Abstract

We show the functional extension of a standard ray tracer to be capable of tracing light fields of different degrees of coherence through complex optical systems. The light fields are represented by spherical waves. An approximate reconstruction of the optical field is possible at arbitrary positions in an optical system under investigation. Therefore, we can calculate the intensity distribution as well as the complex degree of coherence between two points at arbitrary positions. Simulations of the coherence properties of basic optical systems, which can be described analytically, show excellent agreement with theory. Furthermore, we show simulations of the coherence properties of a two-tandem-array microlens beam homogenizer under illumination with fully and partially coherent light.

© 2009 Optical Society of America

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References

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2009 (1)

2007 (1)

1995 (1)

1987 (1)

1985 (1)

1984 (1)

1983 (1)

1972 (1)

1970 (1)

1966 (3)

W. H. Steier, “The ray packet equivalent of a Gaussian light beam,” Appl. Opt. 5, 1229-1233 (1966).
[CrossRef] [PubMed]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550-1567 (1966).
[CrossRef] [PubMed]

K. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1962 (1)

1954 (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
[CrossRef]

1940 (1)

W. H. McCrea and F. J. W. Whipple, “Random paths in two and three dimensions,” Proc. R. Soc. Edinburgh 60, 281-298(1940).

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785-795 (1938).
[CrossRef]

1934 (1)

P. H. van Cittert, “Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene,” Physica 1, 201-210 (1934).
[CrossRef]

1920 (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Proc. Nat. Acad. Sci. U.S.A. 6, 474-475 (1920).
[CrossRef]

Agarwal, G.

Antos, R.

Arnaud, J.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).

Brunner, R.

Burkhardt, M.

Collins, S. A.

Craggs, G.

Douglas, N. G.

Glassner, A. S.

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2006).

Gross, H.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005).

Habermann, T.

T. Habermann and P. Wellner-Bou, Department of Mathematics and Natural Sciences, University of Applied Sciences Darmstadt, Schoefferstrasse 3, D-64295 Darmstadt, Germany (personal communication).

Herloski, R.

Jones, A. R.

Keller, J. B.

Kogelnik, H.

Lee, T. C.

Lemmer, U.

Li, T.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Marshall, S.

McCrea, W. H.

W. H. McCrea and F. J. W. Whipple, “Random paths in two and three dimensions,” Proc. R. Soc. Edinburgh 60, 281-298(1940).

Meuret, Y.

Michelson, A. A.

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Proc. Nat. Acad. Sci. U.S.A. 6, 474-475 (1920).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic , 2005).

Riechert, F.

Singer, W.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005).

Steier, W. H.

Thienpont, H.

Totzeck, M.

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005).

van Cittert, P. H.

P. H. van Cittert, “Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene,” Physica 1, 201-210 (1934).
[CrossRef]

Van Giel, B.

van Hoesel, F. J.

Verschaffelt, G.

Wellner-Bou, P.

T. Habermann and P. Wellner-Bou, Department of Mathematics and Natural Sciences, University of Applied Sciences Darmstadt, Schoefferstrasse 3, D-64295 Darmstadt, Germany (personal communication).

Whipple, F. J. W.

W. H. McCrea and F. J. W. Whipple, “Random paths in two and three dimensions,” Proc. R. Soc. Edinburgh 60, 281-298(1940).

Wolf, E.

E. Wolf and G. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541-546 (1984).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Yura, H. T.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785-795 (1938).
[CrossRef]

Zook, J. D.

Appl. Opt. (7)

IEEE Trans. Antennas Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Nuovo Cimento (1)

E. Wolf, “Optics in terms of observable quantities,” Nuovo Cimento 12, 884-888 (1954).
[CrossRef]

Physica (2)

P. H. van Cittert, “Die Wahrscheinliche Schwingungsverteilung in Einer von Einer Lichtquelle Direkt Oder Mittels Einer Linse Beleuchteten Ebene,” Physica 1, 201-210 (1934).
[CrossRef]

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785-795 (1938).
[CrossRef]

Proc. Nat. Acad. Sci. U.S.A. (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Proc. Nat. Acad. Sci. U.S.A. 6, 474-475 (1920).
[CrossRef]

Proc. R. Soc. Edinburgh (1)

W. H. McCrea and F. J. W. Whipple, “Random paths in two and three dimensions,” Proc. R. Soc. Edinburgh 60, 281-298(1940).

Other (7)

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2006).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).

K. Okamoto, Fundamentals of Optical Waveguides (Academic , 2005).

A. S. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

T. Habermann and P. Wellner-Bou, Department of Mathematics and Natural Sciences, University of Applied Sciences Darmstadt, Schoefferstrasse 3, D-64295 Darmstadt, Germany (personal communication).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the investigated free space propagation setup.

Fig. 2
Fig. 2

(a) 3D plot of the simulated degree of coherence regarding the center of the detector plane for the free space propagation setup. (b) Cut along the x axis of the plot in (a). The solid curve indicates the theoretical description using the van Cittert–Zernike theorem.

Fig. 3
Fig. 3

Schematic of the investigated single lens system setup.

Fig. 4
Fig. 4

(a) 3D plot of the simulated degree of coherence regarding the center of the detector plane for the single lens setup. (b) Cut along the x axis of the plot in (a). The solid curve indicates the theoretical description using ABCD-matrix formalism.

Fig. 5
Fig. 5

Schematic of the investigated two- tandem-array microlens beam homogenizer setup.

Fig. 6
Fig. 6

3D plot of the degree of coherence regarding the center of the fully coherent illumination spot on the first tandem array of the beam homogenizer.

Fig. 7
Fig. 7

3D plot of the simulated degree of coherence regarding the center of the detector plane for the two-tandem- array microlens beam homogenizer for fully coherent, quasi- parallel illumination.

Fig. 8
Fig. 8

3D plot of the degree of coherence regarding the center of the spatially partially coherent illumination spot on the first tandem array of the beam homogenizer.

Fig. 9
Fig. 9

3D plot of the simulated degree of coherence regarding the center of the detector plane for the two-tandem- array microlens beam homogenizer for spatially partially coherent, quasi-parallel illumination.

Equations (13)

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Γ ( x 1 ; x 2 ; τ ) = U ( x 1 , t 1 ) U * ( x 2 , t 2 ) e ,
γ ( x 1 ; x 2 ; τ ) = Γ ( x 1 ; x 2 ; τ ) Γ ( x 1 ; x 1 ; 0 ) Γ ( x 2 ; x 2 ; 0 ) = Γ ( x 1 ; x 2 ; τ ) I ( x 1 ) I ( x 2 ) ,
U j = A j e i ϕ j
U l = j A j l e i ( 2 π / λ opl j l + ϕ j ) ,
U l U m * = j k A j l A k m * e 2 π λ i ( opl j l opl k m ) e i ( ϕ j ϕ k )
U l U m * = 1 N n = 1 N j k A j l A k m * ¯ e 2 π λ i ( opl j l opl k m ) independent of n · e i ( ϕ j n ϕ k n ) .
n = 1 N e i ( ϕ j n ϕ k n )
n = 1 N e i ( ϕ j n ϕ k n ) = N + n = 1 , j k N e i ( ϕ j n ϕ k n ) .
lim N 1 N n = 1 N e i ( ϕ j n ϕ k n ) = δ j k ,
lim N U l U m * = j A j l A j m * ¯ e 2 π λ i ( opl j l opl j m ) .
γ l m = U l U m * U l U l * · U m U m * = j A j l A j m * ¯ e 2 π λ i ( opl j l opl j m ) j | A j l ¯ | 2 · j | A j m ¯ | 2 .
γ l m j A j l ¯ A j m ¯ * e 2 π λ i ( opl j l opl j m ) j | A j l ¯ | 2 · j | A j m ¯ | 2 .
γ l m j A j l A j m * e 2 π λ i ( opl j l opl j m ) j | A j l | 2 · j | A j m | 2 .

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