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.