Abstract

Novel compact beam expanders that could be useful for applications such as providing light to flat panel displays are presented. They are based on a planar configuration in which three spatially linear gratings are recorded on one transparent substrate, so as to expand a narrow incoming beam in two dimensions. We present the design and recording procedures along with results, showing a relatively uniform intensity of the wide output beam. Such expanders can serve for illuminating flat panel displays.

© 2002 Optical Society of America

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References

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  1. M. Chapman, N. R. Heckenberg, “Use of a beam expanding telescope in grating-tuned waveguide CO2 laser,” Int. J. Infrared Millim. Waves. 8, 783–791 (1987).
    [CrossRef]
  2. M. Bea, A. Giesen, M. Huonker, H. Hugel, “Flexible beam expanders with adaptive optics—a challenge for modern beam delivery,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 84–95 (1995).
    [CrossRef]
  3. M. Takahashi, T. Ohntoshi, “Finite-difference time-domain analysis of laser diodes integrated with tapered beam-expanders,” IEEE Photon Tech. Lett. 11, 524–526 (1999).
    [CrossRef]
  4. F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
    [CrossRef]
  5. S. K. Gayen, M. E. Zevallos, B. B. Das, R. R. Alfano, “Time-sliced, two-dimensional, near-infrared imaging of normal and malignant human breast tissues,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 230–231 (1998).
  6. P. Hariharan, “Beam expander for making large rainbow holograms,” Appl. Opt. 26, 1815–1818 (1987).
    [CrossRef] [PubMed]
  7. Wu Jiang, D. L. Shealy, K. M. Baker, “Physical optics analysis of the performance of a holographic projection system,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. LeeProc. SPIE2404, 227–234 (1995).
    [CrossRef]
  8. S. B. Odinokov, M. V. Borisov, “Optical system of the device for making a hologram matrix,” in Diffraction and Holographic Technologies, Systems, and Spatial Light Modulators, I. Cindrich, S. H. Lee, R. L. SutherlandProc. SPIE3633, 279–284 (1999).
    [CrossRef]
  9. F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
    [CrossRef]
  10. M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
    [CrossRef]
  11. R. Neubert, L. Grunwaldt, G. Sesselmann, M. Steinbach, “An innovative telescope system for SLR,” in Laser Radar Ranging and Atmospheric Lidar Techniques II (Europto), U. Schreiber, C. Werner, eds., Proc. SPIE3865, 83–89 (1999).
    [CrossRef]
  12. V. Neuman, C. W. Pitt, L. M. Walpita, “Guided-wave holographic grating beam expander–fabrication and performance,” Electron. Lett. 17, 165–167 (1981).
    [CrossRef]
  13. R. K. Kustuk, M. Kato, Y. T. Huang, “Substrate mode holograms for optical interconnects,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 168–171.
  14. A. A. Friesem, Y. Amitai, “Planar diffractive elements for compact optics,” in A. ConsortiniTrends in Optics (Academic, San Diego, 1996), pp. 125–144.
    [CrossRef]
  15. I. Shariv, Y. Amitai, A. A. Friesem, “Compact holographic beam expander,” Opt. Lett. 18, 1268–1270 (1993).
    [CrossRef] [PubMed]

1999 (1)

M. Takahashi, T. Ohntoshi, “Finite-difference time-domain analysis of laser diodes integrated with tapered beam-expanders,” IEEE Photon Tech. Lett. 11, 524–526 (1999).
[CrossRef]

1993 (1)

1992 (1)

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

1990 (1)

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

1988 (1)

F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
[CrossRef]

1987 (2)

P. Hariharan, “Beam expander for making large rainbow holograms,” Appl. Opt. 26, 1815–1818 (1987).
[CrossRef] [PubMed]

M. Chapman, N. R. Heckenberg, “Use of a beam expanding telescope in grating-tuned waveguide CO2 laser,” Int. J. Infrared Millim. Waves. 8, 783–791 (1987).
[CrossRef]

1981 (1)

V. Neuman, C. W. Pitt, L. M. Walpita, “Guided-wave holographic grating beam expander–fabrication and performance,” Electron. Lett. 17, 165–167 (1981).
[CrossRef]

Alfano, R. R.

S. K. Gayen, M. E. Zevallos, B. B. Das, R. R. Alfano, “Time-sliced, two-dimensional, near-infrared imaging of normal and malignant human breast tissues,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 230–231 (1998).

Amitai, Y.

I. Shariv, Y. Amitai, A. A. Friesem, “Compact holographic beam expander,” Opt. Lett. 18, 1268–1270 (1993).
[CrossRef] [PubMed]

A. A. Friesem, Y. Amitai, “Planar diffractive elements for compact optics,” in A. ConsortiniTrends in Optics (Academic, San Diego, 1996), pp. 125–144.
[CrossRef]

Baker, K. M.

Wu Jiang, D. L. Shealy, K. M. Baker, “Physical optics analysis of the performance of a holographic projection system,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. LeeProc. SPIE2404, 227–234 (1995).
[CrossRef]

Bea, M.

M. Bea, A. Giesen, M. Huonker, H. Hugel, “Flexible beam expanders with adaptive optics—a challenge for modern beam delivery,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 84–95 (1995).
[CrossRef]

Borisov, M. V.

S. B. Odinokov, M. V. Borisov, “Optical system of the device for making a hologram matrix,” in Diffraction and Holographic Technologies, Systems, and Spatial Light Modulators, I. Cindrich, S. H. Lee, R. L. SutherlandProc. SPIE3633, 279–284 (1999).
[CrossRef]

Chapman, M.

M. Chapman, N. R. Heckenberg, “Use of a beam expanding telescope in grating-tuned waveguide CO2 laser,” Int. J. Infrared Millim. Waves. 8, 783–791 (1987).
[CrossRef]

Christensen, F. E.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Das, B. B.

S. K. Gayen, M. E. Zevallos, B. B. Das, R. R. Alfano, “Time-sliced, two-dimensional, near-infrared imaging of normal and malignant human breast tissues,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 230–231 (1998).

Frederiksen, P.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Friesem, A. A.

I. Shariv, Y. Amitai, A. A. Friesem, “Compact holographic beam expander,” Opt. Lett. 18, 1268–1270 (1993).
[CrossRef] [PubMed]

A. A. Friesem, Y. Amitai, “Planar diffractive elements for compact optics,” in A. ConsortiniTrends in Optics (Academic, San Diego, 1996), pp. 125–144.
[CrossRef]

Gayen, S. K.

S. K. Gayen, M. E. Zevallos, B. B. Das, R. R. Alfano, “Time-sliced, two-dimensional, near-infrared imaging of normal and malignant human breast tissues,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 230–231 (1998).

Giesen, A.

M. Bea, A. Giesen, M. Huonker, H. Hugel, “Flexible beam expanders with adaptive optics—a challenge for modern beam delivery,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 84–95 (1995).
[CrossRef]

Grundsoe, P.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Grunwaldt, L.

R. Neubert, L. Grunwaldt, G. Sesselmann, M. Steinbach, “An innovative telescope system for SLR,” in Laser Radar Ranging and Atmospheric Lidar Techniques II (Europto), U. Schreiber, C. Werner, eds., Proc. SPIE3865, 83–89 (1999).
[CrossRef]

Hall, C.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Hariharan, P.

Heckenberg, N. R.

M. Chapman, N. R. Heckenberg, “Use of a beam expanding telescope in grating-tuned waveguide CO2 laser,” Int. J. Infrared Millim. Waves. 8, 783–791 (1987).
[CrossRef]

Horntrup, A.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Huang, Y. T.

R. K. Kustuk, M. Kato, Y. T. Huang, “Substrate mode holograms for optical interconnects,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 168–171.

Hugel, H.

M. Bea, A. Giesen, M. Huonker, H. Hugel, “Flexible beam expanders with adaptive optics—a challenge for modern beam delivery,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 84–95 (1995).
[CrossRef]

Huonker, M.

M. Bea, A. Giesen, M. Huonker, H. Hugel, “Flexible beam expanders with adaptive optics—a challenge for modern beam delivery,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 84–95 (1995).
[CrossRef]

Hussy, C. D.

F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
[CrossRef]

Jacobsen, E.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Jiang, Wu

Wu Jiang, D. L. Shealy, K. M. Baker, “Physical optics analysis of the performance of a holographic projection system,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. LeeProc. SPIE2404, 227–234 (1995).
[CrossRef]

Kato, M.

R. K. Kustuk, M. Kato, Y. T. Huang, “Substrate mode holograms for optical interconnects,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 168–171.

Kustuk, R. K.

R. K. Kustuk, M. Kato, Y. T. Huang, “Substrate mode holograms for optical interconnects,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 168–171.

Lawrence, G. N.

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

Lewis, R.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Martinez, F.

F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
[CrossRef]

Neubert, R.

R. Neubert, L. Grunwaldt, G. Sesselmann, M. Steinbach, “An innovative telescope system for SLR,” in Laser Radar Ranging and Atmospheric Lidar Techniques II (Europto), U. Schreiber, C. Werner, eds., Proc. SPIE3865, 83–89 (1999).
[CrossRef]

Neuman, V.

V. Neuman, C. W. Pitt, L. M. Walpita, “Guided-wave holographic grating beam expander–fabrication and performance,” Electron. Lett. 17, 165–167 (1981).
[CrossRef]

Nilsson, C.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Odinokov, S. B.

S. B. Odinokov, M. V. Borisov, “Optical system of the device for making a hologram matrix,” in Diffraction and Holographic Technologies, Systems, and Spatial Light Modulators, I. Cindrich, S. H. Lee, R. L. SutherlandProc. SPIE3633, 279–284 (1999).
[CrossRef]

Ohntoshi, T.

M. Takahashi, T. Ohntoshi, “Finite-difference time-domain analysis of laser diodes integrated with tapered beam-expanders,” IEEE Photon Tech. Lett. 11, 524–526 (1999).
[CrossRef]

Orup, P.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Payne, F. P.

F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
[CrossRef]

Pitt, C. W.

V. Neuman, C. W. Pitt, L. M. Walpita, “Guided-wave holographic grating beam expander–fabrication and performance,” Electron. Lett. 17, 165–167 (1981).
[CrossRef]

Schnopper, H. W.

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Scholl, M. S.

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

Sesselmann, G.

R. Neubert, L. Grunwaldt, G. Sesselmann, M. Steinbach, “An innovative telescope system for SLR,” in Laser Radar Ranging and Atmospheric Lidar Techniques II (Europto), U. Schreiber, C. Werner, eds., Proc. SPIE3865, 83–89 (1999).
[CrossRef]

Shariv, I.

Shealy, D. L.

Wu Jiang, D. L. Shealy, K. M. Baker, “Physical optics analysis of the performance of a holographic projection system,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. LeeProc. SPIE2404, 227–234 (1995).
[CrossRef]

Steinbach, M.

R. Neubert, L. Grunwaldt, G. Sesselmann, M. Steinbach, “An innovative telescope system for SLR,” in Laser Radar Ranging and Atmospheric Lidar Techniques II (Europto), U. Schreiber, C. Werner, eds., Proc. SPIE3865, 83–89 (1999).
[CrossRef]

Takahashi, M.

M. Takahashi, T. Ohntoshi, “Finite-difference time-domain analysis of laser diodes integrated with tapered beam-expanders,” IEEE Photon Tech. Lett. 11, 524–526 (1999).
[CrossRef]

Walpita, L. M.

V. Neuman, C. W. Pitt, L. M. Walpita, “Guided-wave holographic grating beam expander–fabrication and performance,” Electron. Lett. 17, 165–167 (1981).
[CrossRef]

Wylangowski, G.

F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
[CrossRef]

Zevallos, M. E.

S. K. Gayen, M. E. Zevallos, B. B. Das, R. R. Alfano, “Time-sliced, two-dimensional, near-infrared imaging of normal and malignant human breast tissues,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 230–231 (1998).

Appl. Opt. (1)

Electron. Lett. (2)

F. Martinez, G. Wylangowski, C. D. Hussy, F. P. Payne, “Practical single-mode fibre-horn beam expander,” Electron. Lett. 24, 14–16 (1988).
[CrossRef]

V. Neuman, C. W. Pitt, L. M. Walpita, “Guided-wave holographic grating beam expander–fabrication and performance,” Electron. Lett. 17, 165–167 (1981).
[CrossRef]

IEEE Photon Tech. Lett. (1)

M. Takahashi, T. Ohntoshi, “Finite-difference time-domain analysis of laser diodes integrated with tapered beam-expanders,” IEEE Photon Tech. Lett. 11, 524–526 (1999).
[CrossRef]

Int. J. Infrared Millim. Waves. (1)

M. Chapman, N. R. Heckenberg, “Use of a beam expanding telescope in grating-tuned waveguide CO2 laser,” Int. J. Infrared Millim. Waves. 8, 783–791 (1987).
[CrossRef]

Opt. Eng. (1)

M. S. Scholl, G. N. Lawrence, “Diffraction modeling of a space relay experiment,” Opt. Eng. 29, 271–278 (1990).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

F. E. Christensen, A. Horntrup, P. Frederiksen, C. Nilsson, P. Grundsoe, P. Orup, E. Jacobsen, H. W. Schnopper, R. Lewis, C. Hall, “A beam expander facility for studying x-ray optics,” Rev. Sci. Instrum. 63, 1168–1171 (1992).
[CrossRef]

Other (7)

Wu Jiang, D. L. Shealy, K. M. Baker, “Physical optics analysis of the performance of a holographic projection system,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. LeeProc. SPIE2404, 227–234 (1995).
[CrossRef]

S. B. Odinokov, M. V. Borisov, “Optical system of the device for making a hologram matrix,” in Diffraction and Holographic Technologies, Systems, and Spatial Light Modulators, I. Cindrich, S. H. Lee, R. L. SutherlandProc. SPIE3633, 279–284 (1999).
[CrossRef]

M. Bea, A. Giesen, M. Huonker, H. Hugel, “Flexible beam expanders with adaptive optics—a challenge for modern beam delivery,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 84–95 (1995).
[CrossRef]

S. K. Gayen, M. E. Zevallos, B. B. Das, R. R. Alfano, “Time-sliced, two-dimensional, near-infrared imaging of normal and malignant human breast tissues,” Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 230–231 (1998).

R. Neubert, L. Grunwaldt, G. Sesselmann, M. Steinbach, “An innovative telescope system for SLR,” in Laser Radar Ranging and Atmospheric Lidar Techniques II (Europto), U. Schreiber, C. Werner, eds., Proc. SPIE3865, 83–89 (1999).
[CrossRef]

R. K. Kustuk, M. Kato, Y. T. Huang, “Substrate mode holograms for optical interconnects,” in Optical Computing, Vol. 9 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 168–171.

A. A. Friesem, Y. Amitai, “Planar diffractive elements for compact optics,” in A. ConsortiniTrends in Optics (Academic, San Diego, 1996), pp. 125–144.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Planar configuration of a beam expander in one dimension.

Fig. 2
Fig. 2

Required diffraction efficiency as a function of lateral spatial coordinate on the second DOE; a.u., arbitrary units.

Fig. 3
Fig. 3

Top view of the planar configuration for expanding a narrow beam in two dimensions.

Fig. 4
Fig. 4

Light distribution as a function of the distance along the ξ axis of H2. a.u., arbitrary units.

Fig. 5
Fig. 5

Diffraction efficiency as a function of exposure energy for Shipley’s 1805 photoresist.

Fig. 6
Fig. 6

Recording through a locally varying mask.

Fig. 7
Fig. 7

Output light intensity as a function of location on the second grating. a.u., arbitrary units.

Fig. 8
Fig. 8

Diffraction efficiency of the zero order as a function of wavelength.

Fig. 9
Fig. 9

Diffraction efficiency of the zero order as a function of angle.

Fig. 10
Fig. 10

Output light intensity as a function of location on the third grating for constant diffraction efficiency; a.u., arbitrary units.

Fig. 11
Fig. 11

Output light intensity as a function of location on the third grating for varying diffraction efficiency; a.u., arbitrary units.

Fig. 12
Fig. 12

Images of the output of the beam expander with a narrow input.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

μm=μ11-m-11+αμ1,
Φ1=-2πλv sin βξ,
Φ2=-Φ1+2πλv sin βη=2πλv sin βξ+v sin βη= 2πλ22 v sin β22 ξ+22 η+2πλ22 v sin β22 ξ+22 η.
Φ3=-2πλv sin βη.
Φ1+Φ2+Φ3=0.

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