Three dimensionaloptical transfer functionsfor hi gh aperture systemswith non-symmetric pupils
Matthew R. Arnison, Colin J. R. Sheppard
Physical Optics Laboratory, School of Physics, andKey Centre for Microscopy and Microanalysis
http://www.physics.usyd.edu.au/physopt/[email protected]
High aperture Fourier optics
x
z
Point spread function
High aperture Fourier optics
m
s
x
z
FT
Ö
• Frieden (JOSA, 57, p56, 1967) scalar 3D OTF assumed paraxial rays
• Sheppard (JOSA A, 11, p593, 1994) assumed a radially symmetric pupil function
• Sheppard (Optik, 107, p79, 1997) was vectorial but results were 2D projections
Point spread function Optical transfer function
Why asymmetric?
• Vectorial focussing is not radially symmetric.
• Aberrations are modeled as phase functions across thepupil which are often not radially symmetric.
• We therefore need to generalise the 3D transfer functionintegrals to deal with arbitrary pupil functions.
Fourier optics
2D Fourier transform 3D Fourier transform
P(m,n)pupil
E(x,y)PSF
P∗P*OTF
EE*IPSF
Q(m,n,s)pupil
E(x,y,z)PSF
Q∗Q*OTF
E•E*IPSF
Fourier optics
2D Fourier transform 3D Fourier transform
P(m,n)pupil
E(x,y)PSF
P∗P*OTF
EE*IPSF
Q(m,n,s)pupil
E(x,y,z)PSF
Q∗Q*OTF
E•E*IPSF
Fourier optics
3D Fourier transform
P(m,n)pupil
E(x,y)PSF
P∗P*OTF
EE*IPSF
Q(m,n,s)pupil
E(x,y,z)PSF
Q∗Q*OTF
E•E*IPSF
2D Fourier transform
Fourier optics
3D Fourier transform
Q(m,n,s)pupil
E(x,y,z)PSF
Q∗Q*OTF
E•E*IPSF
Fourier optics
3D Fourier transform
Q(m,n,s)pupil
E(x,y,z)PSF
Q∗Q*OTF
E•E*IPSF
Fourier optics
3D Fourier transform
Q(m,n,s)pupil
E(x,y,z)PSF
Q∗Q*OTF
E•E*IPSF
0.729724
1.37194
-0.322877
0.322877
-1.16338
1.16338
3D vectorial pupils
Sineapodisationα=π/3(NA 0.87)Input beamx-polarised
Qz
QyQx
m
s n
Mansuripur JOSA A, 1986.Sheppard & Larkin, Optik, 1997.
Correlation of 2D pupil functions
m1
n1
K(m,n)
Correlation of 3D pupil functions
�1
0
1m
�1
�0.5
0
0.5
1
n
�1
�0.5
0
s
�1
0
1m
�1 0 1m
�1
�0.5
0
0.5
1
n
�1�0.50s
�1
�0.5
0
0.5
1
n
α=π/2
Correlation of 3D pupil functions
-1
0
1m
-1
-0.5
0
0.5
1
n
-1-0.8-0.6-0.4
s
-1
0
1m
-1 0 1m
-1
-0.5
0
0.5
1
n
-1-0.8-0.6-0.4
s
-1
-0.5
0
0.5
1
n
α=π/3
Circle of intersection
22 nml +=
Correlation inte gral
Generalpupil
Auto-correlation
Correlation inte gral
Projectedpupil
Auto-correlation
PolarisationDeclination Apodisation Complex pupil mask
Normalisation
Correlation inte gral
Projectedpupil
Auto-correlation
Circleofintersection
Widefield vectorial OTF
-1
0
1m
-1
0
1
n
-0.5
0
0.5s
-1
0
1m
Widefield vectorial OTF
Herschelapodisationα=2π/5(NA 0.95)
VOTF: axial slices
�2 �1 0 1 2�0.6�0.4�0.2
00.20.40.6
Cx T 0D +2 Pi/s5
0. 1370.
�2 �1 0 1 2�0.6�0.4�0.2
00.20.40.6
Cy T 0D +2 Pi/s5
�37.2 22.5
�2 �1 0 1 2�0.6�0.4�0.2
00.20.40.6
Cz T 0D +2 Pi/s5
�504. 266.
Cx
Cz
Cy
Herschelapodisationα=2π/5n=0
C
m
s
�2 �1 0 1 2�0.6�0.4�0.2
00.20.40.6
C T 0D +2 Pi/s5
�410. 1350.
Cx
m
n
sSineapodisationα=2π/5(NA 0.95)
Cx
m
n
sSineapodisationα=2π/5(NA 0.95)
Cx
m
n
s
Cx
m
n
s
Cx
m
n
s
Cx
m
n
s
Cx
m
n
s
Cy
m
n
s
Cz
m
n
s
Cz
m
n
s
Cz
m
n
s
Cz
m
n
s
Cz
m
n
s
Cz
m
n
s
C =Cx+ Cy+ Cz
m
n
s
C
m
n
s
C
m
n
s
C
m
n
s
C
m
n
s
C
m
n
s
C
m
n
s
What does it all mean?
• The vectorial OTF is simply the frequency spectrum of thevectorial intensity point spread function: perhaps “transferfunction” is a misnomer
• For weak scattering or fluorescence imaging, if the objectresponse is insensitive to polarisation, you could use this foraccurate modeling
• Might also be useful for analysing high NA polarisationmicroscopy
• Another way of exploring the symmetries of vectorial focusing
Conclusion
• We have developed a general vectorial 3D OTF forarbitrary pupil functions.
• It’s straightforward to calculate for any point infrequency space - no massive FFT arrays required!
• So we have a Fourier version of vectorial focusingtheory, suitable for high aperture analysis.
Stay tuned to Optics Communications for details.
