of refractive angles as a prism does.

A broad angular range of flat lens6 can be achieved by PhCs 8 or metamaterals 9 with negative refraction at all angles, which can provide an effective negative refractive index close to n = -1. However, the losses and the short focal lengths are still one of the challenges of PhCs or metamaterials with negative refraction till date. Therefore, PhCs with anomalous diffraction are chosen as our material and a Gaussian beam is considered as our source. The beam focusing in transmission using 3D PhCs (a woodpile structure) has been demonstrated by my colleague Lina Maigyte 10 and the same principle can be also applied for reflection. The near field lens proposed by her can reach the focal distances extending to far field zone. The maximum focal distance for a beam at wavelength 570 nm with the beam waist 0= 1.1 mm is 60 mm which is larger than the Rayleigh length Zr = 6.6 mm. This is a near field lens which can get a focusing in the far field.

Figure 2.4 shows that a beam source on one side of the flat lens results in a perfect beam image on the other side of the flat lens. At the beginning, the beam starts to diverge and the phase front turns from flat to convex in free space with a normal diffraction marked with blue lines in Fig. 2.4b. When such a beam enters into a modulated material with concave dispersion marked with red lines in Fig. 2.4b, the phase delay is compensated in an opposite way: evolving to a flat phase front or even surpassing it to

Page 8 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

become a concave phase front. When the beam with such a phase front enters into the free space again, it starts to focus, forming the image of the initial beam distribution at the position where the phase front is flat as shown in Fig. 2.4c.

For a given thickness of the flat lens, the amount of phase delay in the anomalous diffraction, to compensate the phase in normal diffraction, is fixed. As a result, the sum of lobject and limage is constant and it also corresponds to the Eq. (2.2.3).

A short focal length and a simple design, i.e. flat surface, are often desirable for the photonic integrated circuits. The engineered subwavelengh gratings have been proposed for the photonic integrated circuit but these specialized designs eventually cannot avoid the existence of an optical axis.

Another problem is that these far-field lenses have a minimum focal length, limited by the geometries of conventionally spherical lenses (curved mirrors) or the local geometries of SWGs. To conclude the chapter, the flat lenses with anomalous diffraction are near-field lenses and, importantly, they have no optical axes. Such flat lens is applied for a beam rather than a point source because of its limited working angular range, known as the numerical aperture (NA). The principle of the flat lens with anomalous diffraction can be also realized in reflection. The discipline of the design of the flat focusing mirror is to keep the lateral invariant and only modulate the phase of the far field in transversal direction.

As long as the phase shift of the angular components in reflection is calculated, the slope of the first derivative or the second derivative of the angular phase dispersion can indicate where the focalization is possible. In order to obtain the phase shift on the far field components at kx or the angles q, either the phase of each angular component should depend on the angles, or some surface waves should be excited to affect the lateral shift. Both cases are considered in chapters 3 and 4.

CHAPTER 2 Reference 1. Goodman, J. W. in Introduction to Fourier Optics (Roberts & Company, Colorado, 2004).

2. Hecht, E. in Optics (Addison Wesley, Reading, 2001).

3. Quimby, R. S. in Photonics and Lasers: An Introduction (Wiley, Hoboken, 2006).

4. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966-3969 (2000).

5. Fattal, D., Li, J., Peng, Z., Fiorentino, M. & Beausoleil, R. G. Flat dielectric grating reflectors with focusing abilities. Nat. Photonics 4, 466-470 (2010).

6. Jiang, X. Y. et al. An ultrathin terahertz lens with axial long focal depth based on metasurfaces. Opt. Express 21, 30030-30038 (2013).

7. Liu, Z., Lee, H., Xiong, Y., Sun, C. & Zhang, X. Far-field optical hyperlens magnifying sub-diffraction-limited objects. Science 315, 1686 (2007).

8. Cubukcu. E., Aydin, K., Ozbay, E., Foteinopoulou, S., Soukoulis, C. M. Electromagnetic waves: Negative refraction by photonic crystals. Nature 423, 604-605 (2003).

9. Smith, D. R., Pendry, J. B., & Wiltshire, M. C. K. Metamaterials and negative refractive index. Science 305, 788-792 (2004).

10. Maigyte, L. et al. Flat lensing in the visible frequency range by woodpile photonic crystal. Opt. Lett. 38, 2376-2378 (2013).

Initially I was also confused about the definitions of the near/far field focusing until my professor, K. Staliunas, explained me the idea. However, people from conferences, referees of journals, colleagues always asked me the same question. It is reasonable that they think the focal length is too far to be considered as near field focusing.

That’s why I decided to dedicate a chapter to its explanation.

I think it is important and interesting for those who are dealing with focusing devices in both transmission and reflection. Here, we prefer to explain the flat lensing from the concepts of the anomalous diffraction by PhCs rather than with the negative index of refraction n = -1 by metamaterials. Such flat PhC lens in transmission has been realized with 3D PhCs by my colleague, Lina Maigyte, who is one of the best PhD students of K. Staliunas.

We worked and studied together and our prof.

Staliunas always shows us his ideas and derivations on his hand written drafts. When we received his drafts, we studied those fine and scratchy words until we could understand them completely.

Then we burned them in order that nobody else takes the idea. Some drafts remain not completely burned are shown on the right hand side…

Page 9 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 3

First guess of the flat focusing mirror

The design of a focusing mirror without any optical axis should consist of a structure, invariant in the transversal direction. The simplest pattern is a uniform distribution in the transversal direction but a modulation in the longitudinal direction. The focusing mirror must work as a reflector, for example, by working in a bandgap 1 region where light cannot pass through and is reflected.

Therefore, a multilayer structure is one of suitable models for the flat focusing mirror.

Page 10 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

Chapter3

Flat focusing mirrors with multilayer thin film structure

Page 11 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

CHAPTER 3 3.1 Anomalous chromatic dispersion of a chirped mirror Fig. 3.1 The operation principle of a chirped mirror. (a) An initial pulse propagating in dispersive homogeneous media becomes (b) a positive chirped pulse. When such a chirped pulse enters into the chirped mirror at normal incidence, the different frequency components are reflected by different layers where the corresponding band gap appears. (c) After reflection, the negative dispersion introduced by the negative chirped mirror compensates the positive chirp of pulse so that the pulse is restored to its original shape.

(a)

f

f



 E (t , z)e

tf

dt = A(, z )e - io t = A(, z ) ei ( , z ) e -iot - (3.1.2) where =- o and  is the phase of spectral components.

We also define the group delay Tdelay as it

Tdelay =

 = GD  GDD·  ... 

(3.1.3)

The group delay (GD) is related to the position of each frequency component along the pulse. The constant term is called the GD and the coefficient at  is called the group delay dispersion (GDD). If the GDD is non-zero, it means that the different frequency components of the pulse dephase, and the pulse is broadening and getting chirped in propagation.

Starting from Maxwell’s equations, a wave equation for the propagation of the pulse in a dispersive media can be obtained. If we retain terms up to second order in dispersion, we obtain the parabolic equation for the envelopes of pulses, in the reference frame moving with the group velocity or the pulse:

A G 2  2 A i =0 z 2 t 2

(3.1.4)

where G2 is the group velocity dispersion coefficient. This equation is equivalent to the one obtained for beam

Page 12 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

t





f

f

f

t

(3.1.1)

t

 t (c)

where A is the complex envelope,  is the phase along the pulse and o is the carrier frequency. The Fourier Transform gives the frequency content of the pulse: E (, z ) =

tt

f

Before starting the analysis of beam reflection from our flat focusing mirror, it is instructive to take a close look at the problem of managing chromatic dispersion for ultrashort pulses. 2 Dispersion occurs in all materials due to the dependence of the refractive index on frequency and affects strongly the evolution of optical pulses. The pulse broaden as they propagate in dispersive medium in the same way as diffraction broadening occurs for an optical beam.

Let us consider a pulse propagating along a given direction in space, z. The pulse is a temporal envelope of a plane wave so we do not consider diffraction effects. Such a pulse can be represented as

E (t , z ) = A(t , z )e - io t = A(t , z ) ei ( z ,t ) e - io t

(b)

t

diffraction in paraxial approximation if we substitute time t by transverse coordinate  x . In a completely analogous way, as we did previously, we can obtain the solution of this equation in the frequency domain as:

t

A(, z ) = A(, 0) exp

i

 G2 2 z 2

(3.1.5)

 amplitude This solution shows that the spectral (equivalent to the far-field in diffraction) does not change when a pulse propagates in a dispersive medium but rather the spectral components acquire the phases, which depends quadratically on the frequency. As a consequence of this phase modulation of spectral components, the temporal pulse profile changes its temporal shape, broadening during propagation. In the region of normal GDD (G2 > 0), low frequency components travel faster than the high frequency components so the pulse will be “red” at the leading edge and the “blue” at the trailing edge (Fig. 3.1a). If the material dispersion is anomalous (G2 < 0), we obtain the opposite effect with blue components at the leading edge.

Any ultrashort pulse (with broad spectrum) propagating in a dispersive medium becomes chirped and broadened, which is detrimental in many applications of these pulses.

Pulse compression and chirp elimination is possible by using a media which provides a negative GDD. One of the most important techniques to compensate the dispersion effects of ultrashort pulses is using the chirped Bragg mirror. A 1D Bragg mirror consists of a multiple layer structure of alternating materials with high and low refractive indices. The optical thickness of each layer is close to a quarter-wavelength, nHlH = nLlL = /4, for operating wavelength λ. The chirping 3,4 implies that the periods vary spatially along the longitudinal direction. The different frequencies of a pulse can penetrate and reflect at different depths in this multi-layered structure.

Consequently, each frequency component of the pulse travels different distances and experiences different optical path lengths inside the mirror. If the chirp is correctly designed, anomalous phase modulation can be achieved, where blue frequencies travel shorter distances than red ones. In this way, a flat chirped mirror has the potential to compress the pulse to its initial value which is completely equivalent to suppress diffractions of an optical beam (detailed discussion in Sec. 3.2).

CHAPTER 3 The main problem of the chirped mirror design is the oscillations (or fringes) that appear in the dispersion curve, which results in distortion of the pulse shape. These fringes appear due to the coupling between the forward and backward waves caused by the periodicity of the structure.

The interference of the reflected waves between the first and middle layers is especially significant. This fringing effect can be reduced by eliminating the reflectance from the previous layers by using mechanisms such as a doublechirped mirror 5–7 or an antireflection coating (AR). 8 In order to obtain broader and smoother negative GDD dispersion for ultra-short pulses, the frequency regions of “dispersive mirrors” becomes more appropriate due to complicated combinations of resonance and penetration depth effects. The resonant structure, a Gires-Tournois interferometer (GTI), 9 comprises of two high- and lowreflecting structures separated by a cavity of thickness close to half wavelength of the incident light. Such nanoscale GTI embedded in the multilayer structure can introduce a large slope of GDs at selected windows of wavelengths to obtain large negative GDD values. However, the performance with a single GTI is satisfactory only in a narrow spectral range. Many advanced structures such as chirped mirrors with a resonance 10 or double GTI 11, a combination of two resonances with Bragg mirrors, can be used to achieve larger negative dispersion in a broader spectral region than that of single a GTI structure.

Nowadays, complicated dispersive mirrors are designed with the help of powerful numerical algorithms 12 and may have dozens of layers.

Figure 3.2a shows the spectral region covered by specially designed dispersive mirrors for different optical frequencies. The phase shift imposed by the chirped mirror in the bandgap becomes roughly a quadratic function, so that the GD and GDD are obtained as a linear trend and with constant value. There are unavoidable oscillations, but small enough to be neglected as shown in Fig.3.2c.

(a)

(b)

Fig. 3.2 The characteristics of dispersive mirror for different spectra. 13 (a) The reflectance for different dispersive mirror. (b) The phase shift of each dispersive mirror. (c) A case of a dispersive mirror in UV spectra.14 The reflectance, GD and GDD are plotted as green, blue and red lines. A linear slope of GD and a constant value of GDD are obtain for a high dispersive mirror. The bandwidth of negative GDD can achieve around 70 nm. The oscillation of GDD could be neglected.

OptiLayer software, my collegue Simonas Kicas uses, is a unique mathematical know-how and has the most comprehensive software implementation.

Its outstanding computational performance enables OptiLayer users to find the best possible coating designs and to successfully produce optical coatings.

Page 13 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 3

3.2 Anomalous angular dispersion of a chirped mirror

In the previous chapter, it is shown that chirped mirrors can manipulate the properties of optical pulses on reflection at normal incidence. They are applied for solving the problem of chromatic dispersion of the pulses by compensating the phase delay for different frequency components.

“Think differently. How about the angular dispersion?”

“Can we manipulate the angular dispersion by the chirped mirror? “ “what happens when narrow focused beams of monochromatic wave reflect from such chirped Bragg mirrors?”

The beam can be considered as a superposition of different plane waves propagating at different angles. For example, the wave vectors k0 = (k x , k z ) of a monochromatic beam at a given frequency  = c|k0| lie on a ring in k-space.

The important magnitude for the propagation in a onedimensional layered structure is the longitudinal component of the wave vector kz = |k0|cos(q), where q is the angle from the normal to the mirror surface. Thus, the projection to the normal direction of each angular plane wave component of the beam, will correspond to a different effective frequency  = c|k0|cosq as illustrated in Fig. 3.3 by red, yellow and blue arrows. Therefore, the principle of compensation of chromatic dispersion by chirped mirrors can equivalently be applied for the compensation of spatial dispersion (diffraction) of the beam.

When a beam is totally reflected from a flat interface between two semi-infinite dielectric media, a lateral shift of the reflected beam with respect to the incident position is observed, known as the Goos-Hanchen (GH) shift 13.

Normally, the GH shift is positive and proportional to the incidence angle as illustrated in Fig. 3.3a. On the other hand, for a chirped mirror providing an anomalous slope, the angular components at the larger angles experiences smaller lateral shift and this effect allows to bring the beam into focus (or form an image) in reflections as shown in Fig. 3.3c. Therefore, the chirped mirror structure can compensate the spatial chirp of the beam, i.e. manipulate the curvature of the beam’s wave front by modulating the penetration depth for different angular components of the beam. Figure 3.3d shows that the flat focusing mirror can modulate the phase of far field of beam at the position before A2 and after mirror A3, and eventually the position of focal point in reflection, A4. At the end, it leads to focalization of the beam in reflection.

Page 14 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

Fig. 3.3: The illustrations of focusing due to the anomalous angular dispersion of chirped mirror. (a) In usual situation, the lateral shift of reflecting beam increases with angle. (b) Focusing appears at the incidence angles when the lateral shift of reflecting wave is decreases for increasingly incidence angle (anomalous slope) as shown in green color. (c) A chirped mirror structure can achieve the anomalous dispersion because the plane waves at larger angles reflect in advance than those at smaller angles. (d) The illustration of amplitude and phase of a beam before and after reflection in free space A(x) and their far fields A(q).

CHAPTER 3 3.3 Lateral shift and Focal length The chapter 2 proposed that effective propagation length L that can compensate a diffracted beam turning its front into a flat profile is determined by the curvature of the spatial dispersion line. The more curved is the anomalous dispersion, the more far away is the focal length because it needs more distance in free space to compensate the negative phase profile. It can be proven that the concave curvature provides a negative diffraction, with the diffraction coefficient adiffr (corresponding to the curvature)

adiffr = -

1  2 2 k x 2

(3.3.1)

Eq. (3.3.1) controls how the angular components dephase during the propagation in the unit length. In the case of reflections, the diffraction coefficient can also be considered and the focal length can be calculated by the formula in term of lateral shift s = -  kx .

F = k0

 2 s = - k0 2 k x k x

(3.3.3)

It brings to two consequences: Firstly, the second order derivative, equivalent to the curvature of dispersion, should be positive as we mentioned in sec. 3.2. Secondly, the slope of lateral shift should be negative.

A simple geometrical approach is proposed for a insight of lateral shift and focal length. To simplify the treatment, the bandgap of a chirped mirror is supposed to sweep linearly with the longitudinal distance z as shown by black dashed

line in Fig. 3.3c. With such approximation, a numerical lateral shift and diffraction length can be calculated: 16 s=

-d 2 1 k2 k = ( k - k0 - x ) x dk x a C 2 k k

L = -k

-d 2 2 3 k x2 = ( k k ) 0 aC 2 k dk x2

(3.3.4)

(3.3.5)

The analysis of equation (3.3.4) shows that the diffractive propagation length L is always positive at small angles (as the optical axis kx = 0) for a chirped structure. However, it can become negative at some large incidence angles only for a positive chirped mirror. This means that the beam reflecting at some sufficiently large angle should experience zero or negative diffraction. Therefore, the sign of the chirp needed for the flat focusing mirror is proposed to be positive, opposite to that of the chromatic dispersion compensating chirped mirror used for pulses, spreading in normal dispersion media.

The oscillation effect in the chirped mirror does not appear in estimations (Eq. 3.3.4 and 3.3.5) in the simple geometric approach. It comes out that due to this fringing, the negative diffraction length can be obtained for both types of chirped mirrors at the angular values where the resonant peaks appear, which is very different from the prediction of the analytical model. In fact, the fringes even increase the effect of the negative diffraction at particular angular ranges where Focal length becomes strongly positive, also well beyond the estimations of our rough analytical model.

Page 15 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 3 (a) Lateral shift (mm-1)

10 10

55

TE TM Prediction

00

5-5 -10 10 -60 -60

s=-40 -40

 q -20 -20

00

20 20

40 40

60 60

Angle of incidence (degree)

(b) Focal length (mm-1)

The lateral shift s and focal length F given by a multilayer structure can be also calculated by using a transfer matrix method by considering a real chirped structure. From the calculated reflectance (complex value) r for each incidence angle we obtain the phase of the reflected light as the imaginary part of the reflection coefficient argument to obtain the phase of reflected light. In this way, the lateral shift and diffraction length by the derivation of the phase with respect to the incidence angle can be calculated as shown in Fig. 3.4. It is noted that kx is a function of incidence angle .

From these calculations, we obtain the focusing regions corresponding tokxthe where the diffraction length or = kangles 0 sin(q ) the slope of the lateral shift is negative. For the particular case of the chirped mirror in Ref. 16, the focusing may appear at incidence angles 22o, 38o and 52o for TE polarization as shown with blue areas in Fig. 3.4. The tendency of lateral shifts and focal lengths matches with the geometric approach as the modulation of fringes is neglected.

These oscillations can be eliminated by introducing an AR coating at the front face of the chirped mirror so that the tendency of dispersion can match the simple geometric approach. However, the effect of anomalous diffraction will be reduced as the oscillation is eliminated. Although the simple prediction is not sufficient to full description of the anomalous diffraction, the idea shows the physical principle.

300 300

TE TM Prediction

200 200 100 100

00 -100 -100

-200 -200 -300 -300 -60 -60

F = k0 -40 -40

 2  2q

-20 -20

00

20 20

40 40

60 60

Angle of incidence (degree) Fig. 3.4 Angular dependence of the reflection properties of the chirped mirror structure: (a) lateral shift s (b) Focal length F. Results for the TE and TM polarizations are given by blue and red-dashed lines respectively. Analytical estimations of geometric approach are depicted by green-dotted lines. The chirped structure is the same as that in Ref. 16.

The transfer-matrix method is a method used in optics and acoustics to analyze the propagation of electromagnetic or acoustic waves through a stratified (layered) medium. This method is widely used, for example, in the design of anti-reflective coatings and dielectric mirrors.

Born, M.; Wolf, E., Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Oxford, Pergamon Press, 1964.

Page 16 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 3 3.4

Numerical Aperture of the focusing mirror

The analysis of the phase in the angular spectrum not only shows the working angles of focusing but also indicate the minimum width of the incident beam. It is obvious that the fringing of the dispersion curve limits the angular bandwidth of the focusing range. Therefore, the size of a beam is limited since its angular bandwidth should be smaller than the available angular region. In fact, these fringing effects can increase the value of the focal length (F) for particular angles but, unavoidably, result in a smaller angular bandwidth of focusing range. This angular bandwidth can be also regarded as an angular numerical aperture (NA) of the focusing mirror. It is important to manipulate the fringing in order to obtain a high value of focal length in a broad working range.

The origin of the fringes is due to the resonances of the Fabry-Perot cavity 17 formed between the entrance face and the reflector. Then the separation between the fringes is kz  2dmirror  2 and the relation between the cavity length dmirror and k z is obtained:

kz =  dmirror

Mathematica code

(3.4.1)

Note that the effective cavity length dmirror can increase due to several round trips inside the structure.

For example, at an angle 45o, the maximum lateral shift would be smax = 2dmirror as the wave propagates in the whole length dmirror and reflects. It is assumed that k x = k0, and the focal length is obtained as:

F = -k0 s kx  2dmirror

(3.4.2)

It means that the maximum focal distance can be of order of the thickness of the mirror structure. From Eq.(3.4.1), s  2dmirror = 2 kx at 45o ( kz = kx ). The NA of a Gaussian beam in the angular spectrum is q = kz k0 =  (d  k0 ) . The focal length can be shown as: 2 F = -k0 s kx = -k0 (-2 kx ) = 4dmirror  (3.4.3)

Eq. (3.4.3) shows that the focal distance can be larger due to longer effective cavity length but the NA becomes smaller. The estimated relation F  NA = 2dmirror can be used to indicates that the focal distance F and numerical aperture NA are limited by the length of a round trip of the mirror 2dmirror. In order to increase the negative slope of lateral shift without reducing the angular bandwidth, the only way is to increase the length of chirped mirror dmirror.

One of solutions if that adding more layers (up to thousand layers) but it not only cost highly in fabrication, but also increases enormously fabrication errors. Another idea is to introduce a defect layer in a chirped mirror that can make multiple internal reflections for a longer effective cavity and for a strongly anomalous dispersion. However, the improvement of the focal length is still not promising due to small NA. One of the solutions could be to introduce more defect layers, but it is too complicated to estimate. A helpful way is to use an algorithm to optimize multilayers as a common procedure of a dispersive mirror.

Page 17 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 3 3.5

Optimized structure for larger focal length

In the previous section, the idea of a focusing mirror was presented on example of a linearly chirped mirror. It showed the physics of focusing by calculating the phase of the angular spectrum in reflection from the chirped structure. Next, we will discuss an approach to obtain an optimized focusing performance. As previously commented, the introduction of a defect in the chirped structure should contribute to enhancement of the anomalous dispersion.

This idea is described and calculated in the article18 which appear already after the preparation of this thesis. It is also note that a stronger dispersion can also be obtained by introducing several defect layers but the system will be too complicated to be analyzed systematically. In this section, we followed the opposite approach. The desired dispersion relation is fixed as a target function and a random structure is iteratively engineered to match the desired dispersion.

For example, a target function for the lateral shift and the focal length can be set as a straight line with a certain slope and a constant, respectively. Two main criteria are considered in the adopted optimization procedure. First, due to technological limitations of fabrication, the number of layers was restricted up to 100. Second, the control of the phase is hardly achieved during the manufacturing process due to their extreme sensitivity to the deposition errors. The sensitivity of the coating to manufacturing errors must then also be considered in the design problem 19-21.

Here, we used a robust synthesis method 22, appropriate for the design of a mirror with low sensitivity to manufacturing errors, without using any specific starting predesign. The proposed method of the robust synthesis can be considered as a generalization of a very efficient needle optimization method 23,24 and gradual evolution techniques 25. The method is based on a simultaneous optimization of spectral characteristics of multiple designs located in a small neighborhood of a main pivotal design. Finally, the structure was designed only for TE polarization 26 but it is also possible to allow tailoring of dispersion for both polarizations simultaneously.

The optimization procedure converged to a specific designed structure, consisting in our particular case of 98 layers, which provided a segment of linear angular dispersion of lateral shift around 8 degrees, corresponding to the spectral range of around 20 nm.

In fact, the final optimized structure can be seen as a complicated combination of several GTI structures, as shown in Fig. 3.5a. Although the structure in this optimized case does not look like a chirped structure, the behavior is the same as a chirped mirror, which can reflect different angular components at different layers, as shown in Fig. 3.5c. The focusing mirror works at 46o, slight larger than the angle 45o where the field penetrates the deepest layer.

The focusing appears when the lateral shift decreases (negative slope of lateral shift).

Following the previously described procedure, the mirror can be designed for any wavelength, for any incidence angles (except for small angles) and for any

Page 18 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

incident linear polarization state or hopefully for both polarizations simultaneously.

Intensity distributions of the reflected beam were calculated using the paraxial model (a part of the code is on the inset). A beam is incident at the angle corresponding to the middle of the negative slope of the angular dispersion curve. Fig. 3.6 shows the focusing effect at the particular incidence angles before and after the optimization process for TE polarization. After optimization, the image distance obtained is limage = 160 mm, and the incident beam was focused at a distance lobject = 100 mm. This results in a focal distance of up to 260 mm, which is a significant increase from the un-optimized case. The beam with a beam waist diameter of 8 mm corresponds to a Rayleigh range of 94 mm for ď Ź = 532 nm and 80 mrad total divergence angle.

Therefore, such optimized mirror can successfully realize a near field focusing in the far field zone which can brings unprecedented applications such as non-dispersive waveguide by two parallel flat focusing mirrors 25.

Fig. 3.5 The calculated optimized structure. (a) The thicknesses of each layer of optimized structure for total 98 layers. (b) The target function (red dashed line) of the lateral shift and the fitting results for both polarizations. (c)The distribution of field inside the optimized structure. The penetration depth depends on incidence angles.

CHAPTER 3

x (mm)

(a)

(b)

(c)

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0 0.2

TE 0.1 0 z(mm)

-0.1

0 0.2

Reference

TM 0.1

0 z(mm)

-0.1

0 0.2

0.1 0 z(mm)

-0.1

Fig. 3.6 The intensity distributions of the beam reflected from the optimized structure for (a) TE and (b)TM polarizations. (c) The reference is the beam reflecting from the flat metallic mirror; its width is shown in green dashed lines in (a) and (b).

Mathematica code

Page 19 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

CHAPTER 3

3.6

Experiment of beam focusing in reflection

The experimental demonstration of the focusing effect of the chirped and the optimized structures requires the appropriate experimental setup, such as the one shown in Fig. 3.7. Some requirements that should be met for recording the beam profiles are: 1) The effect is presented for a given angular spectrum, so properly focused beams must be used. The required focused beam size at the entrance of the mirror, of the order of few mm, can only be achieved by using high quality microscope objectives.

2) A high quality imaging system is most critical, in order to accurately measure the beam profiles at distances on the micrometre scale. For seeing the beam profile of few microns size, we need a magnification system to obtain an enlarged image onto the CCD camera. But at the same time, we also need to record the change in the beam profile, different distances from the mirror surface, in steps of the order of 1 mm. To achieve this order of resolution we have to take into consideration two important aspects: the translation stage of the imaging system must have a resolution of 1 mm or less so the depth of field of the imaging system must be kept to the shortest possible value.

The latter allows us to be able to better resolve the differences in the beam profile for such small distance steps. For example, our imaging objective X50 which meet these requirements has the depth of focus (longitudinal resolution), 0.9 mm and a resolving power (transversal resolution), 0.5 mm.

CCD

lens

3) In order to observe the beam profiles at such small distances with respect to the mirror surface, the size of the optical and mechanical components of the system is crucial.

4) To measure the reflected radiation at different angles, the system should be mounted on a rotation stage.

The experimental setup was designed taking into consideration above requirements. In order to solve the size limitations, long working-distance microscope objectives were used. This allows using working distances of centimetres instead of the millimetre distances typical of conventional microscope objectives. To control the positioning of the focused beam and imaging plane we used closed-loop step motors PI Micos PLS-85 with 0.5 mm precision. To focus the incident beam an infinity -corrected, apochromatic, long-working distance X20 Mitutoyo objective was selected. The imaging system consisted of a conventional microscope system. An infinity-corrected, apochromatic, long working distance X50 Mitutoyo objective was used to image the desired plane. The depth of focus of this objective was 0.9 mm with a resolving power of 0.5 mm. To get the image of the plane at the working distance of the objective at finite distance we used a tube lens of focal length 200 mm and the resulting intermediate image was finally imaged into the CCD sensor (Spiricon SP620-U) with 4.4 pixel size using a x4 objective to obtain a final magnification of approximately 100 onto the CCD. By translating the X50 infinitycorrected objective, we could image different planes and consequently monitor the evolution of the pulse from the mirror surface.

20X

50X

ď Ź/2

Laser ď Ź = 532 nm

y Sample x z Fig. 3.7 The experimental set up of beam reflections from the flat focusing mirror. The measurement scheme consisted of a CW solid state laser at 532 nm, polarization control, x20 focusing objective and imaging system with x50 objective mounted on a step motor, and a tube lens of focal length 20 cm.

Page 20 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

CHAPTER 3

The experimental setup allowed us to measure the reflected beam profile of a set of incidence angles ranging from 43 to 52 degrees. The imaging system recorded the transverse profiles of the beam onto the CCD camera.

From the elliptical 2D images, the information about the focusing in x direction and the diverging beam in y direction can be obtained simultaneously, so the reflected focusing beam and the reference beam can be obtained in one measurement by taking horizontal (x) and vertical (y) cuts at different positions. The evolution of the beam profile is recorded by scanning the imaging objective along the propagation direction (z).

In fact, in the experimental measurement, a critical point is to define the position of the mirror surface and to control the position of the incident focused beam onto the surface. In order to fix the surface plane of the mirror with respect to the imaging system, we first imaged some dust particle on the mirror surface. However, the finite size of the dust particles could introduce an error in the measurement of at least 20 mm. To solve this problem a ring structure of 10 mm diameter was written by direct laser writing during the structure fabrication process. By imaging the ring structure we could fix clearly the position of the surface.

To set the distance lobject from the mirror to the incident beam waist we followed the following procedure: the imaging system was fixed to observe the plane of the mirror surface. We translated the focusing objective of the incident beam until the image of the CCD camera showed the minimum vertical dimensions of the beam. At this point we were sure that the beam is focused exactly on the surface.

Therefore, we could control the distance lobject by moving the focusing microscope objective back by the desired distance. In this way, lobject was set to 40±5 mm and the imaging objective was translated every 5 mm to record the propagation dynamics of the reflected beam.

A typical recorded trace at a particular incidence angle, corresponding to a transverse cross-section of the beam profile along the x direction as a function of the distance from the mirror surface, is shown in Fig. 3.8a. The green line corresponds to the transverse cross-section along the y direction, as a reference of the reflected beam size from a conventional mirror.

By changing the incidence angle we could retrieve similar traces for each case. The measured focal length position for each angle is shown as a red dot with error bars in Fig. 3.8b. The dashed blue line corresponds to the simulation for the optimized mirror while the light blue area shows the variation of the focal length when the

fabrication error for the thicknesses of layers is considered to be around 1.5%.

The discrepancy of measured focal length is mainly due to fabrication errors. A randomization of the width of the layers by 1.5% (which is the estimated fabrication precision of our samples 25) results in the reduction of the focal distance by around 50–100 mm. The small oscillation appearing at lateral shift also destructs the beam slightly, being another reason for the reduction of the focal length.

It is suggested that, as a long-term objective, the calculation of the focal length with paraxial propagation method could be included into the optimization process.

One of the main problems of the 1D focusing mirrors is that the focusing is always obtained at some non-zero angle and never at normal incidence. At zero angle, the lateral shift is always zero and thus we don’t obtain a negative slope for it. The focusing around zero angle is only possible if we can obtain a negative GH shift s < 0. In our case, using materials with a positive refraction and without modulation in the transverse direction, it is impossible to get negative GH shift. Such an effect could possibly be realized using metamaterials with negative refraction.

Another possibility is to create a modulation in the transverse direction by using, for example, diffraction gratings. It will be presented in the next chapter.

(a)

(b)

41 42 43 44 45 46 47 48 49 50

Fig. 3.8 (a)The measured beam profile for TE polarization. A total focal length of F =150 mm is thus obtained experimentally. (b) The numerical calculation of focal with and without the consideration of the error by the dashed line and light blue area.

The red dots are measured results for different angles.

Page 21 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 3

Reference 1. Walmsley, I., Waxer, L. & Dorrer, C. The role of dispersion in ultrafast optics. Rev. Sci. Instrum. 72, 1-29 (2001).

2. Szipocs, R., Ferencz, K., Spielmann, C. & Krausz, F. Chirped multilayer coatings for broadband dispersion control in femtosecond lasers. Opt. Lett. 19, 201 (1994).

3. Zhou, J. et al. Pulse evolution in a broad-bandwidth Ti:sapphire laser. Opt. Lett. 19, 1149-1151 (1994).

4. Kärtner, F. X. et al. Design and fabrication of double-chirped mirrors. Opt. Lett. 22, 831-833 (1997).

5. Matuschek, N., Kartner, F. X. & Keller, U. Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics. IEEE J. Quantum Electron. 35, 129-137 (1999).

6. Kärtner, F. X. et al. Ultrabroadband double-chirped mirror pairs for generation of octave spectra. J. Opt. Soc. Am. B 18, 882–885 (2001).

7. Matuschek, N., Gallmann, L., Sutter, D. H., Steinmeyer, G. & Keller, U. Back-side-coated chirped mirrors with ultra-smooth broadband dispersion characteristics. Appl. Phys. B 71, 509-522 (2000).

8. Gires F. & Tournois P. Interféromètre utilisable pour la compression d’impulsions lumineuses modules en frèquence. C. R. Acad. Sci. Paris 258, 6112-6115 (1964).

9. Li, Y.P., Chen, S.H. & Lee, C.C. Chirped-cavity dispersioncompensation filter design. Appl. Opt. 45, 1525-1529 (2006).

10. Golubovic, B. et al. Double Gires-Tournois interferometer negative-dispersion mirrors for use in tunable mode-locked lasers.

Opt. Lett. 25, 275-277 (2000).

11. Furman, S. A. & Tikhonravov, A. V. in Basics of Optics of Multilayer Systems. (Editions Frontière, Paris, 1992).

12. Pervak, V., Razskazovskaya, O., Angelov, I. B., Vodopyanov, K. L. & Trubetskov, M. Dispersive mirror technology for ultrafast lasers in the range 220–4500 nm. Adv. Opt. Technol. 3, (2014).

13..Goos, F. & Hänchen, H. Ein neuer und fundamentaler Versuch zur Totalreflexion. Ann. Phys. 436, 333-346 (1947).

14. Razskazovsaya, O. et al. HfO2/SiO2 chirped multilayer mirrors for broadband dispersion management in the ultraviolet spectral range. in Dig. XVIII International Conference on Ultrafast Phenomena, 36B, THU.PIII.17 (2012).

15.Garmire, E., Hammer, J. M., Kogelnik, H. & Zernike, F. in Integrated Optics. (Springer-Verlag, Berlin Heidelberg New York, 1975.).

16. Cheng, Y. C. et al. Beam focusing in reflection from flat chirped mirrors. Phys. Rev. A 87, 045802 (2013).

17. Hernandez., G. in Fabry Perot Interferometers. (Cambridge University Press, Cambridge, 1988) 18. Cheng, Y. C., Kicas, S. & Staliunas, K. Flat focusing in reflection from a chirped dielectric mirror with a defect layer. to be published in J. Nanophotonics (2015).

19. Tikhonravov, A. V., Trubetskov, M. K., Amotchkina, T. V. & Tikhonravov, A. A. Application of advanced optimization concepts to the design of high quality optical coatings. Proc. SPIE 4829, 19th Congress of the International Commission for Optics: Optics for the Quality of Life, 1061 (2003).

20. Nohadani, O., Birge, J. R., Kärtner, F. X. & Bertsimas, D. J. Robust chirped mirrors. Appl. Opt. 47, 2630-2636 (2008).

21. Birge, J. R., Kärtner, F. X. & Nohadani, O. Improving thinfilm manufacturing yield with robust optimization. Appl. Opt. 50, C36-C40 (2011).

22. Pervak, V., Trubetskov, M. K. & Tikhonravov, A. V. Robust synthesis of dispersive mirrors. Opt. Express 19, 2371–2380 (2011).

Page 22 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

23. Tikhonravov, A. V., Trubetskov, M. K. & DeBell, G. W. Application of the needle optimization technique to the design of optical coatings. Appl. Opt. 35, 5493-5508 (1996).

24. Tikhonravov, A. V., Trubetskov, M. K. & DeBell, G. W. Optical coating design approaches based on the needle optimization technique. Appl. Opt. 46, 704-710 (2007).

25. Dobrowolski, J. A. Completely automatic synthesis of optical thin film systems. Appl. Opt. 4, 937-946 (1965).

26. Cheng, Y. C. et al. Flat focusing mirror. Sci. Rep. 4, 6326 (2014).

CHAPTER 4

Chapter4

I

Flat focusing mirrors from subwavelength gratings

remember, on the first day of my arrival, my advisor of K. Staliunas made an informal test to explain why does a Gaussian beam diffract? My answers were as delayed as my jet lag from Taiwan to Spain.

He looked at me and simply smiled…for around ten seconds. He used to say sentences ending with a “no?”. Especially, one of the most frequent questions that he likes to ask me is “Do you understand, no?!” and then he directly explained to me without waiting for my answer.

What I have his serious face in my mind is drawn by me on the blackboard…

“Magic Mirrors” was assigned as the topic for my PhD. I had to design them with different configurations such as multilayers structure (in chapter 3) and the periodic gratings in this chapter. Most importantly, the optical axis of all designs should be absent. As challenging as it is, the flat focusing mirror is of much interest for many optical micro-devices.

Chapter4

Page 23 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 4

4.1 Focusing with subwavelength gratings In this chapter, the phenomenon of focusing in reflection is proposed for a configuration working at normal incidence while simultaneously keeping the transversal invariance. The idea is to modulate the surface in the transverse rather than in the longitudinal direction such as a diffraction grating. In fact, the transverse modulations may break this invariance so we keep the modulation periodic for invariance larger than a period. The diffraction grating is made of high index rods grown on a low index background, with a grating period d. Light incident on the grating surface is scattered into several diffraction “orders ”of both transmitted and reflected waves. The diffraction angles q with respect to the normal of the incident light follow ml  2d sin(q ), where m  0, 1, 2... , l is the wavelength and d is the grating period.

When the period of gratings is reduced, the diffracted angles increase until they become larger than 90o thus meaning that the diffracted components are in the direction of the grating periods as evanescent waves. It can also be shown that as the period is reduced to approximately the wavelength of the incident light, only the 0th order reflected and transmitted modes are present in the far field 2.

Thanks to the advances in micro-fabrication technology, these tiny subwavelength gratings have been greatly investigated due to their anti-reflection characteristics 3 , and their polarization dependence for many possible optical applications. The focusing effect from the subwavelength gratings is also one of the hot topics. The common idea is to modulate the phase in transversal direction  ( x) and to obtain the desired concave wave front for the focusing. To modulate the phase, the local period or the geometry of local bar could be engineered to achieve the desired phase profile. However, such engineered structures lose their lateral invariant. In fact, a simple and periodic SWG also can obtain a concave wave front for a focusing without modulating.

Why and how a periodic SWG can modulate phase? For a row of dielectric rods with a period d < l, the character of reflection in the lowest order is analogous to the reflection from a plane interface: the reflected beam propagates as if it was emitted by an image source positioned symmetrically

Page 24 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

on the opposite side of the grating. This simple approach of a flat semi-transparent grating does not take into account the possibility that a part of the light scattered by a given rod is directed to the neighboring rods, acting as a secondary scattering source. As a consequence, the wave reflection is not only directly from the modulated surface but also after diffusion to the neighbor sites through lateral scattering, due to which a secondary image sources appear.

The character of the reflected radiation strongly depends on the phase delay between the direct and secondary scattering or image sources). The reflected beam will focus or defocus depending on the phase shift between the primary and the secondary image sources. For example, under the condition l 2  d  l , the phase of the secondary image is advanced in comparison to the primary one. Therefore, the phase fronts of the primary and secondary image sources become concave, which results in the focusing of the reflected beam as shown in Fig. 4.1.

The numerical analysis of this problem starts from the angular distribution of a Gaussian beam reflected from the structure where only the primary and secondary scattering events are considered. By applying the paraxial wave propagation theory, an effective diffraction length can be calculated as 4

lref  2k0 sd 2 p1 sin(2 d / l )

(4.1.1)

The distance corresponds to the beam propagation length in free space. A negative diffraction length lref can compensate the diffraction of an incident beam which propagated in free space. Therefore, a focusing of the beam can be observed where the condition of negative diffraction length lref < 0 is met.

There are three conditions in the above expression for focal length. First, the reflection coefficients s should be weak enough so that it only works for dielectric gratings.

Second, the focusing is only valid for small angles.

According to above estimation, the focal distance can reach the value of several wavelengths, so the effect is observable for sufficiently narrow beams.

CHAPTER 4

4.2 The comparisons between near field and far field lens It is noted that the phase fronts become curved for narrow beams and not for plane waves, as it generally holds for near-field focusing (in contrast to far-field focusing).

A more detailed analysis is demonstrated by finitedifference time-domain (FDTD) calculations. The focusing effect is obtained in a frequency range from f = 0.59 ~ 0.61, where f is the normalized frequency. The focusing case at f = 0.6 is shown in Fig. 4.1d.The full width half-maximum (FWHM) of the source is 5 mm and the position of the source from the grating is lobject = 3 mm. A narrower focusing beam (FWHM = 3.6 mm) can be obtained in reflection at a distance of limage = 10 mm. Focusing, in practice, means that at the same point, the size of the reflected beam in reflection is smaller than that of a beam reflecting from a conventional flat (metallic or dielectric) mirror. It is unexpected that the focal spot of the near field focusing is smaller than its original source as their angular bandwidths of a beam are fixed. In particular cases, however, the reflected focused spot can be slightly smaller than the width of the initial beam as the bandwidth of the reflected beam is broadened by the gratings 4.

A metallic SWG have also been of interest in the control of phase modulations by the distribution of optical elements.

For example, it was proposed the use of metallic optical antennas to modulate the phase of transmitted light, as reported in Science in 2011 11. These artificial optical antennas can be implemented as an ultrathin focusing lens with plasmonic metasurfaces 12. The mentioned article claims that special optical phase arrays also can result in focusing in reflection but there is still no related paper published until date.

Considering the focusing effect in reflection by our periodically SWGs with metallic material, we have experimentally checked that a metallic SWG cannot focus a beam in reflection. The reason is that its reflection coefficient s is too high as explained by Eq. (4.1.1).

However, the FDTD calculation (not published) also indicates that, instead of using pure metallic gratings, coating a thin metallic layer on the top of dielectric gratings for higher reflection coefficient s can slightly increase the focal length.

S2 S1

(c)

l 2  d  0 l 2d l

0d l 2  l  d  3l 2

2nd 1st

S2

Source Images

(d)

2nd

Source

1st 2nd

Focusing

Fig. 4.1 Near-field focusing effect with a SWG. (a) Primary scattering S1 (grey line) and secondary scattering S2 (light blue line) are considered. The phase of the primary source image is (b) advanced and (c) delayed. (d) The advanced primary source image results in a concave phase front and brings it to a focus in reflection.

25

z (mm)

15 10

20

5

15

0 10

-5

5

-10

-15

0 -10

-5

0 x (mm)

5

Norm. field intensity |H|2

It should be noted that the focusing of the periodic SWG is near field mirrors (NFM), different from the one of far field mirrors (FFM) composed of high-index-contrast subwavelength gratings (HCG) . HCG was firstly proposed for strong reflection and anti-reflection with both narrowand broadband spectral profiles 5,6. As fabrication and simulation techniques for such devices have become more sophisticated, the focusing effect can be realized by introducing some non-periodic variation into the HCG structure. It became possible to create a focusing effect for either the reflected or transmitted beam 7–10. The comparison of the focusing effects with periodic and nonperiodic SWGs is given in Table 1.

(b)

(a)

10

Fig. 4.2 Far-field focusing effect with an engineered HCG structure. A plane wave is focused in both transmission and reflection, simultaneously. (a)

Fig. 4.3 Far-field focusing effect with engineered antennas. (a) The phase can be modulated by the individual antennas. (b) The engineered antenna’s structure can achieve the (c) focusing effect.

Page 25 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 4 The table 1 is a summary of flat focusing mirrors with dielectric subwavelength gratings.

Reference

1

2

3

4

5

Year

2010

2010

2013

2014

2014

University

HewlettPackard Laboratories

Berkeley

University of York

Hanyang

Universitat Politecnica de Catalunya

City

California

California

York

Seoul

Barcelona

Authors

D. Fattal et al.

F. Lu et al.

A. B. Klemm et al.

J.H. Lee et al.

Y.C. Cheng et al.

Silicon grating on SiO2

amorphous silicon on a glass substrate

Si grating ridges

photoresist IP-Lon a glass structure

980 nm

1.55 mm

532 nm

Structure

Material

Wavelength

Amorphous silicon on quartz substrate 1.55 mm

Beam waist

200 mm

1.55 mm 30 mm

1 mm

5.4 mm

4 mm

Focal spot

140 mm

0.86 mm

5 mm

1.09 mm

3.6 mm

NA

0.45

0.81

0.37

Focal length

17.2 mm

10.3 mm

100 mm

16.76 mm

13 mm

Polarization

TM

TM

TE/TM

TM

TE

Reflectance

98%

93%

90%

41.1%

30%

NFM or FFM

FFM

FFM

FFM

FFM

NFM

From the comparison between the specialized HCG focusing mirror and our periodic SWG we can formulate the following conclusions:

6.

7.

1.

2.

3.

4.

5.

The symmetrically specialized patterns of the HCG break the transverse invariance unlike our flat focusing mirrors.

Roughness and fabrication limitations have to be taken into account. For example, the specialized grating could hardly be fabricated with low-cost mass production techniques such as nano-imprint lithography.

The non-periodic gratings are always designed for infrared 1.55 mm due to their complicated phase control which is hardly designed for visible light.

It is hard to achieve large enough phase shifts for focusing in a short distance so that this kind of modulated grating cannot perform focusing for a very narrow beam. Our gratings can achieve focusing even for beam diameters of 1 mm.

Compared with the modulated gratings, the reflection of our SWGs is only 30 % so it is still far from applications.

Page 26 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

8.

0.2

Our gratings can perform a focal spot smaller than that of the source only in a specific condition but the width reduction is not obvious.

Our grating performs a good focusing only in a narrowband region of frequency. The focusing can be only observed at a small range of normalized frequencies from f = 0.59 to f = 0.61, (f = a/l), where a is the period of grating and l the operating wavelength. It is noted that the focal lengths vary with operating wavelength.

Normally, the SWG works on TM polarization.

However, our structure works on TE polarization when the electric field oscillates along the lines in the grating. It is because that the field of this polarization can be scattered more efficiently to the direction perpendicular to the direction of propagation of incident field. The scattering is always much more efficient in the plane perpendicular to the electric field vector. It is reminded that, for focusing effects, the scattering in this plane (the secondary scattering through the nearest neighbors) is crucial.

CHAPTER 4 4.3 Fabrication and Measurement of SWG

IP-DIP

Fig. 4.4 The illustration of two photon polymerization or multiphoton polymerization. A rapid prototyping technique can be utilized for making arbitrary structures.

(a)

(b)

Laser P l/2

1 mm

10X

In order to observe the focusing with SWGs in visible frequncy (l = 532 nm) the following parameters were used.

The period of gratings is around 313 nm, which can be achieved by two-photon absorption direct-laser writing (DLW) 13,14 as illustrated in Fig. 4.4. It used a focused femtosecond laser beam (130 fs pulse duration, 780 nm wavelength, 100 MHz repetition rate) to fabricate nanostructures, in a commercial fabrication workstation (Photonic Professional, Nano-scribe GmbH).

The SWG structures were fabricated using a negative photoresist, IP-Dip (from Nanoscribe GmbH) on top of a glass substrate both with refractive index n = 1.5. A sample translation writing speed of 100 μm∕s and laser power of 3.05 mW (in front of the objective) were chosen. The writing laser beam was circularly polarized. A droplet of photoresist was set on the clean glass substrate placed on 3D piezo translation stages. The photoresist was dipped in contact with a 100× objective (NA 1.3), which focused the laser beam on the interface between the glass substrate and photoresist. After scanning the laser beam, the structure was solidified and the non-polymerized photoresist was washed away during development by immersing in isopropanol.

The photoresist ridges forming the grating have a semielliptical profile, with a diameter of 2r = 87±3 nm (r = 0.14d) and a depth of 190±12 nm 15. Although the thin bar and the shallow depth of the grating result in a weak modulation of refractive index, the focusing depends critically on the periods of gratings so a focusing effect is still expected.

The experimental set up is similar to the one used for 1D focusing mirror concerning the focusing of the incident beam as shown in Fig. 4.5. A diode-pumped solid state laser of visible wave-length l= 532 nm was used as a light source. The linear polarization of the beam incident onto the SWG was controlled by a half-wavelength plate. A 10× microscope objective, mounted on a motorized translation stage with 0.1 μm precision, was used to focus the light at normal incidence at some distance in front of the sample.

The reflected beam was separated from the incoming beam using a beam splitter. The intensity distribution of the reflected beam was recorded using an imaging system consisting of a 20× microscope objective and a tube lens, which images the field at a desired plane (the distance z from the grating surface) into a CCD camera. The position of the SWG plane at z = 0 μm was defined where a clear CCD image of the SWG surface was obtained in the CCD camera. The reflected beam was recorded by moving imaging objective (20×) with a step of 1 μm to scan the distance from z = 3 μm to z = 30 μm.

The imaging system used to record the light distribution as a function of the distance from the surface as shown in Fig. 4.6b and it is compared with the numerical results as shown in Fig. 4.6a. In this experiment we explored the reflected beam at normal incidence, so a small cube beam splitter is introduced in the previous experimental set up, just in front of the sample in order to separate the reflected beam, which is imaged into the CCD. From the experimental measurements obtained in this setup, we

observed that the focusing appears at a distance limage = 10 mm when the source (focused spot of the incidence beam) is placed at a distance of lobject = 3 mm in front of the mirror as predicted by the numerical simulation. The numerical and experimental intensity profiles of the reflected beams and their transversal cross-sections at focal points (limage = 10 mm) are shown in Fig. 4.6c and 4.6d.

From our measurement results, the numerical aperture (NA) is estimated 0.2. However, since the focal distance of NFM is also given by the formula F = lobject + limage, the distance from the focusing peak to the gratings can be adjusted to be smaller by increasing the distance from the source to the grating.

10 mm

TE

x Lens

z

Fig. 4.5 (a) Experimental setup for the measurement of reflected beam at normal incidence. (b) Top-view SEM image of the sample.

Page 27 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 4 It is noted that our SWG is a near field mirror (NFM) giving a focusing in the near field. The far field is defined as the distance from the focal point which is significantly greater than the Rayleigh range, which is proportional to W2/λ, where λ is the wavelength and W is the largest dimension in the aperture. Under these definitions, all our theory and experiments are in the near field area. In particular, for the beams of width W = 4 mm (the width of aperture or of source) the border between near and far field is around W2/λ = 42/0.532 = 30 mm. Therefore, the focal distance of our SWG limage = 10 mm is still in the regime of the near field. Generally the NFM can restore the initial beam-width, but cannot provide beams narrower at the image than the original beam. However, as found theoretically 4 , under certain conditions the angular distribution of the reflected beam is broadened resulting in a focused beam, narrower than the initial one. Since the angular components are not mixing as in normal lenses or curved mirrors, this can only happen when the central angular components (on axis components) are attenuated.

However, the effect is very weak, although can be observed numerically, could not be obtained in the experiment. For example, the measured FWHM of the focal spot is 5 mm when the source is focused to a FWHM of 4.8 mm. The quite restrictive conditions make it hard for the effect to be observed in our experimental setup, because the inaccuracies of fabrication and the measurement resolution

(c) 1

x (mm)

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0 -10

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Fig. 4.6. (a)Theoretical and (b) experimental profiles of reflected beam for TE polarization at normalized frequency f = 0.59 (d/λ). The green dashed line indicates the beam width reflecting from the flat mirror as a reference beam and the white dashed lines indicate the intensity peaks of beam reflecting from the SWG. (c) Numerically and (d) experimentally obtained transversal distributions of the reflected beam from SWG and the plane indicated by the white dashed lines are compared with the one of the source.

-5

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mask the effect. For example, the optimal focusing depends critically on the width and distance of the source beam or the radius and height of the gratings. These parameters are not very precisely controlled in our fabrication and measurement so it is difficult to obtain the optimum focusing conditions.

In conclusion, SWGs were firstly proposed for their use as broadband high reflectance mirrors 16, and the focusing effect was demonstrated recently with a modified structure 7–10. However, the axisymmetric structure proposed lacked transversal invariance. Therefore, in this chapter, we propose a new structure to obtain focusing in reflection from such SWGs. The advantages of our structure are: the absence of optical axis, a simpler design for fabrication and easier alignment than the traditional one. The critical problem is that the reflection is very low, around 30%, so its potential technological applications are still far from realization. The reflection could be enhanced by adding more grating rows, with separation l/2, or changing the duty cycle of the gratings. The analysis of periodic SWGs can also be extended to cover 2D periodicities for 2D focusing. For example, concentric rings, periodic grating of square symmetry and quasi-periodic gratings of octagonal symmetry are able to obtain focusing in reflections as shown in Fig. 4.7. These structures can give still potential applications in the fields of integrated optoelectronic devices and near field optical devices.

0 -10

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Page 28 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

7.5

Fig. 4.7 Reflection of a Gaussian beam from 2D gratings of different symmetries as obtained by semi-analytic study. (a) The reference beam (reflection from the flat surface) for comparison; (b) one-dimensional periodic array of cylinders; (c) periodic grating of square symmetry; (d) quasiperiodic grating of octagonal symmetry, where the intensity distributions are calculated at 5.5λ in front of grating; (e) intensity distributions calculated by FDTD with the same parameters

CHAPTER 4 The maximum efficient coupling with the backward wave occurs for the incidence angle: sin(a )  l d x  1 .

The back-propagating mode, corresponding to the leaky mode of the waveguide, is excited since it is a diffraction order (−1) of the incoming beam. The leaky mode propagates backward along the structure, thus the guided field diffuses. The leaky mode also diffracts on the modulation of the wave-guiding structure: two diffraction orders (−1) and (+1) coincide with the directions of the transmitted initial beam and of the reflected initial beam (red line) as shown in Fig. 4.8b. Our structure can be regarded as a very short Fabry–Perot resonator made of two semi-transparent mirrors, where the wave can propagate at larger angle to the optical axis creating multiple partial reflections.

As a result, the structure shows negative GH shifts for a relatively large range of incidence angles (of around 10o).

The magnitude of the negative shift can reach ≈ 6λ. The GH shift depends on the incident angles and a negative slope of GH shift leads to focusing effects similar to that from the 1D chirped mirrors. A focusing with negative GH is shown in Fig. 4.9b, where the focal length is found to be 5 μm at an incidence angle 41o where the slope of the GH shift is negative [Fig. 4.9c]. The principle of focusing is the same as that of the 1D chirped mirror of chapter 3, as shown in Fig.3.3. As expected, the beam widths are smaller than that of the reference beam in the range of negative slopes of the GH shift. In addition to the reported negative GH shift, a focusing of the reflected beam is also predicted.

Although the focal length is relatively short for realistic structures, we still numerically observe the focusing.

Both effects (the negative GH shift and the beam focusing) can be enhanced by optimizing the structure. For example, by introducing point defects or changing to metallic gratings one can achieve larger slope of GH shift.

However, we cannot get focusing from the gratings in Ref. 22, which can achieve giant GH shifts, because the GH shift only appears at the resonant angle.

4.4 2D two parallel waveguide-like SWGs The main idea of a NFM is to modulate the phase of the reflected beam in the far field to get focusing in the near field. However, the principle of focusing with SWGs 4,15 is slightly different from focusing with multilayers structure 17,18. For example, the focusing of multilayer structure mainly relies on the appearance of a negative slope of the lateral shift upon reflection from the surface. The principle of focusing with SWGs is that the period of the gratings causes the phase displacement of the reflected beam. The lateral shift, or so called Goos–Hänchen shift, from the SWG is too little to be observed at normal incidence.

However, the GH shift of SWGs may appear in a positive or negative direction at a nonzero incidence angle.

For example, a giant negative GH shift has been studied by a diffraction grating in optics 19 and has also been experimentally achieved in acoustics 20. The guided mode resonance (GMR) 21 is proposed to obtain a giant GH shift in total internal reflection 22. However, in schemes of total internal reflection, the shifted beam can only propagate inside the material rather than in free space. Another limitation of the negative GH shift in Ref. 21 is that it works only at a resonant angle. A negative GH shift was also measured in reflections from electron plasma in metallic surfaces, modulated structures with negative refraction such as photonic crystals or metamaterials.

However, they are usually too technologically complicated for fabrication.

Here, we introduce another configuration, as shown in Fig. 4.8a, in a planar waveguide-like structure composed of two parallel dielectric subwavelength diffraction gratings 23.

It shows an observable GH effect with a thin, tiny, and structurally simple structure. Here, especially, a negative GH shift can also be obtained.

Two parallel diffraction gratings, i.e., two rows of periodic arrays of dielectric cylinders, are assumed to support the forward and backward leaky modes with wavenumber kmod e  k0 . The mode can be excited at the wave vector matching condition at k0 sin(a)  q  kmod e , where the signs correspond to the forward backward mode, respectively, a is the incidence angle, and q  2 d x is the wave number of the grating.

(a)

Incident beam

(b)

Reflected beam

dx Backward Mode (BM)

Forward Mode (FM)

k+1 dz

kBM  k0 a

k0 , k-1

Transmitted beam

q

Incident, transmitted

Fig 4.8 (a) The illustration of negative GH shifts in reflection from two parallel SWGs, (b) the wave-vector matching condition for the excitation of backward guided mode.

Page 29 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

CHAPTER 4 Shortly concluding, the magnitude of the negative GH can reach up to 3.6 mm, approximately six times that of the wavelength (l = 633 nm) for realistic structures. This is an experimentally observable effect. In addition to the reported negative GH shift, the focusing of the reflected

beam is also predicted. Although the focal length is relatively short for realistic structures, we still numerically observe focusing as the slope of lateral shift is negative, which makes a good correspondence with our 1D focusing mirrors.

15

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Fig. 4.9 (a) The intensity profile of a reflected beam with a negative GH. (b) a focused reflected beam with a negative GH. (c) The GH depends on the incidence angles for different widths of the source W= 2, 3 and 4 mm. (d) The normalized width of reflected beam at the transversal cut at z = 10 mm depends on incidence angles.

Reference 1. Chuang, S. L. in Physics of Optoelectronic Devices. (John Wiley & Sons, New Jersey, (2009).

2. Peters, D. W., Kemme, S. A. & Hadley, G. R. Effect of finite grating, waveguide width, and end-facet geometry on resonant subwavelength grating reflectivity. J. Opt. Soc. Am. A 21, 981-987 (2004).

3. Gaylord, T. K., Baird, W. E. & Moharam, M. G. Zeroreflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings. Appl. Opt. 25, 4562-4567 (1986).

4. Cheng, Y. C., Redondo, J. & Staliunas K. Beam focusing in reflections from flat subwavelength diffraction gratings. Phys. Rev. A 89, 33814 (2014).

5. Kontio, J. M., Simonen, J., Leinonen, K., Kuittinen, M. & Niemi, T. Broadband infrared mirror using guided-mode resonance in a subwavelength germanium grating. Opt. Lett. 35, 2564-2566 (2010).

6.Kärtner, F. X. et al. Ultrabroadband double-chirped mirror pairs for generation of octave spectra. J. Opt. Soc. Am. B 18, 882-885 (2001).

7.Fattal, D., Li, J., Peng, Z., Fiorentino, M. & Beausoleil, R. G. Flat dielectric grating reflectors with focusing abilities. Nat. Photonics 4, 466-470 (2010).

8.Lu, F., Sedgwick, F. G., Karagodsky, V., Chase, C. & ChangHasnain, C. J. Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings.

Opt. Express 18, 12606-12614 (2010).

9.Klemm, A. B. et al. Experimental high numerical aperture focusing with high contrast gratings. Opt. Lett. 38, 3410-3413 (2013).

10.Su, W., Zheng, G., Jiang, L. & Li, X. Polarization-independent beam focusing by high-contrast grating reflectors. Opt. Commun.

325, 5-8 (2014).

11.Yu, N. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333-

Page 30 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

337 (2011).

12.Aieta, F. et al. Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces. Nano Lett. 12, 4932-4936 (2012).

13.Maruo, S. & Fourkas, J. t. Recent progress in multiphoton microfabrication. Laser Photonics Rev. 2, 100-111 (2008).

14.Deubel, M. et al. Direct laser writing of three-dimensional photonic-crystal templates for telecommunications. Nat. Mater. 3, 444-447 (2004).

15.Cheng, Y. C. et al. Beam focalization in reflection from flat dielectric subwavelength gratings. Opt. Lett. 39, 6086-9 (2014).

16.Huang, M. C. Y., Zhou, Y. & Chang-Hasnain, C. J. A surfaceemitting laser incorporating a high-index-contrast subwavelength grating. Nat. Photonics 1, 119-122 (2007).

17.Cheng, Y. C. et al. Beam focusing in reflection from flat chirped mirrors. Phys. Rev. A 87, 045802 (2013).

18.Cheng, Y. C. et al. Flat focusing mirror. Sci. Rep. 4, 6326 (2014).

19.Tamir, T. & Bertoni, H. L. Lateral displacement of optical beams at multilayered and periodic structures. J. Opt. Soc. Am. 61, 1397-1413 (1971).

20.Teklu, A., Breazeale, M. A., Declercq, N. F., Hasse, R. D. & McPherson, M. S. Backward displacement of ultrasonic waves reflected from a periodically corrugated interface. J. Appl. Phys.

97, 084904 (2005).

21.Wang, S. S., Moharam, M. G., Magnusson, R. & Bagby, J. S. Guided-mode resonances in planar dielectric-layer diffraction gratings. J. Opt. Soc. Am. A 7, 1470-1474 (1990).

22.Yang, R., Zhu, W. & Li, J. Giant positive and negative GoosHänchen shift on dielectric gratings caused by guided mode resonance. Opt. Express 22, 2043 (2014).

23.Cheng, Y.-C. & Staliunas, K. Negative Goos-Hänchen shift in reflection from subwavelength gratings. J. Nanophotonics 8, 084093 (2014).

CHAPTER 5 a g i c

m i r r o r

Chapter5

Conclusion

In this last chapter, we summarize the main results presented in this collection of my articles, also highlighting possible future applications. In this thesis, focusing effects of a beam reflected from different modulated structures have been studied theoretically and experimentally. These include, beam focusing from a 1D chirped mirror, improvement of beam focusing by adding a defect layer, extremely long focal length of an optimized mirror, beam focalization at normal incidence with subwavelength gratings, negative GH shift and beam focusing with two parallel subwavelength gratings. The majority of the studied phenomena can be explained by the anomalous lateral shift at certain incidence angles and most of them have been proved by experimental results fitting well with the analytical and numerical predictions.

Page 31 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

CHAPTER 5 a g i c

m i r r o r

Before describing in detail each of the studied effects, an overview of all flat focusing lenses and mirrors has been presented in Chapter 1. The interesting applications of such flat focusing mirrors and the disadvantages of todayâ&#x20AC;&#x2122;s focusing devices make our idea of flat focusing mirrors more relevant. The near field lens or mirror that does not have any optical axis is an important goal in the field of photonics. It has been realized by flat or perfect lenses with negative refraction or negative (anomalous) diffraction using PhCs or metamaterials. We have proposed a similar near field focusing effect, based on the principle of flat lens with anomalous diffraction, but without the need of effective negative index of refraction. There are two types of configurations for the flat focusing mirrors: multilayer structure and subwavelength gratings, which modulate the phase of reflected beam in longitudinal and transversal directions, respectively.

The different properties and principles between far field and near field lens/mirrors are discussed in Chapter 2. The diffraction of a Gaussian beam is explained in terms of the superposition of many plane waves. A modulation of far field components of the beam is proposed in order to suppress the beam divergence and even to obtain beam focusing. Therefore, the phenomena of anomalous diffraction directly leads to the understanding of the effects of focusing in transmission or reflection. The physical ideas of near field mirrors based on the principle of flat lensing with negative (anomalous) diffraction, similar to super-prism effect, instead of negative refractions. It is also mentioned that the challenge of a near field lens/mirror is to obtain focusing in the far field zone which is one of the main objectives for my PhD.

I

n the third chapter, beam focusing in reflection is realized with multilayer thin film structures. The idea of solving the chromatic dispersion through the temporal compensation of phase delays by a chirped mirror is analogous to the idea of the beam focusing by the angular compensation of phase difference. However, the oscillations of the dispersion relation from a chirped mirror make focusing unperceivable. These oscillations also bring the restriction of the minimum of the working beam size (Numerical aperture in an angular spectrum).

The methods to improve the focal length are proposed by adding a defect layer into a chirped mirror or optimizing multilayer structures for a stronger and a broader dispersion relation. Finally, a promising and new near field focusing mirror which can successfully bring a near field focusing to the far field zone is demonstrated numerically and experimentally.

Page 32 | Flat Focusing Mirrors | Yu-Chieh Chengâ&#x20AC;&#x2122;s Thesis | January 2015

CHAPTER 5

a g i c

m i r r o r

I

n the fourth chapter, the different configuration of a flat focusing mirror is proposed with periodic SWGs. Importantly, they are periodical in a transversal direction which keep a transversal invariance. The scheme of focusing with SWGs is proposed with the principle of SWGs which only a zero order diffraction occurs. By considering the phase of angular components is advanced or delayed by the secondary scatters, the concave wave front could be obtained due to the phase shifts between the primary and secondary scatters. Eventually, a focusing effect in reflection is obtained by SWGs which is very different from those spatially engineered SWGs. The differences between near field mirrors (periodic SWGs) and far field mirrors (engineered SWGs) are discussed. Their limitations are also mentioned. A two-dimentionally (2D) focusing was also demonstrated numerically with 2D configurations such as square symmetry and quasi-symmetry. Another type of waveguide-like SWGs can also perform beam focusing in reflection with an accompanied effect of negative Goos-Hänchen. Two parallel SWGs are assumed to support the forward or backward wave and, surprisingly, the slope of lateral shifts depending on angle of incidence can be negative which results in beam focusing in reflection.

An illustration of a comapct pulse laser device with two flat focusing mirror. A laser can be passively Q-swithched by placing a saturable absorber like a graphene layer inside the flat foucsing mirror. The result is an intense burst of light in a self-Q-switched pulse.

F

inally, the appendix consists of a collection of my related published articles.

The ‘highlight’ pages of each article introduce the motivations and short stories of the paper.

The first idea of flat focusing mirror with a chirped mirror has been demonstrated numerically and experimentally in the journal of Physics Review A (PRA). Simultaneously, the theoretical demonstration of the idea of flat focusing mirror with subwavelength gratings, which can achieve focusing at normal incidence, has been published in PRA in 2013. In 2014, the experimental part of subwavelength gratings was published in the journal of Optics Letters. The best performance of the flat focusing mirror with an optimized multilayer structure has been published in Scientific Reports, a primary research publication from the publishers of Nature. Additionally, an observation of negative GH shift which accompanies, as a side effect, the beam focusing is demonstrated numerically in the special issue of the Journal of Nanophotonics. All these publications about flat focusing mirror comprise my PhD thesis. Apart from many potential applications of the flat focusing mirror for photovoltaic cells, optical filters, and detectors, being one of the contributions to apply the theory of flat lens for different fields is more meaningful for my PhD.

Page 33 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

APPENDIX I

Appendix Collection of Publications

In this appendix ,five journal articles of my publications are introduced. Firstly, there are two journal papers about the beam focusing from 1D multilayer structures which are the main stem of the chapter III. The article “Beam focusing in reflection from flat chirped mirrors” shows the idea of the beam focusing from a simple chirped mirror. A better focusing performance with the optimized structure is presented as the title “Flat focusing mirror” in Scientific Reports.

The following papers about flat focusing mirrors from the subwavelength gratings are also presented. The theoretical and experimental results are published with titles “Beam focusing in reflections from flat subwavelength diffraction gratings” and “Beam focalization in reflection from flat dielectric subwavelength gratings” in Physic Review A and Optics Letters, respectively. The content of these two papers are mainly described in chapter IV. Finally, the flat focusing mirror with waveguide-like subwavelength gratings published in journal of Nanophtonics with the title “Negative Goos-Hänchen shift in reflection from subwavelength gratings” is presented.

For each publication, a highlighting page of the publications is presented before each paper. These cover pages describe some short stories about our funny research lives and the short biography of contributing authors.

I am really lucky that I do my PhD here with many people´s help and their accompanies. The professors here cooperate intensively and they share their ideas, students, equipment among others. My professor, Kestutis Staliunas provides his fancy ideas and then I perform them in a fast and efficient way. My professor, Jose Trull helps me a lot in the lab about the structure of the experimental set up and the data analysis. There was a postdoc from Lithunia, Martynas Peckus, who worked together on the measurement. One PhD student, Simonas Kicas, who fabricated the sample of the thin film structures for us is under the direction of his supervisor Ramutis Drazdys in Lithuania. Without everyone’s help, I cannot complete the work.

I still remember the day when we observed the focusing effect after I returned from Dublin. I had been so sick in Dublin but still eager to make the measurement. Therefore, I took my professor Kestutis’s advice that Irish Whisky can help me recover so that I could work more efficiently

a public sculpture by Indian-born British artist Anish Kapoor

T

Beam Reflection THINK BACK TO THE LAST TIME WE SUBMIT TO NATURE PHOTONICS (NPs) . We happily opened a bottle (or few bottles) of wines to celebrate it after we observed the focusing effect. Although it was too early to open the wine for this work, I am grateful that I can work with many people here and never feel lonely during my research life.

Unfortunately, the firt attempt to the journal, Nature Photonics, was not successful. A referee from NPs commented: “The authors mentioned that this is a promising result, hinting that spatial dispersion can be reduced to zero or even made negative but this language is weak and vague. The paper feels half-baked. Why didn't they go further?” At least, we are still happy for the editor’s affirmation. Yes, we admit that the chirped mirror can only show “physically” the principle although the focusing effect was not obvious.

Afterwards, we improved our work, solved those problems that referees pointed out and try to submit it to NPs again.

From our point of views, the story of how a chirped mirror modulate the divergence of beam is really novel and suprising. The chirped

mirror is a well-known devices for a pulse but it unexpectly can be also used for a beam focusing. The strucutre is not fancy enough to attarct much attentions but the simple idea is really beautiful!

To be honest, the application is not so appealing due to its limitations, i.e. The focus works only at nonzero angles, only for a beam and only for one direction.

The other difficult thing is that the unavoivable fringes effect which results in a narrow angular bandwidth . Therefore, the focusing is too weak to be observed. My final aim is to design a flat focuisng mirror with a revalent focal length which will make the referrees feel well-baked or even over-baked.

A famous sculpture in Chicago, a bean reflection, is taken as the highlighting picture here. This giant bean mirror is so eye catching just like how interesting our work is.

Surprising, a novel idea is realized with a common device, a chirped mirror. As long as we consider the beam in the angular space which we are not used to, we can see a outstanding view form such magic mirrors.

COAUTHOR LIST: Martynas Peckus

is senior researcher at Vilnius University, Laser Research Center, Lithuania.

His major is in spatial light modulation, linear and nonlinear Mini-Resonators, photonic crystals.

Simonas Kicas

is a PhD student at the CPST in the field of thin film coating deposition technologies and dispersive multilayer systems. He is in charge of design and fabrication of flat focusing mirror.

Ramutis Drazdys

is the supervisor of Simonas Kicas at the CPST. He is major in femtolaser, optical thin film and coating technique.

Crina

Cojocaru is a associate professor at UPC. Her activity covers, periodic materials, photonic crystals and random nonlinear media and ultrashort pulse, etc.

Ramon

Vilaseca is professor at UPC, and coordinator of the research group DONLL. He works in nonlinear optics and dynamics and in photonic crystals.

Appendix I

Appendix I

Appendix I

Appendix I

APPENDIX II fter one year, the well-baked flat focusing mirror with long focal length is demonstrated successfully with the 1D optimized multiple mirror. We have tried to submit again to NPs but the editor rejected it directly.

At the end, this paper was still published with the title ¨Flat focusing mirrors ¨ in a sub journal of Nature, Scientific Reports 4, 6326, 2014.

Compared to the 1D chirped mirror, the focal length and the numerical aperture of the flat lens are generally enhanced by optimizing the multiple layers. The focal length is increased up to 250 mm numerically and 150 mm experimentally. The optimized mirror can bring the near field focusing to the far field zone. Such design of this special mirror based on the optimization of the spatial dispersion.

The most surprising and interesting points of the flat focusing mirror are its lateral translational invariance and its long focal length. This device can largely increase the applicability of structured photonic materials for light beam propagation control in small-dimension photonic intergraded circuits. It is noted that it is still hard to realize the focusing at zero angle with such structure. Therefore, another configuration of flat focusing mirrors is proposed by subwavelength gratings which can focus light beam at normal incidence.

A

-

A WELL-BAKED FLAT FOCUSING MIRROR IS READY TO BE PUBLISHED

Appendix II

AUTHOR LIST:

Yu-Chieh Cheng

is a joint PhD student at the Universitat Politecnica de Catalunya (UPC), Barcelona, Spain and European Laboratory for Non-Linear Spectroscopy (LENS), Florence, Italy. She received her B.D. and M.D. from National Central University, Taiwan, and continues her PhD in Europe, sponsored by the EuroPhotonics program of Erasmus Mundus Joint Doctorate (EMJD). Her PhD study is on photonic crystals for beam modulations and resonant cavities. Presently she has published 8 journal papers.

Kestutis Staliunas,

is a full professor of ICREA in Universitat Politecnica de Catalunya (UPC), Barcelona, Spain. He received his PhD degree in physics from the Vilnius University (Lithuania) in 1989. He is the author of more than 200 journal papers and of a monograph (K.Staliunas, and V.J.Sanchez-Morcillo "Transverse Patterns in Nonlinear Optical Resonators", Springer, 2003). His current research interests include light pattern formation in nonlinear optical systems, in photonic crystals, metamaterials, and others.

Jose Trull is a associate professor in Universitat politècnica de Catalunya (UPC), Barcelona, Spain. He reived his phD degree in Physiscs from Universitat Politècnica de Catalunya 1n 1999. He is author of more than 60 journal papers and responsible of the Nonlinear Optics and Lasers laboratory of the research gourp “Nonlinear Dynamics, Nolinear Optics and Lasers” (UPC). His current research interests include light pattern formation in nonlinear optical systems, ultrashort pulse characterization techniques and others.

Wiersma Diederik is my supervisor in Italy and currently runs a research group in LENS. His research interests lie in the fundamental optical properties of photonic materials, in particular materials with periodic, random, or quasicrystalline structure. He has authored many papers on this topic, amongst which 31 in Nature (including 3 cover stories) and the Physical Review Letters.

APPENDIX III

The idea

of the flat focusing mirror with subwavelength gratings was proposed by Prof. Javier Redondo in Gandia. He observed the sound wave can be focused by a layer of periodic scatters and we try to realized this idea in optics. I love to visit the group in Gandia for their beautiful beach, nice weather, delicious seafood and so on. With the accompany of such nice environment, I also learned how to simulate sound waves propagation with the commercial software, COMSOL there. I mainly worked with a PhD student, Alejandro Cebrecos, who is one of the Spanish hardest worker I have ever seen. We have stayed lately in the lab several times until the security came to kick us out.

We have tried to establish experiments in acoustics such as the focusing of evanescent wave in transmission based on fluid-like scatters. The experiment of sound waves is very attractive because I can see the structure of a real photonic/sonic crystals. Build them with my hands and my eyes without any precise instrument or microscope is real impressive.

Appendix III forphotonic Integrated Optics circuit For integrated circuit

Our flat focusing mirrors are not only realized in 1D Bragg-type optimized chirped mirror but also in subwavelength gratings which has more potential applications in photonic integrated circuits. The subwavelength gratings can be written in glass substrates or photoresists via multi-photon absorption using a focused short-pulsed laser.

The subwavelength gratings usually acts as a highly reflective mirror, a high-Q resonator, a focusing element or lens, a vertical in-plane coupler, and even as a slow-light waveguide.

Our subwavelength gratings are designed for visible frequency so that a ultra precise fabrication is required. At first, the period was too small to be fabricated by the group in Lithuania. During my period in my secondary university, LENS, I worked with a Chinese PhD student, Hao Zeng, who has an open mind and admits Taiwan is a country. We cooperated very well and successfully fabricated the structures. The measurement were then carried out when I came back to UPC. The measurement part was not easy and we spent several months trying to realized it.

Everything worked at the limit because we had to place the beam splitter in a very narrow area between the objectives and the sample. To avoid equipment from colliding with each other, the focal distance of objectives should be long enough or the beam splitter and the substrate of the sample should be small enough. Objectives with long-working distances are quite expensive (1.5 times my monthly salary) but it is really worthy to have one. All my work could be done because of these long-working distance objectives. After we found the proper implementation, we observed focusing very soon.

Finally, the theoretical and experimental papers of flat focusing mirrors with subwavelength gratings were published in Phys. Rev. A 89, 033814 (2014) and in Opt. Lett. 39 pp. 6086 (2014)

COAUTHOR LIST:

Javier Redondo

is a professor in UPV in Valencia, Spain and his interests relate to acoustics engineering, numerical modeling and so on. As a romantic musician, he even conceives scientific projects from his music.

The flat focusing mirror with subwavlength gratings is his idea and has been firstly demonstrated in acoustics .

Hao Zeng

is a PhD student at LENS. He has distinctive skills of microfabrication, particularly in smart materials like Liquid Crystal Elastomers. His research fields relate to micro robots, tunable photonics, etc. He has authored 6 scientific papers till date.

Mangirdas Malinauskas continues investigation on threedimensional laser structuring of polymers for applications in micro-optics, photonics and biomedicine. Current interests are lithographic micro- or nanofabrication techniques and additive manufacturing technologies.

Tomas Jukna

is a researcher in Kaunas University of Technology. His major is on the development of special charge based method for resistance measuring of thin foils deposited by thermo-evaporation techniques.

Appendix IV

Appendix IV

Appendix IV

APPENDIX V

Waveguide-like SWGs Negative Goos-Hänchen Shift

Negative Goos-Hänchen (GH) shift has been one of popular topic since the idea of trapped rainbow has been proposed. The light hits the media interfaces with an angle while propagating down the waveguide, and experiences a positive or negative lateral displacement.

Especially, the ray experiencing the negative GH shift can be permanently trapped at the certain condition such as a double light cone ('optical clepsydra').

It is well known that GH shift can be obtained from flat dielectric or metallic interfaces. The negative GH one has to utilize a material with negative refractive index such as meta-material or photonic crystals.

Here we also can observe a negative GH shift with a very simple

structure, two parallel subwavelength gratings (SWGs). The waveguide-like SWGs can excited the backward mode in the separation among two SWGs and make a clear reflected beam shift in a opposite direction. The GH shift depends on incidence angles and, the most importantly, the slope of GH shift depending on incidence angle is negative which brings a focusing effect.

At the begging, I looked for the focusing effect with a whole 2D Photonic Crystals (PhCs) and I noted that the behavior of reflected beam is strange as the transversal period is smaller than wavelength (working in subwavelength region).

For different wavelength of the incident light, the beam penetrate and bend anomalously. And at certain

condition, a clear negative GH shift is observed but we didn’t know how it worked. We gradually reduce the rows of PhCs and the effect still exist.

Surprisingly, only two rows of PhCs also can perform the same effect with the whole 2D PhCs. Afterward, we realized the physics of GH shifts from such waveguide-like SWGs. The tendency of GH shifts depending on incidence angles can perform a near field focusing corresponding to the principle of our flat focusing mirror.

Although the idea is only demonstrated theoretically in J. Nanophoton. 8(1), 084093 (2014), I am happy to complete one more configuration of the flat focusing mirror.

Negative GH shift SWGs

Negative GH shift by two SWGs.

Focusing

Chirped 2D PhCs Negative GH shift by 2D PhCs

SWGs

The near field focusing effect by SWGs

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Copyright 2015 Yu-Chieh Cheng All rights reserved. No part of this book may be reproduced in any form. Short and brief quotations may be accepted but please pay attention to article errors.

Concluding, please write your PhD thesis by yourself.

Printed in Spain by Singulars, Terrassa Printed in Taiwan by Yi Sheng, Taipei

Copy your data in different hard disks. It is possible that someone may accidently format your hard disk and cause the loss of one and half years of hard work.

Dedicated to them, whom I love the most Milu Gorina Cheng & Oriol Gorina Parra

EUROPHOTONICS Erasmus Mundus Doctorate (EMJD) program Universitat Politècnica de Catalunya. BarcelonaTech.

LENS - European Laboratory for Non-Linear Spectroscopy / University of Florence