11 barbero 10 2013

Power-adjustable optical systems for optometry applications Sergio Barbero1, Jacob Rubinstein2 (1) Instituto de Óptica (Consejo Superior Investigaciones Científicas) (2) Department of Mathematics (Technion-Israel Institute of Technology)

Spanish Optical Designer’s Meeting Barcelona, October, 2013

Uncorrected refractive errors ď Ž

Global causes of blindness

ď Ž

Around 150 million visually impaired (8 million blind) caused by uncorrected refractive errors Main cause of low vision and second cause of blindness in the world

Resnikoff - Bulletin of the World Health Organization - 2008

Subjective refraction

“Subjective refraction is the term applied to the technique of comparing one lens against another, using changes in vision as the criterion, to arrive at the dioptric lens combination that results in maximum visual acuity�

W. J. Benjamin and I. M. Borish, Borish's clinical refraction (Butterworth Heinemann/Elsevier, St. Louis, Missouri, 2006).

Trials lenses

Trial lens frame

Trial lens set

Typically as many as 266 lenses

Phoropters (refractors)

W. J. Benjamin and I. M. Borish, Borish's clinical refraction (Butterworth Heinemann/Elsevier, St. Louis, Missouri, 2006).

Self refraction in the past The use of eyeglasses (1623) Benito Daza de Valdes

“We explain a rule for anyone to quantify the amount of vision imperfection and to ask for the pertinent eyeglasses where they are manufactured�

Based on the near point location in myopes Vazquez, D., A. Gonzalez-Cano, et al. (2012). History of optics: a modern teaching tool. SPIE Proceeding.

Self refraction in the past Self refraction at the beginning of the XX century

“Daza de Valdés en la oftalmología” Tesis doctoral Javier Jiménez, 2013, p. 295-296

Our goals ď Ž

ď Ž

To develop novel power-adjustable systems for measuring (refraction) and correction (spectacles) of common refractive errors: myopia, hypermetropia, presbyopia and astigmatism.

To create a complete optical design methodology for designing such systems

Cubic-type surfaces x3 ( 3

2

xy )

Monkey Saddle (Alvarez)

L. W. Alvarez, “Two-element variablepower spherical lens" Patent, 1967.

x3 ( 3

x3

A. W. Lohmann, "A New Class Of Varifocal Lenses," Appl. Optics 9, 1669-1671 (1970).

xy 2 ) W. E. Humphrey, " Variable anamorphic lens and method for constructing lens“ Patent, 1973

Adjustable power with lateral shift Neutral position (0 D)

Positive power addition

Negative power addition

Sphero-cylindrical refraction Sphere (Sph) Cylinder Power (C) Cylinder Axis (A)

Mean Sphere (S) Cross-Cylinder (Cx) Cross-Cylinder (C+) Diopter Units

S C Cx

Cx S C

Dioptric matrix Sphero-cylindrical refraction described in matrix formalism

Spherical system (paraxial appr.) u ( x, y)

x3 A( 3

xy 2 )  

3

v( x, y)

A(

x 3

δ: Lateral shift in the X direction + δ first lens, - δ second lens

xy 2 )

Heissan matrix 2

K (0, 0) (n 1)

Thin lens approximation

2

u x2

u xy

2

2

u xy

u y2

S

2

(1 n)

2

v x2

v xy

2

2

v xy

4 A (n 1)

v y2

4A (n 1) 0

0 4A

Cylindrical system (paraxial appr.) u ( x, y)

x3 A( 3

xy 2 )

δ: Lateral shift in the Y direction + δ first lens, - δ second lens

 

v( x, y)

x3 A( 3

xy 2 )

Thin lens approximation

Heissan matrix 2

K (0, 0) (n 1)

2

u x2

u xy

2

2

u xy

Cx

u y2

2

(1 n)

2

v x2

v xy

2

2

v xy

4 A (n 1)

v y2

0 (n 1) 4A

4A 0

Humphrey Vision Analyzer u3 ( x, y)

3

u2 ( x, y) u1 ( x, y)

x3 A( 3

A(

x 3

xy 2 )

xy 2 )

x3 A( 3

xy 2 )

Z

Y Cross-Cylinder (C+)

Cross-Cylinder (Cx)

Mean Sphere (S)

X W. E. Humphrey, “Remote subjective refractor employing continuously variable sphere-cylinder corrections,” Optical Engineering, 15(4), 286-291 (1976).

Sphero-cylindrical two lenses system

Mean Sphere (S)

u ( x, y)

x3 A( 3

xy 2 ) Cross-Cylinder (C+)

Cross-Cylinder (Cx)

v( x, y)

x3 A( 3

xy 2 )

New system: Paraxial optics u ( x, y)

x3 A( 3

xy 2 )

δux: Lateral shift in X δvx, δvy : Lateral shift in X & Y

 

v( x, y)

K (0, 0) (n 1)

2(n 1)

A

u x2

u xy

2

2

u xy B

ux

B

2

vy

x3 B( 3

2

(1 n)

u y2 B

vx

A

ux

xy 2 )

2

v x2

v xy

2

2

v xy

Thin lens approximation

2

v y2

...

S

2(n 1) A

vy

B

vx

ux

C

2(n 1) B

vx

Cx

2(n 1) B

vy

Computation of dioptric matrix Computation of quadratic point eikonal O

â—?

Optical Path Difference

Γ

Propogation of localized quadratic wavefronts: Differential geometry Barbero S & J Rubinstein (2011): Adjustable-focus lenses based on the Alvarez principle. Journal of optics 13: 125705.

Sphero-cylindrical refractor

Barbero, S. and J. Rubinstein (2013). "Power-adjustable sphero-cylindrical refractor comprising two lenses." Optical Engineering 52(6): 063002-063002.

Surface design parameters

R( x 2 y 2 )

u ( x, y ) 1 p1 x  



3

p2 xy

1 R(Q 1)( x 2

p3 y

3

p4 yx

2

2

2

y) p5 xy p6 x p7 y

R, Q : General shape control p1, p2, p3, p4 : Variable power and astigmatism p5, p6, p7 : Thickness and prism

Merit function j N

MF

j 1

SE

1j

CE

1j

CE

1j x

i 3

l m

i 2

l 1

SE il CE il CExil

N 6m

 i: Three configurations. Only front lens is moved (i=1); only back lens is moved: x direction (i=2) or y-direction (i=3).  j: x-lateral shift front lens (N=21).  l: Back lens shift: x-direction (i=2) or y-direction (i=3). m=17.  SEij : Power error configuration i and lateral shift j.  CE+ij and CExij: Power error of C+ and Cx respectively, for configuration i and lateral shift j.

Sequential optimization Nelder-Mead simplex method R2

MF 2 MF 1

18 design parameters!

p31 , R1 , p32 , R 2 Q1 , Q 2

MF 1 p11 , p12 , p31 , p14 , R1 ,

MF 1

p12 , p22 , p32 , p42 , R 2 p11 , p12 , p31 , p14 , p61 , p81 , p91 , R1 , Q1 p12 , p22 , p32 , p42 , p61 , p81 , p91 , R 2 , Q 2

Sphero-cylindrical refractor: Sphere X front lens shift (δvx=0, δvy=0) Sphere C+ Cx

Pre-design

Optimized

-----.-

Sphero-cylindrical refractor: C+ X back lens shift (δux=0, δvy=0) Sphere C+ Cx

Pre-design

Optimized

-----.-

Sphero-cylindrical refractor: Cx Y back lens shift (δux=0, δvx=0) Sphere C+ Cx

Pre-design

Optimized

-----.-

Sphero-cylindrical refractor X front lens shift (δvx=-1.25, δvy=3.125) Sphere C+ Cx

Pre-design

Optimized

-----.-

3D eye movements: implications in lens design

26

1D dimensional movements

ď Ž

Horizontal, vertical and cyclotorsion movements described by rotations movements with respect to vectors

T. Haslwanter, "Mathematics of 3-dimensional eye rotations," Vision Research 35, 1727-1739 (1995) 27

3D eye position 





Horizontal and vertical rotation of the eye, in a well-defined sequence, uniquely defines the gaze direction Adding the cyclotorsion (rotation around the line of sight) define the 3D eye position Mathematically, rotation vectors describe efficiently a 3D eye position: the direction of the vector gives the axis of rotation and its module its size 28

Listing’s law 





A gaze direction sets the cyclotorsion magnitude This value is independently of the path the eye has followed to reach such position!

All rotation vectors characterizing 3D eye position lie in a plane.

29

Dioptric matrix: Listing’s law y

Rotation vector

a

a

(

r2 , r1 , 0) r12

r22

Rotation angle

z

cos(

)

z

x Rodrige’s Rotation formula ( r1 , r2 , r3 )

K ( x, y)

K( , )

Barbero, S. and J. Rubinstein (2013). "Power-adjustable sphero-cylindrical refractor comprising two lenses."

Optical Engineering 52(6): 063002-063002.

Sphere (S) errors: off-axis δux=-5, δvx=0, δvy=0

δux=-5, δvx=2.5, δvy=2

Cylinder (C+) errors: off-axis δux=-5, δvx=0, δvy=0

δux=-5, δvx=2.5, δvy=2

Cylinder (Cx) errors: off-axis δux=-5, δvx=0, δvy=0

δux=-5, δvx=2.5, δvy=2

Lens manufacturing

34

Manufactured lenses Front lens

Rear lens

Barbero, S. and J. Rubinstein (2013). "Power-adjustable sphero-cylindrical refractor comprising two lenses." Optical Engineering 52(6): 063002-063002.

Manufacturing technique Five-axis Freeform Generator

Moore Nanotech速 350 FG

Lens form accuracy (front lens) Laser line scanning technology Nominal

Measured

Mean absolute error: 8.1 Âľm

Errors

Lens form accuracy (back lens) Nominal

Measured

Errors

Mean absolute error: 8.7 Âľm Tolerance in ophthalmic lens: 0.06 D ~ 80 Âľm Savoie, M. Surface quality in the freeform process, Satisloh, Technical report.

Surface roughness Scanning white light interferometer

Ra (average deviation from the mean) Front lens: 6 nm Rear lens: 8.25 nm

NewViewTM 6300 Zygo速

Mechanical set-up

Spectacles design

41

Updated 2012

Commercial models

Focusspec (focus-on-vision.com)





Minus model (-1 to -5 D) Plus model (+0.5 to +4.5 D)

Eyejusters (eyejusters.com)





Minus model (0 to -5 D) Plus model (0 to +4.5 D)

Adlens Emergensee (adlens.com)



Unique model (-6 to +3 D)

Design trade-off δ A

δ Lateral shift A Maximum thickness

Optical power addition is proportional to δ and A

A ↓ and δ ↑ Large lateral dimension A ↑ and δ ↓ Large axial dimension

Lens thickness control  

The lens is divided into two zones The outer part, mechanical part, connects the optical (inner) part to the frame such that the entire surface is smooth and the edge thickness is fixed

Alvarez “pure” lens

Thickness reduction with linear term

Thickness with new technique

Barbero, S. and J. Rubinstein (2011). "Adjustable-focus lenses based on the Alvarez principle."

Journal of Optics 13(12): 125705.

Thickness distribution

Optical quality in spectacles ď Ž

Spectacle lens design optimizes optical performance for different gaze directions (conventionally up to 30Âş ~ 10 mm area in the spectacle)

Off-axis optical tolerances

Only found at the 1972 ANSI Z80.1 standard

Prismatic errors ď Ž

Chief ray through Alvarez lenzes

Positive power

Negative power

Barbero S & J Rubinstein (2011): Adjustable-focus lenses based on the Alvarez principle.

Journal of optics 13: 125705.

Merit function MF

PR j j

i: j: PRj : PEij : wij(PEij) : Aij : wij(Aij) :

( wij ( PEij ) *PEij j

wij ( Ai ) * Aij )

i

Gaze directions Lateral Shift between lenses Central Prismatic Error Power Errors Weights to Power Errors Astigmatism Errors Weights to Astigmatism Errors

Sequential optimization Nelder-Mead simplex method

R2

MF 2 MF 1

p31 , R1 , p32 , R 2 Q1 , Q 2

MF 1 1 1

1 2

1 3

1 4

1

p , p , p , p ,R ,

MF 1

MF 1

p12 , p22 , p32 , p42 , R 2 p61 , p81 , p91 , Q1 , p62 , p82 , p92 , Q 2 p11 , p12 , p31 , p14 , p61 , p81 , p91 , R1 , Q1 p12 , p22 , p32 , p42 , p61 , p81 , p91 , R 2 , Q 2

18 design parameters!

Example of spectacles design



 



Design of eyeglasses covering presbyopia or hypermetropia from +0.5 D to +5 D Maximum lateral shift 4 mm Optical analysis in a window of 20º of eye rotations Refractive index assumed 1.586 (polycarbonate)

Barbero, S. and J. Rubinstein (2011). "Adjustable-focus lenses based on the Alvarez principle."

Journal of Optics 13(12): 125705.

Power error (D) Pre-Design

Optimized Lens

Astigmatism (D) Pre-Design

Optimized Lens

Next steps

ď Ž

ď Ž

To construct spectacles prototypes to study the functioning of the optical performance To design mechanical system to provide the lens movements

Mechanical frame

ď Ž

ď Ž

Power change is achieved by moving lenses vertically with a set of special screws.

An horizontal movement mechanism , formed by two pinion-rack assemblies, adjust interpupillary distance.

A. Zapata and S. Barbero, "Mechanical design of a power-adjustable spectacle lens frame," Journal of Biomedical Optics 16, 055001-055006 (2011).

Mechanical frame Rack Pinion 9.5ยบ pinion rotations provides 0.5 mm

Special screws: (a) Threaded part (b) Rounded with a circular groove

Other mechanical frame ď Ž

ď Ž

Lenses are moved independently and horizontally

Interpupillary distance correction possible

Other designs

Courtesy of Ori Yaffe