Advanced Optics Lab – Spring 2012
Point Spread Function Engineering - prelab
Consider the imaging system depicted in Figure 1. The focal lengths of the lenses are f L1  50mm ,
f L2  f L3  f  100mm , a  100mm. In the Fourier plane there is a transmission spatial light modulator
(SLM) with 512x512 pixels and an array size of 7.68 x 7.68mm.
L1
L2
L3
Object
SLM
CCD
a
b
f
f
f
f
1. What is the magnification of the system?
2. Assuming the limiting aperture is in the SLM, find the numerical aperture of the system. Find the
depth of focus [1].
3. Assume a centered point source placed in the object plane and no phase modulation on the SLM.
Calculate and plot the image on the CCD camera [1].
4. The point source is moved 10mm along the optical axis in steps of 2mm. Calculate the point
spread function and plot the images obtained on the CCD camera. Calculate the modulation
transfer function (MTF) of the system (along the x-direction) for the six input locations [1].


5. A phase mask implementing the complex amplitude function U ( x, y)  exp  i ( xˆ 3  yˆ 3 ) is
loaded on the SLM ( xˆ , yˆ are normalized coordinates) [2]. Repeat part 4 for this new system with
  20 . Explain the result.
6. Consider the Laguerre-Gaussian basis functions defined as follows [3]:
GLm, n (r , )  Cn, m exp(r 2 )  ( 2r )|m| L|(mn||m|) / 2 (2r 2 )  exp(im )
where (r ,  ) are polar coordinate and L|(mn||m|) / 2 are the generalized Laguerre polynomials with the
integers n, m satisfying n | m |, | m | 2, | m | 4, . Cn, m are normalization constants.
Consider the superposition f RPSF  GL1,1  GL3,5  GL5,9  GL7,13  GL9,17 with
2
Plot f RPSF . Show that f RPSF is a rotating PSF based on the location of the modes in the
Laguerre-Gaussian modal plane [4].
7. A phase and amplitude mask implementing f RPSF is loaded on the SLM (choose a proper scaling
to cover as much as possible the aperture). Repeat part 4 for this new system. Explain the result.
References
1.
Introduction to Fourier Optics, Joseph W. Goodman: 2nd Edition (McGraw-Hill, New York, 1996) or 3rd
Edition (Roberts & Company, Englewood CO, 2005)
2.
Edward R. Dowski, Jr. and W. Thomas Cathey, "Extended depth of field through wave-front coding," Appl.
Opt. 34, 1859-1866 (1995)
3.
Greengard, Y. Y. Schechner, R. Piestun, "Depth from rotation", Opt. Lett. 31, 181-183 (2006)
4.
R. Piestun, Y. Y. Schechner, J. Shamir, "Generalized propagation invariant wave fields with finite energy", J.
Opt. Soc. Am. A 17 , 294-303 (2000)