Astigmatic RefractionAstigmatic RefractionUsing Peaks ofUsing Peaks of

InterferogramInterferogram MTF for a MTF for aTalbot MoirTalbot Moiréé Interferometer Interferometer

Ed Sarver, PhDEd Sarver, PhD

Acknowledge coauthorsAcknowledge coauthors

• Tony Y. Van Heugten• Thomas D. Padrick• Max T. Hall

RonchiRonchi Ruling Ruling

Set of opaque, parallel barswith 50% duty cycle

RonchiRonchi Grid Grid

Two Ronchi rulings orientedorthogonal to each other

RonchiRonchi Grid Parameters Grid Parameters

p

p/2

RonchiRonchi Grid as Grid asTransmission FunctionTransmission Function

Wavefrontpropagation G1

d

Observation plane

For simplicity, letFor simplicity, let’’s model amplitudes model amplitudetransmissiontransmission11 as a sinusoid as a sinusoid……

1Goodman, Fourier Optics, 3rd edition, Roberts & Company, p. 88, 2005.

( )

+= xyxT dπ2cos1

2, 1

Fresnel diffraction leads to intensityFresnel diffraction leads to intensitydistributiondistribution22 given by given by……

2Goodman, Fourier Optics, 3rd edition, Roberts & Company, p. 89, 2005.

( )

=

+

+=

dp

A

where

xp

xp

AyxI

2

2

cos

2cos2cos2141,

πλ

ππ

Talbot planeTalbot plane……

= d

pA 2cos πλ

A = ± 1 provides maximum contrast and exact replica of transmission plane.Corresponding d location is called Talbot plane.

Now we place a second Now we place a second RonchiRonchiGrid at a Talbot distance dGrid at a Talbot distance d……

Wavefrontpropagation

G1 G2

d

……And rotate the second gridAnd rotate the second grid

Wavefrontpropagation

G1 G2

d

_

This superposition of grid patterns leads to amoiré pattern. The combination of the moiréeffect with the Talbot distance requirementleads to the name…

Talbot Moiré Interferometer

Fourier domain primary peaks ofFourier domain primary peaks ofG1 areG1 are……

1/p

1/p

u

v

Fourier domain primary peaks ofFourier domain primary peaks ofrotatedrotated G2 are G2 are……

u

v

_

In the spatial domain the propagated transmissionof G1 is multiplied by the transmission function atG2.

The equivalent operation in the Fourier domain isconvolution…

Two primary peaks of MTF ofTwo primary peaks of MTF ofIntensity distribution at G2Intensity distribution at G2

P1

P2

What happens when the incident wavefront atG1 has defocus?

With diverging incident wavefrontWith diverging incident wavefront……

P1P2 = S x P1

S = period scale factor= 1 – (Dxd)/1000

d

Fourier domain for scaled G1Fourier domain for scaled G1……

u

v

No defocus

Diverging wavefront

Converging wavefront

After modulation by G2After modulation by G2……

P1

P2

Apparent rotation due to defocusApparent rotation due to defocus……

P1

P2

Apparent rotation due to defocus..Apparent rotation due to defocus..

P1

P2

Apparent rotation due to defocus..Apparent rotation due to defocus..

P1

P2

Instead of pure defocus, what about astigmaticwavefront?

Astigmatic wavefront aligned withAstigmatic wavefront aligned withsystem axessystem axes……

u

v

No defocus

Diverging wavefront inu axis direction

Converging wavefront inv axis direction

General Astigmatic wavefrontGeneral Astigmatic wavefront

( )sycsxsCsxsycsBsyssxcA

yx

CBBA

yx

cssc

sysx

cssc

yx

22

22

00

''

+=

−=

+=

=

−

−=

Rotate by _

Rotate by -_

Scale by Sx, Sy

Sx and Sy are related to power inthe principal axes.

Given MTF peaks, need to calculateGiven MTF peaks, need to calculatematrix coefficients A, B, and Cmatrix coefficients A, B, and C

Compute Affine TransformCompute Affine Transform……NN R

CBBA

R

='

=

=

=

CBA

x

RRRRRRRR

b

RRRR

RRRR

RRRR

RRRR

A

y

x

y

x

y

x

y

x

yx

yx

yx

yx

yx

yx

yx

yx

,

3'3'2'2'1'1'0'0'

,

330033220022110011000000

Since for each G1 point:

( ) ( )bAAAx TT 1−=Solve

Where…

Then, find Then, find eigenvalueseigenvalues……

( ) ( )2

4 22

2,1BCACA +−±+

=λ

Note: ALWAYS two real eigenvalues.

Scale factors and axisScale factors and axis……

−=

=

−

BAAxis

S

11

2,12,1

tan

1

λ

λ

Note: Use atan2 function to handle zero valued B.

Principal powersPrincipal powers……

( )d

SP

10001 2,12,1

−=

Note: d is the Talbot plane distance.

Results for model eyeResults for model eye

• 5 exams• Sphere

– Mean = -4.26 D– SD = 0.004

• Cylinder– Mean = -2.90– SD = 0.008

• Axis– Mean = 92– SD = 0

Results for 46 year old maleResults for 46 year old male

Spatial domain interferogram MTF with peaks

Results for 10 examsResults for 10 exams

1.290.140.11sd69.10-1.45-0.09mean

69.00-1.23-0.289

68.00-1.26-0.288

68.00-1.44-0.097

69.00-1.55-0.036

70.00-1.38-0.025

71.00-1.47-0.074

71.00-1.61-0.103

67.00-1.580.032

69.00-1.38-0.111

69.00-1.600.020

AxisCylinderSphereTrial

DiscussionDiscussion

• The sphere, cylinder and axis can beextracted from the MTF of theinterferogram of a Talbot moiré wavefrontsensor

• Processing time is very fast: 46 ms on a3.6 GHz PC– Fast enough for real time acquisition and

display

ConclusionConclusion

• MTF calculation method is fast and simplemeans to process Talbot moiré wavefrontsensor images

• May be especially helpful for real-timedisplay of refraction

• May also be helpful in augmenting orproviding a quality control check for a fullwavefront reconstruction

Thank you!