Abstract

In laser projection applications, laser light modules are often combined with rotating diffusers in order to reduce the appearance of speckle on the projection screen. The rotation of a diffuser in a laser beam generates a beam of partially coherent light. Propagation of this light through the different optical components constituting the laser projector is thus essential when investigating the appearance of speckle. In this paper, a computationally efficient simulation model is presented to propagate partially coherent light through a homogenizing rectangular light pipe. The light pipe alters the coherence properties of the light and different consequences are discussed. The outcomes of the simulation model are experimentally verified using a reversing wavefront Michelson interferometer.

© 2013 Optical Society of America

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References

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  • Publication
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    [Crossref]
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    [Crossref]
  7. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–187 (1978).
    [Crossref]
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    [Crossref] [PubMed]
  9. H. Gross, Handbook of Optical Systems: Aberration Theory and Correction of Optical Systems (Wiley, 2007).
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    [Crossref] [PubMed]
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  15. M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005).
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  21. G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
    [Crossref]

2010 (3)

K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. 49, 79–98 (2010).
[Crossref]

J. Hecht, “A short history of laser development,” Appl. Opt. 49, F99–F122 (2010).
[Crossref] [PubMed]

U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID 18, 813–820 (2010).
[Crossref]

2009 (2)

F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees of coherence through complex optical systems,” Appl. Opt. 48, 1527–1534 (2009).
[Crossref] [PubMed]

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

2006 (3)

2005 (1)

1988 (1)

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

1986 (1)

1980 (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

1979 (1)

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

1978 (1)

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–187 (1978).
[Crossref]

1971 (1)

Bellancourt, A.

U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID 18, 813–820 (2010).
[Crossref]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge university press, Cambridge, 1999).

Chellappan, K.

K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. 49, 79–98 (2010).
[Crossref]

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

Craggs, G.

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

DeSantis, P.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Dürr, F.

Erden, E.

K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. 49, 79–98 (2010).
[Crossref]

Farina, J. D.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

Fisher, I.

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005).
[Crossref] [PubMed]

Friberg, A. T.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, Englewood, 2007).

Gori, F.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–187 (1978).
[Crossref]

Grabherr, M.

Gross, H.

H. Gross, Handbook of Optical Systems: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

Guattari, G.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Hecht, J.

Imai, M.

Joyeux, D.

Lemmer, U.

Lowenthal, S.

Mackens, U.

U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID 18, 813–820 (2010).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge university press, Cambridge, 1995).
[Crossref]

Mandre, S.

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005).
[Crossref] [PubMed]

Moench, H.

U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID 18, 813–820 (2010).
[Crossref]

Narducci, L. M.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

Ohtsuka, Y.

Palma, C.

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–187 (1978).
[Crossref]

Peeters, M.

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005).
[Crossref] [PubMed]

Riechert, F.

Rohlfing, U.

Santarsiero, M.

Satoh, S.

Shen, F.

Thienpont, H.

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005).
[Crossref] [PubMed]

Turunen, J.

P. Vahimaa and J. Turunen, “Finite-elementary source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006).
[Crossref] [PubMed]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Urey, H.

K. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. 49, 79–98 (2010).
[Crossref]

Vahimaa, P.

Verschaffelt, G.

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

M. Peeters, G. Verschaffelt, H. Thienpont, S. Mandre, I. Fisher, and M. Grabherr, “Spatial decoherence of pulsed broad-area vertical-cavity surface-emitting lasers,” Opt. Express 13, 9337–9345 (2005).
[Crossref] [PubMed]

Wang, A.

Weichmann, U.

U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID 18, 813–820 (2010).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge university press, Cambridge, 1995).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Cambridge university press, Cambridge, 1999).

Wyrowski, F.

F. Wyrowski, “Field Tracing for Unified Optical Modeling,” in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW4A.1.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

G. Verschaffelt, G. Craggs, M. Peeters, S. Mandre, H. Thienpont, and I. Fisher, “Spatially resolved characterization of the coherence area in the incoherent emission regime of a broad-area vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron. 45, 249–255 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

JSID (1)

U. Weichmann, A. Bellancourt, U. Mackens, and H. Moench, “Solid-state lasers for projection,” JSID 18, 813–820 (2010).
[Crossref]

Opt. Commun. (4)

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional light beams,” Opt. Commun. 27, 185–187 (1978).
[Crossref]

P. DeSantis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model sources,” Opt. Commun. 67, 245–250 (1988).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Other (6)

H. Gross, Handbook of Optical Systems: Aberration Theory and Correction of Optical Systems (Wiley, 2007).

F. Wyrowski, “Field Tracing for Unified Optical Modeling,” in Frontiers in Optics Conference, OSA Technical Digest (online) (Optical Society of America, 2012), paper FW4A.1.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, Englewood, 2007).

J. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge university press, Cambridge, 1995).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Cambridge university press, Cambridge, 1999).

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Figures (8)
Fig. 1
Fig. 1 Schematic of the investigated setup that consists of a single-mode laser in combination with a rotating diffuser, closely followed by a rectangular glass light pipe with a length of 13cm. We also denote in the figure the axes orientation used in the modelling.

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Fig. 2
Fig. 2 Intensity distribution in the transverse direction of the elementary Gaussian modes constituting the total source field. The source model consists of a Gaussian weighted linear superposition of spatially shifted but identical, fully coherent elementary Gaussian-shaped modes.

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Fig. 3
Fig. 3 Top: propagation of rays from an elementary mode in the light pipe. Bottom: equivalent propagation of the same elementary mode through free-space propagation in glass and afterwards transforming/mapping the field to incorporate the internal reflections.

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Fig. 4
Fig. 4 Schematic of the reversing wavefront Michelson interferometer used to measure the spatial coherence area at the end of the light pipe.

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Fig. 5
Fig. 5 Degree of coherence |γ[(0, 0), (x, 0), 0]| between the beam’s center position and a transverse point at a distance x from the center (for a single mode laser beam passing through a rotating diffuser) for free-space propagation over a distance of 8.67cm after the diffuser. Blue marks: simulation, green curve: analytical value. The profile of the degree of coherence is Gaussian and the coherence diameter is 75μm.

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Fig. 6
Fig. 6 Intensity map of the degree of coherence |γ[(−x, −y), (+x, +y), 0]| measured at the CCD of the interferometer setup for free-space propagation over 8.67cm. The coherence diameter is found to be 75μm.

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Fig. 7
Fig. 7 The degree of coherence for the light of our partially coherent source that has propagated through the 13cm long light pipe. It exhibits a spiked profile with a slowly-varying envelope of width 75μm.

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Fig. 8
Fig. 8 The magnitude of the degree of coherence at the exit facet of the light pipe, measured at the detector of the interferometer setup (a) and a cross-section taken at the center (b).

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Tables (1)

Tables Icon

Table 1 Experimental verification of the far-field coherence function period in the case a light pipe is used.

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Equations (13)

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w = 2 λ π θ diff ,
U ( x , y , z ) = z j λ ∬ aperture U ( ξ , η , 0 ) e j k r r 2 d ξ d η ,
Γ ( r 1 , r 2 , τ ) = 〈 U ( r 1 , t ) U * ( r 2 , t + τ ) 〉 ,
γ ( r 1 , r 2 , τ ) = Γ ( r 1 , r 2 , τ ) Γ ( r 1 , r 1 , 0 ) Γ ( r 2 , r 2 , 0 ) = Γ ( r 1 , r 2 , τ ) I ( r 1 ) I ( r 2 ) ,
Γ lm = ∑ elem . modes U l U m *
γ lm = Γ lm Γ ll Γ mm .
| γ | = [ I t 0 − I 0 H − I 0 V ] 2 + [ I t λ / 4 − I 0 H − I 0 V ] 2 2 I 0 H I 0 V
θ coh = arctan ( D coh / 2 z ) ,
W = 2 λ π θ coh
NF int ∝ exp ( − a x 2 ) * [ ∑ k δ ( x − k D ) ]
FF coh ∝ ℱ { NF int } ,
FF coh ∝ ℱ { exp ( − a x 2 ) } ⋅ [ 1 D ∑ k δ ( ξ − k D ) ] ,
x grid = 1 2 λ z D ,

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