Oc lect 03

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30 September 2014

Taiz University, YEMEN

ďƒ˜ Fiber types according to dimensions and mechanism of propagation: Single-mode Step Index Fibers Types of Fibers

Multimode Graded Index Fiber

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The refractive index profile and ray transmission in step index fibers: (a) multimode step index fiber; (b) single-mode step index fiber 30 September 2014

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• The optical fiber with a core of constant refractive index n1 and a cladding of a slightly lower refractive index n2 is known as step index fiber. • This is because the refractive index profile for this type of fiber makes a step change at the core-cladding interface as indicated in previous Figure which illustrates the two major types of step index fiber. • The refractive index profile may be defined as: đ?‘›1 đ?‘› đ?‘&#x; = đ?‘›2 30 September 2014

đ?‘&#x; < đ?‘Ž (đ?‘?đ?‘œđ?‘&#x;đ?‘’) đ?‘&#x; ≼ đ?‘Ž (đ?‘?đ?‘™đ?‘Žđ?‘‘đ?‘‘đ?‘–đ?‘›đ?‘”)

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• Figure shows a single mode or monomode step index fiber which allows the propagation of only one transverse electromagnetic mode and hence the core diameter must be of the order of 8-12 ¾m. • The advantage of the propagation of a single mode within an optical fiber is that the signal dispersion caused by the delay differences between different modes in a multimode fiber may be avoided. Thus achieving a large bandwidth. 30 September 2014

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• In describing SMF, a parameter known as mode-field diameter (MFD) or spot size is used. In a SMF light travels mostly within the core and partially within the cladding. MFD is a function of the wavelength. • Spot size given as: 3  wo  0.65  1.619V 2  2.879V 6 a

• It can be shown that for small values of V (normalize frequency), the light beam extends into the cladding. Therefore susceptible to bend losses. To reduce this, V is made normally between 2 - 2.4 30 September 2014

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• Mode-field diameter (MFD)

• Distribution of the beam’s intensity in a single mode fiber. Core diameter 8.3 μm and MFD is typically 9.3 μm. 30 September 2014

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• The beams travel at distinct propagating angles ranging from zero to critical value. • These different beams are called modes. • The smaller the propagating angle, the lower the mode. • The mode traveling precisely along the axis is the zero-order mode or the fundamental • Hence for the transmission of a single mode the fiber must be designed to allow propagation of only one mode, whilst all other modes are attenuated by leakage or absorption. • This may be achieved through a suitable choice of normalized frequency, V for the fiber. For single mode operation, only the fundamental TE01 mode can exist. The cut-off normalized frequency for the TE01 mode occurs at V = 2.405. Thus single mode propagation is possible over the range:

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0  V  2.405 •

This parameter can also be expressed in terms of numerical aperture and relative refractive index difference Δ as:

V 

2a

( NA) where a = core radius

 1 2a V  n1 ( 2) 2  V 

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2a

n1  n2 2

2

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• Figure shows a multimode step index fiber with a core diameter of around 50 µm or greater, which is large enough to allow the propagation of many modes within the fiber core

• Multimode step index fibers allow the propagation of a finite number of guided modes along the channel. • The number of guided modes is dependent upon the physical parameters (i.e. relative refractive index difference, core radius) of the fiber and the wavelength of the transmitted light. 30 September 2014

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•

•

It can be shown that the total number of guided modes (or mode volume) Ms, for a step index fiber is related to the V value for the fiber by the approximate expression: đ?‘‰2 đ?‘€đ?‘ ≅ 2 which allows an estimate of the number of guided modes propagating in a particular multimode step index fiber.

Lower bandwidth applications multimode fibers have several advantages over single mode fibers: 1. The use of spatially incoherent optical sources (e.g. most light emitting diodes) which cannot be efficiently coupled to single mode fibers. 2. Larger numerical apertures, as well as core diameters, facilitating easier coupling to optical sources 3. Lower tolerance requirements on fiber connectors.

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Taiz University, YEMEN

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•

•

GRIN fibers do not have a constant refractive index in the core but a decreasing core index n(r) with a radial distance from a maximum value of n1 at the axis to a constant value n2 beyond the core radius, a in the cladding. This index variation may be represented as:

1   2 n1 1  2r / a  ; r  a (core) nr    1 n 1  2 2  n ; r  a (cladding) 2  1 30 September 2014

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• Δ is the relative refractive index difference and α is the profile parameter which gives the refractive index profile of the fiber core. • The equation above is a convenient method of expressing the refractive index profile of the fiber core as a variation of α allows representation of • Step index profile when α = ∞, a parabolic profile when α = 2 and a triangular profile when α = 1. • This range of refractive index profiles is illustrated in next figure 30 September 2014

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• Possible fiber refractive index profile for different values of ι

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• The refractive index profile and ray transmission in a multimode GRIN fiber

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• The graded index profiles which at present produce the best results for multimode optical propagation have a near parabolic refractive index profile core with α= 2. • A multimode graded index fiber with a parabolic index profile core is illustrated in previous figure. It may be observed that the meridional rays shown appear to follow curved paths through the fiber core. • Using the concepts of geometric optics, the gradual decrease in refractive index from the center of the core creates many refractions of the rays as they are effectively incident on a large number of high to low index interfaces. 30 September 2014

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•

This mechanism is illustrated in below Figure where a ray is shown to be gradually curved, with an ever-increasing angle of incidence, until the conditions for total internal reflection are met, and the ray travels back towards the core axis again being continuously refracted.

An expanded ray diagram showing refraction at the various high to low index interface within a graded index fiber given an overall curved ray path 30 September 2014

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• Parameter defined for the step index fiber may also be applied to graded index fibers and give a comparison between them. • However, for graded index fibers the situation is more complicated since the numerical aperture is a function of the radial distance from the fiber axis. • Graded index fiber therefore accept less light than corresponding step index fibers with same relative refractive index difference. • To support single mode transmission in a graded index fiber, the normalized frequency is:

2  Vg  2.405 1     30 September 2014

Taiz University, YEMEN

1 2

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For the parabolic profile, the numerical aperture is given by; 1 đ?‘&#x; 2 đ?‘ đ??´ = đ?‘›1 (2đ?›Ľ)2 1 − ( ) đ?‘Ž This shows that the NA is a function of the radial distance from the fiber axis (r/a).

For n1 = 1.48 and n2 = 1.46, the axial NA is 0.24 and this is equal to the NA of a step index fiber with same n1 and n2. The NA drops to zero at the edge of the core 30 September 2014

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• Therefore, it is possible to determine fiber parameters which will give single mode operation. • For multimode graded index fibers, the total number of guided modes, Mg is also related to the V value for the fiber by the approximate expression: Mg

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2    V       2  2

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   

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Example (1)

A multimode step index fiber with a core diameter of 80 μm and a relative index difference of 1.5% is operating at a wavelength of 0.85μm. If the core refractive index is 1.48, estimate: (a) the normalize frequency of fiber; (b) the number of guided modes. Answer: (a) 75.8, (b) 2873 Example (2)

A graded index fiber has a core with a parabolic refractive index profile which has a diameter of 50μm. The fiber has a numerical aperture of 0.2. Estimate the total number of guided modes propagating in the fiber when it is operating at wavelength of 1 μm Answer: V=3.14, Mg=247 guided modes 30 September 2014

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Example (3)

A graded index fiber with a parabolic refractive index profile core has a refractive index at the core axis of 1.5 and a relative index difference of 1%. Estimate the maximum possible core diameter which allows single-mode operation at a wavelength of 1.3 μm Answer: V=2.4(2)0.5, a=3.3 μm, the diameter= 6.6 μm

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• The number of modes Ms , transmitted by a step-index fiber, depends on NA, core diameter D, and the wavelength of light, λ

V  2a.NA   D  2 2 Ms   0.5  0 . 5 n  n    1 2 2       2

2

2

• Maximum allowable core diameter D for single-mode transmission is 2.405

D

 n1  n2 2

2

• To ensure single-mode transmission  Making the core thin enough  Making the refractive index difference between core and cladding small enough  Using a longer wavelength 30 September 2014

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• The cut-off wavelength is wavelength above which a particular fiber becomes single mode. If Vc is the cutoff normalized frequency and V is the normalized frequency corresponding to wavelength Îť, then the cut-off wavelength Îťc, is given by: đ?œ†đ?‘? đ?‘‰ = đ?œ† đ?‘‰đ?‘? • For a step index fiber the cut-off wavelength is given by: đ?‘‰đ?œ† 2đ?œ‹đ?‘Ž. đ?‘ đ??´ đ?œ†đ?‘? = = 2.405 2.405 30 September 2014

Taiz University, YEMEN

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Example (1)

Determine the cutoff wavelength for a step index fiber to exhibit single mode operation when the core refractive index and radius are 1.46 and 4.5 μm, respectively, with the relative index difference being 0.25%. Answer: λc= 1214 nm Hence the fiber is single-moded to a wavelength of 1214 nm Example (2)

What is the maximum allowable core diameter for a step-index singlemode fiber operating at 1.3 μm, with core index of 1.5 and cladding index of 1.0003? If the operating wavelength is 1.5 μm, will the fiber designed above still be operating as single-mode? Answer: D=0.89 μm If the operating wavelength is 1.5 μm, the above fiber is still single-mode

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