Abstract

Reconfigurable optical interconnects constructed by recording dynamic holograms onto spatial light modulators may be crucial elements in all-optical networks. The extremely low cross-talk level of such free-space holographic switches was shown by an analytic approximation and verified experimentally. The fiber-to-fiber switch utilizes the spatial filtering properties of single-mode fibers, and its cross-talk noise is limited to the sidelobe power as a result of diffraction of the clipped Gaussian beam at the hologram aperture edges, provided that all higher orders are avoided. Greater than 45-dB cross-talk isolation has been measured at transverse-axis locations, and locating a fiber port at off-transverse-axis directions promises to double this level if aberrations are negligible.

© 2001 Optical Society of America

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References

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  1. J. A. Neff, R. A. Athale, S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
    [CrossRef]
  2. D. C. O’Brien, R. J. Mears, “Computer generated holograms optimized for illumination with partially coherent light using a silicon backplane spatial light modulator as the recording device,” in Optics for Computers: Architectures and Technologies , G. J. Lebreton, ed., Proc. SPIE1505, 32–37 (1991).
    [CrossRef]
  3. D. C. O’Brien, R. J. Mears, T. D. Wilkinson, W. A. Crossland, “Dynamic holographic interconnects that use ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).
    [CrossRef] [PubMed]
  4. S. T. Warr, M. C. Parker, R. J. Mears, “Optically transparent digitally tunable wavelength filter,” Electron. Lett. 31, 129–130 (1995).
    [CrossRef]
  5. H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).
  6. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  7. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  8. K. L. Tan, S. T. Warr, I. G. Manolis, T. D. Wilkinson, M. M. Redmond, W. A. Crossland, R. J. Mears, B. Robertson, “Dynamic holography for optical interconnections. II. Routing holograms with predictable replay location and in-tensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001).
    [CrossRef]
  9. M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.
  10. E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
    [CrossRef]
  11. J. A. Davis, S. W. Connely, G. W. Bach, R. A. Lilly, D. M. Cottrell, “Programmable optical interconnections with large fan-out capability using a magneto-optic spatial light modulator,” Opt. Lett. 14, 102–104 (1989).
    [CrossRef] [PubMed]
  12. I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
    [CrossRef]
  13. A. Yariv, Optical Electronics, 4th ed. (Saunders, London, 1991), pp. 705–707.
  14. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, London, 1994), p. 938.
  15. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 177–184.
  16. W. B. Joyce, B. C. Deloach, “Alignment of Gaussian beams,” Appl. Opt. 23, 4187–4196 (1984).
    [CrossRef] [PubMed]
  17. K. L. Tan, W. A. Crossland, R. J. Mears, “A comparison of the efficiency and crosstalk of quaternary and binary phase-only holograms based on ferroelectric liquid crystals (FLC),” Ferroelectrics 213, 233–240 (1998).
    [CrossRef]
  18. A. W. Snyder, R. A. Sammut, “Fundamental modes of graded optical fibers,” J. Opt. Soc. Am. 69, 1663–1671 (1979).
    [CrossRef]

2001 (1)

1998 (1)

K. L. Tan, W. A. Crossland, R. J. Mears, “A comparison of the efficiency and crosstalk of quaternary and binary phase-only holograms based on ferroelectric liquid crystals (FLC),” Ferroelectrics 213, 233–240 (1998).
[CrossRef]

1996 (1)

E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
[CrossRef]

1995 (1)

S. T. Warr, M. C. Parker, R. J. Mears, “Optically transparent digitally tunable wavelength filter,” Electron. Lett. 31, 129–130 (1995).
[CrossRef]

1994 (1)

1990 (1)

J. A. Neff, R. A. Athale, S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

1989 (1)

1984 (1)

1979 (1)

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

1971 (1)

1970 (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).

Al Chalabi, A.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 177–184.

Athale, R. A.

J. A. Neff, R. A. Athale, S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Bach, G. W.

Bayvel, P.

E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
[CrossRef]

Birch, M. J.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Bradford, G.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Churin, E. G.

E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
[CrossRef]

Connely, S. W.

Cottrell, D. M.

Crossland, W. A.

K. L. Tan, S. T. Warr, I. G. Manolis, T. D. Wilkinson, M. M. Redmond, W. A. Crossland, R. J. Mears, B. Robertson, “Dynamic holography for optical interconnections. II. Routing holograms with predictable replay location and in-tensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001).
[CrossRef]

K. L. Tan, W. A. Crossland, R. J. Mears, “A comparison of the efficiency and crosstalk of quaternary and binary phase-only holograms based on ferroelectric liquid crystals (FLC),” Ferroelectrics 213, 233–240 (1998).
[CrossRef]

D. C. O’Brien, R. J. Mears, T. D. Wilkinson, W. A. Crossland, “Dynamic holographic interconnects that use ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).
[CrossRef] [PubMed]

M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Dainty, P.

M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.

Dammann, H.

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).

Davis, J. A.

Deloach, B. C.

Fancy, N. E.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Gloge, D.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, London, 1994), p. 938.

Hall, T. J.

M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.

Hill, A. M.

E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
[CrossRef]

Holmes, M. J.

M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.

Joyce, W. B.

Lantham, S. G.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Lee, S. H.

J. A. Neff, R. A. Athale, S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Lilly, R. A.

Manolis, I. G.

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Mears, R. J.

K. L. Tan, S. T. Warr, I. G. Manolis, T. D. Wilkinson, M. M. Redmond, W. A. Crossland, R. J. Mears, B. Robertson, “Dynamic holography for optical interconnections. II. Routing holograms with predictable replay location and in-tensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001).
[CrossRef]

K. L. Tan, W. A. Crossland, R. J. Mears, “A comparison of the efficiency and crosstalk of quaternary and binary phase-only holograms based on ferroelectric liquid crystals (FLC),” Ferroelectrics 213, 233–240 (1998).
[CrossRef]

S. T. Warr, M. C. Parker, R. J. Mears, “Optically transparent digitally tunable wavelength filter,” Electron. Lett. 31, 129–130 (1995).
[CrossRef]

D. C. O’Brien, R. J. Mears, T. D. Wilkinson, W. A. Crossland, “Dynamic holographic interconnects that use ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).
[CrossRef] [PubMed]

D. C. O’Brien, R. J. Mears, “Computer generated holograms optimized for illumination with partially coherent light using a silicon backplane spatial light modulator as the recording device,” in Optics for Computers: Architectures and Technologies , G. J. Lebreton, ed., Proc. SPIE1505, 32–37 (1991).
[CrossRef]

Midwinter, J. E.

E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
[CrossRef]

Neff, J. A.

J. A. Neff, R. A. Athale, S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

O’Brien, D. C.

D. C. O’Brien, R. J. Mears, T. D. Wilkinson, W. A. Crossland, “Dynamic holographic interconnects that use ferroelectric liquid-crystal spatial light modulators,” Appl. Opt. 33, 2795–2803 (1994).
[CrossRef] [PubMed]

D. C. O’Brien, R. J. Mears, “Computer generated holograms optimized for illumination with partially coherent light using a silicon backplane spatial light modulator as the recording device,” in Optics for Computers: Architectures and Technologies , G. J. Lebreton, ed., Proc. SPIE1505, 32–37 (1991).
[CrossRef]

Parker, M. C.

S. T. Warr, M. C. Parker, R. J. Mears, “Optically transparent digitally tunable wavelength filter,” Electron. Lett. 31, 129–130 (1995).
[CrossRef]

Payne, F. P.

M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.

Redmond, M. M.

Robertson, B.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, London, 1994), p. 938.

Sammut, R. A.

Sillito, R. M.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Snyder, A. W.

Sparks, A. P.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Tan, K. L.

K. L. Tan, S. T. Warr, I. G. Manolis, T. D. Wilkinson, M. M. Redmond, W. A. Crossland, R. J. Mears, B. Robertson, “Dynamic holography for optical interconnections. II. Routing holograms with predictable replay location and in-tensity of each diffraction order,” J. Opt. Soc. Am. A 18, 205–215 (2001).
[CrossRef]

K. L. Tan, W. A. Crossland, R. J. Mears, “A comparison of the efficiency and crosstalk of quaternary and binary phase-only holograms based on ferroelectric liquid crystals (FLC),” Ferroelectrics 213, 233–240 (1998).
[CrossRef]

Underwood, I.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Vass, D. G.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

Warr, S. T.

Wilkinson, T. D.

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, London, 1991), pp. 705–707.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Electron. Lett. (1)

S. T. Warr, M. C. Parker, R. J. Mears, “Optically transparent digitally tunable wavelength filter,” Electron. Lett. 31, 129–130 (1995).
[CrossRef]

Ferroelectrics (1)

K. L. Tan, W. A. Crossland, R. J. Mears, “A comparison of the efficiency and crosstalk of quaternary and binary phase-only holograms based on ferroelectric liquid crystals (FLC),” Ferroelectrics 213, 233–240 (1998).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

E. G. Churin, P. Bayvel, J. E. Midwinter, A. M. Hill, “The influence of aperture size and shape on crosstalk level in free-space grating multiplexes for WDM networks,” IEEE Photonics Technol. Lett. 8, 1337–1339 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Optik (Stuttgart) (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttgart) 31, 95–104 (1970).

Proc. IEEE (1)

J. A. Neff, R. A. Athale, S. H. Lee, “Two-dimensional spatial light modulators: a tutorial,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Other (6)

D. C. O’Brien, R. J. Mears, “Computer generated holograms optimized for illumination with partially coherent light using a silicon backplane spatial light modulator as the recording device,” in Optics for Computers: Architectures and Technologies , G. J. Lebreton, ed., Proc. SPIE1505, 32–37 (1991).
[CrossRef]

M. J. Holmes, F. P. Payne, P. Dainty, T. J. Hall, W. A. Crossland, “Low crosstalk devices for wavelength-routed networks,” in Guided Wave Optical Signal Processing, Digest 95-128 (Institution of Electrical Engineers, London, 1995), paper 2.

I. Underwood, D. G. Vass, R. M. Sillito, G. Bradford, N. E. Fancy, A. Al Chalabi, M. J. Birch, W. A. Crossland, A. P. Sparks, S. G. Lantham, “A high performance spatial light modulator ,” in Devices for Optical Processing , D. M. Gookin, ed., Proc. SPIE1562, 107–115 (1991).
[CrossRef]

A. Yariv, Optical Electronics, 4th ed. (Saunders, London, 1991), pp. 705–707.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, London, 1994), p. 938.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 177–184.

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Figures (15)

Fig. 1
Fig. 1

Free-space holographic optical architecture for a 1×N single-mode fiber-optic switch. The beam-steering function provided by the hologram has been assumed to produce a single replay peak.

Fig. 2
Fig. 2

Cross-sectional view of the beam diameters of the Gaussian input and hologram illuminations and the complex-field profile of the replay beam in a coherent 4f optical configuration. The three planes are labeled, with the Gaussian parameters and lateral axes appropriately denoted.

Fig. 3
Fig. 3

Asymptotic approximation of the replay field of a blank aperture illuminated by a collimated Gaussian beam (wr=5.06 μm for the fiber mode). The inset contains the replay field for an asymmetrically clipped beam, with a 0.1 γ lateral shift of a γ=2.5 aperture. In essence the approximate replay is the superposition of a symmetrical and real field that is due to the γ=2 subaperture and a complex field that is due to the extended 0.5 aperture.

Fig. 4
Fig. 4

Power coupling of the square-aperture replay and the Gaussian fiber mode with an angular tilt and a lateral shift. The overlap integral can be evaluated, equivalently, over the hologram aperture by use of the hologram illumination (solid curve) and the fictitious transform of the output fiber mode (dotted curve). Heavy lines, opaque region; dashed vertical lines, reference planes for the hologram, the lens, and the replay.

Fig. 5
Fig. 5

Coupling intensity into tilted fibers for several Gaussian illumination truncations. Marcuse’s expression, scaled by the on-beam-axis power for γ=2, is plotted by the dashed-dotted curve.

Fig. 6
Fig. 6

Dependence of on-beam-axis coupling intensity on the truncation ratio γ. Solid curve, 2-D symmetrical and equal truncation; dotted curve, illustration that the collimated beam is truncated in one direction only.

Fig. 7
Fig. 7

1-D coupling-intensity profile with use of the asymptotic approximation for wr=5.06 μm. The dotted curves show the further approximation for large lateral offsets, which are the likely cross-talk levels for an asymmetrically truncated hologram illumination, slight angular tilts in the output fiber, or both.

Fig. 8
Fig. 8

Approximated cross-talk level at 30-μm offset due to the replay of a single beam. Dotted line, on-transverse-axis offsets; solid curve, offsets from both transverse axes; dashed–dotted and dashed curves, coupling intensity according to the asymptotic approximation at 30 μm along one and away from both transverse axes, respectively.

Fig. 9
Fig. 9

Numerical integration of the coupling intensity into an output fiber with 1-D lateral offsets and angular tilts and assuming a symmetric γ=2 truncation and f=25 mm for γ=1.55-μm Gaussian beams.

Fig. 10
Fig. 10

Analytic coupling intensity profile along a lateral axis with 0–0.1γ asymmetric clipping of a γ=2 aperture ratio for f=25 mm, λ=1.55 μm, and wr=5.06 μm.

Fig. 11
Fig. 11

Numerical integration results for coupling with up to ±20-mrad angular tilts at 0.05γ asymmetric clipping, where γ=2, f=25 mm, γ=1.55 μm, and wr=5.06 μm. Along each coupling profile with 1-D lateral offsets, the degradation in sidelobe cross-talk isolation is caused by the relative shifts that are due to the angular tilt and asymmetric clipping.

Fig. 12
Fig. 12

Numerical convolution results (solid curves) of the coupling intensity into a SMF with the fiber mode described as Bessel functions and analytic coupling-intensity profiles (dotted curves) according to the approximate asymptotic expression (assuming a Gaussian fiber mode) in a 4f coherent optical configuration with an aperture clipping the illumination with 1-D factors γ from 1.78 to 2.32.

Fig. 13
Fig. 13

Optical setup for transmissive grating replay measurements. The replay orders are scanned by a SMF mounted on stepper-motor-controlled stages, and the intensity is obtained with a powermeter with a 70 dB dynamic range.

Fig. 14
Fig. 14

2-D coupling-intensity profile of the 0 and ±1 orders of the fixed intensity grating replay.

Fig. 15
Fig. 15

Line scans along (X axis) and orthogonal to (Y axis) the dispersion plane of -1, 0, and +1 fixed-intensity grating replay orders. Dashed–dotted curves, the approximation of the solution to the overlap integral, the dotted horizontal lines cut through the 1/e2 intensity points.

Equations (17)

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

R(xr; yr)=FG(xh, yh; wh)rectxhγwh, yhγwh×H(xh, yh),
G(xh, yh; wh)=1wh2/πexp-xh2+yh2wh2,
R(xr, yr)=141wr2πexp-xr2wr22π×-γ/2-jxr/wrγ/2-jxr/wrexp(-t2)dt×exp-yr2wr22π×-γ/2-jyr/wrγ/2-jyr/wrexp(-t2)dt.
R(xr, yr)=1wr2/πR(xr)R(yr),
R(xr)exp-xr2wr2-exp[-(γ2/4)]π(γ2/4+xr2/wr2)×γ2cos(γxr/wr)-xrwrsin(γxr/wr),
|Rc(u, v; θx, θy)|2=--R(xr, yr)G2(xr-u, yr-v; θx, θy; wr2)dxrdyr2--|R(xr, yr)|2dxrdyr--|G2(xr-u, yr-v; θx, θy; wr2)|2dxrdyr.
G2(xr-u, yr-v; θx, θy; wr2)
=1wr22πexp-(xr-u)2+(yr-v)2wr22+jk[θx(xr-u)+θy(yr-v)].
|Rc(u, v; θx, θy)|2
=-(γ/2)wh1(γ/2)wh11wh1wh22π×exp-(xh+fθx)2wh22-xh2wh12+jkuxhf×exp-(yh+fθy)2wh22-yh2wh12+jkvyhfdxhdyh2.
|Rc(0, 0; θx, θy)|2=|Rc(0, 0; θx)|2|Rc(0, 0; θy)|2,
|Rc(0, 0; θx)|2
=exp-12kθxwr22×12erfγ2+kθxwr22+erfγ2-kθxwr222.
|Rc(u, v; 0, 0)|2|Rc(u; 0, 0)|2|Rc(v; 0,0)|2,
|Rc(u, v; 0, 0)|2exp-u22wr2-exp(-γ2/2)π(γ2/2+u2/2wr2)×γ2cos(γu/wr)-u2wrsin(γu/wr)2.
|Rc(u, v; 0, 0)|22π2wr2uv2exp[-(γx2+γy2)].
|Rc(u=0, v; 0, 0)|2(2/π)(wr/v)2×[erf(γx/2)]2exp(-γy2),

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