Astigmatic Refraction Using Peaks of Interferogram MTF for...
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Astigmatic RefractionAstigmatic RefractionUsing Peaks ofUsing Peaks of
InterferogramInterferogram MTF for a MTF for aTalbot MoirTalbot Moiréé Interferometer Interferometer
Ed Sarver, PhDEd Sarver, PhD
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Acknowledge coauthorsAcknowledge coauthors
• Tony Y. Van Heugten• Thomas D. Padrick• Max T. Hall
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RonchiRonchi Ruling Ruling
Set of opaque, parallel barswith 50% duty cycle
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RonchiRonchi Grid Grid
Two Ronchi rulings orientedorthogonal to each other
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RonchiRonchi Grid Parameters Grid Parameters
p
p/2
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RonchiRonchi Grid as Grid asTransmission FunctionTransmission Function
Wavefrontpropagation G1
d
Observation plane
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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
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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
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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.
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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
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……And rotate the second gridAnd rotate the second grid
Wavefrontpropagation
G1 G2
d
_
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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
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Fourier domain primary peaks ofFourier domain primary peaks ofG1 areG1 are……
1/p
1/p
u
v
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Fourier domain primary peaks ofFourier domain primary peaks ofrotatedrotated G2 are G2 are……
u
v
_
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In the spatial domain the propagated transmissionof G1 is multiplied by the transmission function atG2.
The equivalent operation in the Fourier domain isconvolution…
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Two primary peaks of MTF ofTwo primary peaks of MTF ofIntensity distribution at G2Intensity distribution at G2
P1
P2
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What happens when the incident wavefront atG1 has defocus?
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With diverging incident wavefrontWith diverging incident wavefront……
P1P2 = S x P1
S = period scale factor= 1 – (Dxd)/1000
d
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Fourier domain for scaled G1Fourier domain for scaled G1……
u
v
No defocus
Diverging wavefront
Converging wavefront
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After modulation by G2After modulation by G2……
P1
P2
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Apparent rotation due to defocusApparent rotation due to defocus……
P1
P2
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Apparent rotation due to defocus..Apparent rotation due to defocus..
P1
P2
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Apparent rotation due to defocus..Apparent rotation due to defocus..
P1
P2
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Instead of pure defocus, what about astigmaticwavefront?
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Astigmatic wavefront aligned withAstigmatic wavefront aligned withsystem axessystem axes……
u
v
No defocus
Diverging wavefront inu axis direction
Converging wavefront inv axis direction
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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.
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Given MTF peaks, need to calculateGiven MTF peaks, need to calculatematrix coefficients A, B, and Cmatrix coefficients A, B, and C
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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…
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Then, find Then, find eigenvalueseigenvalues……
( ) ( )2
4 22
2,1BCACA +−±+
=λ
Note: ALWAYS two real eigenvalues.
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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.
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Principal powersPrincipal powers……
( )d
SP
10001 2,12,1
−=
Note: d is the Talbot plane distance.
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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
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Results for 46 year old maleResults for 46 year old male
Spatial domain interferogram MTF with peaks
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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
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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
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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
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Thank you!