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Corneal astigmatism compensation by oblique light incidence
Optics Department
University of Alicante
Julián Espinosa
Group of Optics and Vision Science. University of Alicante
David Mas. University of Alicante
Henryk Kasprzak. Wroclaw University of Technology
Collaborators
Funding
Spanish Ministry of Science through project FIS2005-05053
Corneal astigmatism compensation by oblique light incidence
Justification and Objectives
�Oblique incidence induces two main aberrations: coma
and oblique astigmatism.
�Previous works show kappa angle compensates coma
due to crystalline tilt [Tabernero et al., JOSA, 2007]
�Oblique incidence may compensate corneal
astigmatism
�Angle kappa and foveal off-centred play a double role
compensating coma and astigmatism
Outline
�With-the-rule astigmatic Kooijman eye modelization.
�Eye transmittance calculation for oblique incidence.
�Light patterns analysis at retina.
�Merit function. Best PSF for an oblique light incidence.
�Passive compensation of astigmatism.
( ) ( )( ) 2 21
, 1 11
y
y x y
R Q x yz x y
Q R R R
+ = − − + +
Astigmatic Kooijman eye
1.3361.421.33741.3771Refractive
index
16.64.03.050.55Thickness
(mm)
-1.0-3.06-0.25-0.25Conic constant
Q
-6.010.26.5Ry=7.8
Rx=7.9
Radius (mm)
Posterior
lens
Anterior
lens
Posterior
cornea
Anterior
corneaSurface
Surface equation
zret
na n
v
Cornea Lens Retina
A.C.Gros. P.C.
Cornea + cristalline lens Transmittance Propagation
Eye transmittance
( ) ( ) ( ) ( ), , cos ,cos ,cosx y z
v v v v α β γ = = �
Ray vector
( ), , , ,1x y z
z zp p p p
x y
∂ ∂= = ∂ ∂
�
Surface normal vector
( ) ( ) ( )1 1
2 1
2 2
, , cos cosx y z
n nk k k k v p
n nε ε
= = ±
� � �∓
Snell law Refracted ray vector
Eye transmittance under oblique incidence
Transmittance Calculation
Optical path length d, solving:
( ) [ ]2, exp ( , )
tot
it x y d x y
πλ
=
Transmittance
( ), , ,
, ,i i i z i f i x i i y i
i i i
d d dz x y k z x k y k
n n n
+ = + +
( ) ( )( ) ( )
max,
,
, ,
IJ
M RMS
α βα β
α β α β=
Light Patterns Analysis
x
z
y
β
α
γ
Fresnel propagation
( )2
2 20 0
02 2'
0
exp ' expz
x i xzu i DFT u
x N zNµ
µ πλπ µ µλ
∆ ∆ = ∆
( ) ( ) 2
maxmax , max ,
zI I x y u x y = =
Maximum Intensity
Merit function
Light Patterns Analysis
Merit function
( ) ( )( ) ( )
max,
,
, ,
IJ
M RMS
α βα β
α β α β=
( ) ( )2, ,M I x y x y= ℜ∑
Secondary Momentum
( ) ( ) ( )( )
, , ,
, ,
,
x y
I x y x I x y yx y
I x y
ℜ = ℜ ℜ = ∑ ∑
∑Centroid position
Fresnel propagation
( )2
2 20 0
02 2'
0
exp ' expz
x i xzu i DFT u
x N zNµ
µ πλπ µ µλ
∆ ∆ = ∆
x
z
y
β
α
γ
Light Patterns Analysis
Fresnel propagation
( )2
2 20 0
02 2'
0
exp ' expz
x i xzu i DFT u
x N zNµ
µ πλπ µ µλ
∆ ∆ = ∆
( ) ( ), ,
tot j j
j
d x y c Z x y=∑
( )2,
jRMS c x y= ∑
Root Mean Square function
Zernike decomposition of the optical path length
x
z
y
β
α
γ
Merit function
( ) ( )( ) ( )
max,
,
, ,
IJ
M RMS
α βα β
α β α β=
Results: Astigmatic Kooijman eye
First corneal surface Rx= 7.9 mm R
y= 7.8 mm
Merit function
J (α,β)
Rx= 7.9 mm R
y= 7.8 mm γ from 0º to 7º
Results: Astigmatic Kooijman eye
x
y
x
100 µm
86 µm 86 µm
(0,0,1) (cos(85.25º),0,cos(4.75º))
Results: Astigmatic Kooijman eye
Point Spread Function at different incidence angles
µm µm
(cos(85.25º),0,cos(4.75º))
µm µm
(0,0,1)
Corneal heights from Pentacam
1.3361.421.33741.3771Refractive
index
17.34.02.44.524Thickness
(mm)
-1.0-3.06Conic constant
Q
-6.010.2Ry=6.11
Rx=6.31
Ry=7.82
Rx=7.92
Radius (mm)
Posterior
lens
Anterior
lens
Posterior
cornea
Anterior
corneaSurface
( ) ( )( ) 2 21
, 1 11
y
y x y
R Q x yz x y
Q R R R
+ = − − + +
Real astigmatic eye
Surface equation
Results: Real astigmatic eye
First corneal surface Rx= 7.92 mm R
y= 7.82 mm
Merit function
J (α,β)
x
y
x
160 µm
Results: Real astigmatic eye
Corneal heights from PentacamFirst corneal surface R
x= 7.92 mm R
y= 7.82 mm
γ from 0º to 7º
Conclusions
�Oblique incidence can compensate corneal
astigmatism.
�Tilting the incident beam introduce an aberration term
that may cancel the existing one.
�Exists passive compensation of astigmatism.
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