Abstract

The theory of Fresnel diffraction images is applied to Bragg-grating formation in a germanium-doped silica fiber. Fresnel diffraction images arise from the near-field diffraction at a periodic mask. The diffraction images are calculated as a function of the propagation distance for several mask configurations. The average of the diffraction-image intensities is calculated for a single longitudinal repetitive interval, and it is shown that the period of the resulting average intensity field is twice that of the original mask period. In some cases the periodic mask can be predicted for a desired average intensity field by calculation of the magnitude of its Fourier coefficients.

© 1994 Optical Society of America

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References

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  1. K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
    [CrossRef]
  2. K. D. Simmons, S. LaRochelle, V. Mizrahi, G. I. Stegeman, D. L. Griscom, “Correlation of defect centers with a wavelength-dependent photosensitive response in germania-doped silica optical fibers,” Opt. Lett. 16, 141–143 (1991).
    [CrossRef] [PubMed]
  3. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  4. R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
    [CrossRef]
  5. E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
    [CrossRef]
  6. K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  7. J. T. Winthrop, C. R. Worthington, “Convolution formulation of Fresnel diffraction,” J. Opt. Soc. Am. 56, 588–591 (1966).
    [CrossRef]
  8. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  10. G. Reynolds, J. DeVelis, G. Parrent, B. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
    [CrossRef]
  11. G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
    [CrossRef]
  12. K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.
  13. D. Anderson, V. Mizrahi, T. Erodan, A. White, “Phase mask methods for volume manufacturing of fiber phase gratings,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD16-1, pp. 68–70.

1993 (1)

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

1991 (2)

K. D. Simmons, S. LaRochelle, V. Mizrahi, G. I. Stegeman, D. L. Griscom, “Correlation of defect centers with a wavelength-dependent photosensitive response in germania-doped silica optical fibers,” Opt. Lett. 16, 141–143 (1991).
[CrossRef] [PubMed]

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

1990 (2)

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

1989 (1)

1978 (1)

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

1966 (1)

1965 (1)

Abe, M.

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Albert, J.

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Anderson, D.

D. Anderson, V. Mizrahi, T. Erodan, A. White, “Phase mask methods for volume manufacturing of fiber phase gratings,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD16-1, pp. 68–70.

Armitage, J. R.

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

Bayon, F.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Bernage, P.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Bilodeau, F.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Canon, S.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Champion, G.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Davey, S. T.

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

DeVelis, J.

G. Reynolds, J. DeVelis, G. Parrent, B. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Douay, M.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Erodan, T.

D. Anderson, V. Mizrahi, T. Erodan, A. White, “Phase mask methods for volume manufacturing of fiber phase gratings,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD16-1, pp. 68–70.

Fallahi, M.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Fertein, E.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Fujii, Y.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Georges, T.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Glenn, W. H.

Griscom, D. L.

Hibino, Y.

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Hill, K. O.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Johnson, D. C.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Kashyap, R.

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

Kawachi, M.

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Kawasaki, B. S.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

LaRochelle, S.

Legoubin, S.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Malo, B.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

Maritan, C.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Meltz, G.

Mizrahi, V.

K. D. Simmons, S. LaRochelle, V. Mizrahi, G. I. Stegeman, D. L. Griscom, “Correlation of defect centers with a wavelength-dependent photosensitive response in germania-doped silica optical fibers,” Opt. Lett. 16, 141–143 (1991).
[CrossRef] [PubMed]

D. Anderson, V. Mizrahi, T. Erodan, A. White, “Phase mask methods for volume manufacturing of fiber phase gratings,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD16-1, pp. 68–70.

Moore, R.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Morey, W. W.

Niay, P.

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

Pakulski, G.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Parrent, G.

G. Reynolds, J. DeVelis, G. Parrent, B. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Reynolds, G.

G. Reynolds, J. DeVelis, G. Parrent, B. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Shepherd, F.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Simmons, K. D.

Skinner, I.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Stegeman, G. I.

Templeton, I.

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Thompson, B.

G. Reynolds, J. DeVelis, G. Parrent, B. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Vineberg, K. A.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

White, A.

D. Anderson, V. Mizrahi, T. Erodan, A. White, “Phase mask methods for volume manufacturing of fiber phase gratings,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD16-1, pp. 68–70.

Williams, D. L.

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

Winthrop, J. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Worthington, C. R.

Wyatt, R.

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

Appl. Phys. Lett. (2)

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

G. Pakulski, R. Moore, C. Maritan, F. Shepherd, M. Fallahi, I. Templeton, G. Champion, “Fused silica masks for printing uniform and phase adjusted gratings for distributed feedback lasers,” Appl. Phys. Lett. 62, 222–224 (1993).
[CrossRef]

Electron. Lett. (3)

R. Kashyap, J. R. Armitage, R. Wyatt, S. T. Davey, D. L. Williams, “All fibre narrow band reflection grating at 1500 nm,” Electron. Lett. 26, 730–732 (1990).
[CrossRef]

E. Fertein, S. Legoubin, M. Douay, S. Canon, P. Bernage, P. Niay, F. Bayon, T. Georges, “Shifts in resonance wavelengths of a Bragg grating during writing or bleaching experiments by UV illumination within a germanosilicate optical fibre,” Electron. Lett. 27, 1838–1839 (1991).
[CrossRef]

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Other (4)

K. O. Hill, F. Bilodeau, B. Malo, J. Albert, D. C. Johnson, Y. Hibino, M. Abe, M. Kawachi, “Application of phase masks to the photolithographic fabrication of Bragg gratings in conventional fiber/planar waveguides with enhanced photosensitivity,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD15-1, pp. 64–68.

D. Anderson, V. Mizrahi, T. Erodan, A. White, “Phase mask methods for volume manufacturing of fiber phase gratings,” in Conference on Optical Fiber Communication/International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper PD16-1, pp. 68–70.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

G. Reynolds, J. DeVelis, G. Parrent, B. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Physical arrangement for Fresnel-image formation.

Fig. 2
Fig. 2

Fresnel images of a 25% transparent mask at β planes: (a) β = 0.00, (b) β = 0.10, (c) β = 0.20, (d) β = 0.25, (e) β = 0.30, (f) β = 0.3333333, (g) β = 0.40, (h) β = 0.50.

Fig. 3
Fig. 3

Fresnel images of a 50% transparent mask: (a) β = 0.00, (b) β = 0.10, (c) β = 0.20, (d) β = 0.25, (e) β = 0.30, (f) β = 0.3333333, (g) β = 0.40, (h) β = 0.50.

Fig. 4
Fig. 4

Fresnel images of a π phase mask at β planes: (a) β = 0.00, (b) β = 0.10, (c) β = 0.20, (d) β = 0.25, (e) β = 0.30, (f) β = 0.3333333, (g) β = 0.40, (h) β = 0.50.

Fig. 5
Fig. 5

Average intensity of a 25% transparent mask.

Fig. 6
Fig. 6

Average intensity of a 50% transparent mask.

Fig. 7
Fig. 7

Average intensity of a π phase shift mask.

Equations (18)

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

g ( x , z ) = s ( z ) * t ( z ) * f ( x , z ) .
s ( z ) = δ ( z ) .
f ( x , z ) = ( i λ x ) 1 / 2 exp ( i π z 2 n λ x ) .
g ( x , z ) = t ( z ) * f ( x , z ) .
t ( z ) = t c ( z ) * a ( z ) .
g ( x , z ) = t ( z ) * a ( z ) f ( x , z ) .
G ( u , x ) = + g ( z , x ) exp ( i 2 π u z ) d z ,
g ( z , x ) = + G ( u , x ) exp ( + i 2 π u z ) d u ,
G ( u , x ) = T c ( u ) A ( u ) F ( u , x ) ,
g ( x , z ) = + T c ( u ) h = δ ( u h a ) × exp ( i πλ x u 2 n ) exp ( i 2 π u z ) d u .
g ( x , z ) = 1 a h = T c ( h ) exp ( i πλ x h 2 n a 2 ) exp ( i 2 π z h a ) .
g ( x , z ) = 1 a h = ( T h + i T h ) exp ( i πλ x h 2 n a 2 ) exp ( i 2 π z h a ) .
g real ( x , z ) = 2 a h = 1 T h cos ( 2 π h z a ) cos ( 2 πλ x h 2 2 a 2 n ) + 2 a h = 1 T h sin ( 2 πλ x h 2 2 a 2 n ) cos ( 2 π h z a ) + 1 a T 0 ,
g imag ( x , z ) = 2 a h = 1 T h cos ( 2 π h z a ) sin ( 2 πλ x h 2 2 a 2 n ) + 2 a h = 1 T h cos ( 2 π h z a ) cos ( 2 πλ x h 2 2 a 2 n ) + 1 a T 0 .
| g ( z ) | 2 = λ 2 a 2 n 0 2 a 2 n / λ | g ( x , z ) | 2 d x .
| g ( z ) | 2 = 1 a 2 | T 0 | 2 + 4 a 2 h = 1 | T h | 2 cos 2 ( 2 π h z a ) .
cos 2 θ = 1 2 ( 1 cos 2 θ ) ,
| T h | 2 = h π a 2 a / 2 a / 2 | g ( z ) | 2 cos ( 4 π h z a ) d z , | T 0 | 2 = h π a a / 2 a / 2 | g ( z ) | 2 d z 2 h = 1 | T h | 2 .

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