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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 15976–15981
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Reconstructing the phase distribution of two Interfering wavefronts by analysis of their nonlocalized fringes with an iterative method

Ehsan A. Akhlaghi, Ahmad Darudi, and M. Taghi Tavassoly  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 15976-15981 (2011)
http://dx.doi.org/10.1364/OE.19.015976


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Abstract

This paper presents a technique for reconstructing two interfering wavefronts by analyzing their 3D interference field pattern. The method is based on the numerical inverse problem and will present a robust algorithm for reconstructing of wavefronts. Several simulations are done to validate the proposed method.

© 2011 OSA

1. Introduction

Wavefront reconstruction of an optical wave has found wide applications in analyzing and measuring topography, aberrations of optical systems [1

1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

,2

2. D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).

], adaptive optics for astronomy [3

3. R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).

,4

4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).

]. Because of high frequency of optical wave there is no direct method for detecting the phase of CW monochromatic light. Hence, different techniques have been proposed and implemented to reconstruct wavefront indirectly. All wavefront sensing methods fall into two general categories: interferometric methods and noninterferometric methods [4

4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).

].

The noninterferometric methods have been developed on the basis of light intensity measurement e.g. the Foucault knife-edge test [4

4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).

], Shack–Hartmann wavefront sensor [4

4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).

,5

5. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

], intensity transport equation [6

6. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983). [CrossRef]

,7

7. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]

], pyramid wavefront sensors [8

8. J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]

], and Gerchberg–Saxton algorithm [9

9. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

,10

10. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21(12), 842–844 (1996). [CrossRef] [PubMed]

] to name a few. In practice, these methods are applied in situations where high accuracy is not required or high speed data processing is crucial and harsh environment conditions are exist.

Interferometry with laser has become one of the most commonly used methods due to availability of solid-state detectors and fast processors. These interferometric techniques require highly coherent reference wave sources, high spatial resolution detectors, and a stable and clean setup. Generally speaking, the precision of interferometric methods is limited by the quality of the reference wavefront. There are also interferometric methods without reference wave for instance; shearing interferometry [11

11. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef] [PubMed]

], phase conjugate technique [12

12. R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999). [CrossRef]

], three flats method [13

13. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993). [CrossRef] [PubMed]

,14

14. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]

] and diffracted wave interferometry [15

15. R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express 15(7), 3876–3887 (2007). [CrossRef] [PubMed]

]. But these methods also have limitations, they add to complexity of the setup, or need extra measurements, and furthermore they require very good optics.

We have already shown that by analyzing the intensity distribution in an interference field in three dimensions one can reconstruct the wave vectors of the two interfering wavefronts [16

16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000). [CrossRef]

18

18. A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001).

]. Authors have introduced and implemented a new method to reconstruct the wavefront based on nonlocalized two-wave interferometry in which the overall angle of two wavefronts are small and satisfy the paraxial approximation. The algorithm of the method is summarized as following; A) the interference patterns are recorded at two parallel planes at z1 and z2 separated by Δz=z2z1, and perpendicular to the average propagation direction of two waves, represented by z. B) The phase difference distributions (PDDs) Δϕ1and Δϕ2on planes z1 and z2 are reconstructed from the fringe patterns by using available fringe analyzing methods. C) Then, the phase difference gradient (Δϕ)and propagation directions of the interfering wavefronts are determined at enough number of points on the interferograms. D) Finally, the propagation directions and consequently the phase distributions of two interfering wavefronts are reconstructed on each element of the fringe pattern.

The article is organized as follows. The algorithm of the method is described in section two. The simulation results for different wavefronts are presented in section three, and conclusion remarks are given in section four.

2. Two wavefronts reconstruction

2.1. Theory and algorithm

In paraxial approximation a wave that is propagating along z direction can be express as:
Uj,z(x,y)=Aj,z(x,y)eiϕj,z(x,y)j=1,2
(1)
where Aj,z, ϕj,z are the amplitude and phase distributions of the wave on the planes z=z1 and z=z2. In Fig. 1
Fig. 1 Schematic representation of the two interfering waves at planes z=z1 and z=z2.
the traces of these waves on the planes z=z1 and z=z2are schematically represented.

The intensity distribution interference patterns of these two waves at two planes z=z1 and z=z2 can be described as,
Izj(x,y)=A1zj2(x,y)+A2zj2(x,y)+2A1zj(x,y)A2zj(x,y)×cos(Δϕzj),j=1,2
(2)
where

Δϕzj=ϕ2,zj(x,y)ϕ1,zj(x,y),j=1,2.
(3)

The PDD can be obtained from fringe pattern by well known fringe pattern analysis methods [1

1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

,2

2. D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).

], for instance the Fourier or the phase shift fringe pattern analysis methods. To reduce the calculation time, the resolution of experimental phase distributions (Δϕz1(exp) andΔϕz2(exp)) are decreased to N×N meshes, where simulation results have shown that N should be about more than15 and its maximum should be correspond to the lowest required resolution. To start with, we assume one of the wavefronts as a plane wave (ϕ1z1=cte). Therefore, the second wavefront can be given by,

ϕ2z1=ϕ1z1+Δϕz1(exp).
(4)

The two constructed waves propagate distance Δz=z2z1 in the z direction. So the phase difference of the two wavefronts in the second plane is given by,

Δϕz2(cal)=ϕ2z2ϕ1z2.
(5)

(a) We reduce the resolution of experimental data (Δϕz1(exp) andΔϕz2(exp)) to N×N meshes. (b) We choose the initial conditions for ϕ1z1 as a plane wave, ϕ1z1=cte. (c) We use all N×N elements of ϕ1z1 as optimization parameters (input of COST function). (d) We construct the second wavefront (ϕ2z1) by adding experimental Δϕz2 to ϕ1z1. (e) We propagate the both wavefronts distanceΔz in the z direction and construct ϕ1z2, ϕ2z2and calculate the PDD on the second plane. f) We compare the experimental PDD with the calculated one at the second plane, and calculate COST function (Eq. (9). (g) We use an optimization algorithm and the value of the COST function to update the optimization parameters and we repeat the iteration between step (d) and step (g) until the cost function is minimized. (h) End.

2.2. Wavefront propagator

In this method, a wavefront propagator is required to generate the wavefronts in the second plane from the wavefronts in the first plane. There are different free space optical wavefront propagators that are valid under different regimes of operation. The Fourier method that is based on Fresnel-Kirchhoff diffraction integral is used extensively for simulation of wavefront propagation [19

19. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

21

21. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999). [CrossRef]

]. If wave propagation is in geometrical optics regime and the effects of diffraction optics can be neglected, the ray tracing method for wavefront propagation can be used that is simple and need short calculation times [19

19. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

,22

22. M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. I.

].

Δϕ(xx,yy,z2z1)=k0(xx)2+(yy)2+(z2z1)2.
(8)

The new wave field is known on the primed coordinates, which will generally no longer coincide with the initial N×N grid. Therefore, we transformed the rays’ data to the data grid points by Spline 2D interpolation. This algorithm was implemented in MATLAB, and its accuracy was tested by aberrated waves which were generated by ZEMAX [23] and was propagated the same distance. The waves were generated by reflecting a plane wave from a mirror surface that consisted of 20th order Zernike polynomial aberration, Fig. 2(a)
Fig. 2 (a) Phase distribution of a wavefront that is generated by ZEMAX. (b) Propagated wavefront of Fig. 2(a) by 15cm distance. (c) The difference in phase distribution of the propagated wavefront by our algorithm and by ZEMAX.
and the propagation distance was 15cm, Fig. 2(b). The size of detector is taken as 4mm×4mm with 400×400 pixels. As shown in Fig. 2(c), peak-to-valley of the phase error between ZEMAX propagation and the method described in section 2.2 is about 0.01rad (optical path difference OPD error is aboutλ/103) which is quite accurate.

2.3. COST unction

As it was mentioned previously, if the calculated PDD, Δϕz2(cal), matches the measured PDD, Δϕz2(exp), then the estimated answer (ϕ1z1) is the sought wavefront. To compare the results quantitatively a COST functions should be defined (E) which measures the degree of phase matching. For this purpose we compare Δϕz2(cal) to Δϕz2(exp) as follows.

E=i,j|[e]i,j|N2   ,[e]i,j=[Δϕz2(cal)]i,j[Δϕz2(exp)]i,j,iandj=1...N.
(9)

Here e is a 2D array and represents difference between measured and calculated phase distribution at the coordinate z2 (second plane).

3. Simulation results

The spherical and cylindrical wave propagation can be done analytically. A combination of these waves with different radii was used to simulate the wavefronts and their interference patterns in space. The PDDs at both sides of the reference plane were used to find out the individual phase distributions. In this section, the simulation results under two different conditions are presented (Interference of spherical-cylindrical and two cylindrical waves). For both of these simulations, it is assumed that the observation plane is perpendicular to z axis (mean propagation direction) and its size is 5mm×5mm with 500×500 resolution. In order to lowering the calculation time, the resolution of phase distribution is decreased to 50×50 pixels by averaging over 10×10 window. Simulations were performed on a PC with 2.6 GHz clock rate and triple cores processor (AMD Phenom(tm) II X3 710). The optimization time and convergence behavior of the COST function strongly depends on complexity of the wavefronts, grid size of the reconstructed waves and optimization algorithm. Normally with simple optimization algorithm the optimization loop is trapped on local minima and results has less accuracy. We have used the simulated annealing for optimization algorithm which reduce the computation time and also improve the accuracy of the reconstruction.

3.1 Interference of spherical and cylindrical waves

In this case we investigate reconstructing a spherical and a cylindrical wave, whose radii are −1m and 4m, respectively (Figs. 3(a)
Fig. 3 (a) phase distribution of a −1m-radius spherical wave. (b) Phase distribution of a 4m-radius cylindrical wave. (c) Interference patterns of spherical and cylindrical waves at first observation plane (Media 1). (d) Difference of estimated and simulated phase distribution.
and 3(b)). Their interference patterns are inscribed on two planes separated by 70cm from each other.

The difference between the estimated and simulated wavefronts is demonstrated in Fig. 3(d). A 0.7rad peak-to-valley error in PDD and a 6% relative error in the estimated wavefront are achieved in a 52min processing timeframe.

3.2 Interference of two cylindrical waves

In this case we investigate reconstructing two cylindrical waves with −2m and −1m radii (Figs. 4(a)
Fig. 4 (a) phase distribution of a −2m-radius cylindrical wave. (b) Phase distribution of a −1m-radius cylindrical wave. (c) Interference patterns of two cylindrical waves at first observation plane (Media 2). (d) Difference of estimated and simulated phase distribution.
and 4(b)). Their axis is rotated 60° respect to each other in plane of observation. Their interference is inscribed on two planes separated 70cm from each other.

The difference between the estimated and simulated wavefronts is demonstrated in Fig. 4(d). A 0.35rad peak-to-valley error in PDD and a 3% relative error in the estimated wavefront are achieved in a 39min processing timeframe.

4. Conclusion

A technique for reconstruction of two interfering wavefronts has been introduced. The method is based on an iterative algorithm and numerical analysis. The algorithm has been evaluated with some simulated wavefronts such as spherical and cylindrical or two cylindrical wavefronts. The simulation results confirm this method has potentiality to be used for reconstruction of two interfering wavefronts from interference field. Although simulation results show some systematic errors, the attainable accuracy and precision is about λ/10in noiseless condition. The error sources are due to numerical errors in wavefront propagator mostly at corner of the wavefronts and also optimization algorithm which normally trapped in local minima.

For comparison; the algorithm of the reference [16

16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000). [CrossRef]

] is an analytical method which for simple wavefronts is much faster than method of current paper. In order to has accurate calculation of PDD with respect to Δz, there is a limitation for maximum value of Δzwhich will affect the attainable accuracy for complex wavefronts. Also, it is difficult to write a automated program for whole process and in the points of fringe patterns at which gradient of PDD in (x, y) plane become zero, it fails to reconstruct the wavefronts. Current method is a numerical method. It is possible to reconstruct more detail of the wavefronts far from boarders by increasingΔz. At boarders the divergence and convergence of the wave front will limit the accessible information and fails to reconstruct wavefronts accurately. The accuracy depends on grid size and Δz. It can be easily written in a form of automated program. The method has application to find out the two interfering unknown wavefronts or survive for accuracy of reference wavefront, for instance: two reflected waves from front and back surfaces of a transparent plate, finding the position of two coherent optical point sources for range finding and reflection from lens interfaces for finding the lens parameters [17

17. A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001). [CrossRef]

]. Authors believe the presented method open new possibilities for removing the necessity of good reference wavefront in optical testing and other filed of metrology.

References and links

1.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

2.

D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).

3.

R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).

4.

J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).

5.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

6.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983). [CrossRef]

7.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]

8.

J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]

9.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

10.

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21(12), 842–844 (1996). [CrossRef] [PubMed]

11.

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef] [PubMed]

12.

R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999). [CrossRef]

13.

C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993). [CrossRef] [PubMed]

14.

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]

15.

R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express 15(7), 3876–3887 (2007). [CrossRef] [PubMed]

16.

M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000). [CrossRef]

17.

A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001). [CrossRef]

18.

A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001).

19.

M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

20.

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006). [CrossRef] [PubMed]

21.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999). [CrossRef]

22.

M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. I.

23.

http://www.zemax.com

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: March 9, 2011
Revised Manuscript: April 19, 2011
Manuscript Accepted: April 25, 2011
Published: August 5, 2011

Citation
Ehsan A. Akhlaghi, Ahmad Darudi, and M. Taghi Tavassoly, "Reconstructing the phase distribution of two Interfering wavefronts by analysis of their nonlocalized fringes with an iterative method," Opt. Express 19, 15976-15981 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15976


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References

  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  2. D. Malacara, Optical Shop Testing (John Wiley & Sons, 1998).
  3. R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
  4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).
  5. R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).
  6. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983). [CrossRef]
  7. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984). [CrossRef]
  8. J. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]
  9. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  10. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21(12), 842–844 (1996). [CrossRef] [PubMed]
  11. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40(10), 1637–1648 (2001). [CrossRef] [PubMed]
  12. R. Kowarschik, L. Wenke, T. Baade, M. Esselbach, A. Kiessling, G. Notni, and K. Uhlendorf, “Optical measurements with phase-conjugate mirrors,” Appl. Phys. B 69(5-6), 435–443 (1999). [CrossRef]
  13. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32(25), 4698–4705 (1993). [CrossRef] [PubMed]
  14. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45(23), 5856–5865 (2006). [CrossRef] [PubMed]
  15. R. de la Fuente and E. López Lago, “Mach-Zehnder diffracted beam interferometer,” Opt. Express 15(7), 3876–3887 (2007). [CrossRef] [PubMed]
  16. M. T. Tavassoly and A. Darudi, “Reconstruction of interfering wavefronts by analyzing their interference pattern in three dimensions,” Opt. Commun. 175(1-3), 43–50 (2000). [CrossRef]
  17. A. Darudi and M. T. Tavassoly, “Interferometric specification of lens (single and doublet) parameters,” Opt. Lasers Eng. 35(2), 79–90 (2001). [CrossRef]
  18. A. Darudi, “Interferometry without reference wave,” (PhD Thesis, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran, 2001).
  19. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 2003).
  20. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006). [CrossRef] [PubMed]
  21. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999). [CrossRef]
  22. M. Bass, Handbook of Optics (McGraw-Hill, 1995), Vol. I.
  23. http://www.zemax.com

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