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
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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
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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
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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
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Lenses are moved independently and horizontally
Interpupillary distance correction possible
Other designs
Courtesy of Ori Yaffe