Abstract

A phase-step in a phase mask is not copied into the substrate but is split into two half-amplitude phase-shifts in the near-field because of the presence of an additional interference fringe system of the two beams diffracted from the two grating sections separated by the phase-step. In the case of multiple phase-shifts, the split phase-shifts from two adjacent phase-steps can crossover in the propagation without interfere. This paper contributes to understanding the near-field diffraction of irregular phase gratings with multiple phase-shifts, and provides a theoretical base for designing multiple phase-shifted phase masks for high channel-count phase-only sampled fiber Bragg gratings [1,2].

© 2005 Optical Society of America

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References

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  1. Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
    [CrossRef]
  2. J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).
  3. P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. 115, 327–334 (1995)
    [CrossRef]
  4. Z. S. Hegedus, “Contact printing of Bragg gratings in optical fibers: rigorous diffraction analysis,” Appl. Opt. 36, 247–252 (1997)
    [CrossRef] [PubMed]
  5. J. A. R. Williams et al. “The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near-field holography,” ECOC 97, 187–190 (1997)
  6. Y. Qiu, Y. Sheng, and C. Beaulieu, “Optimal phase masks for fiber Bragg grating fabrication,” J. Lightwave Technol. 17, 2366–2370 (1999).
    [CrossRef]
  7. J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
    [CrossRef]
  8. N. M. Dragomir, C. Rollinson, S. Wade, A. J. Stevenson, S. F. Collins, and G. W. Baxter, “Nondestructive imaging of a type I optical fiber Bragg grating,” Opt. Lett. 28, 789–791 (2003)
    [CrossRef] [PubMed]
  9. Y. Sheng, Y. Qiu, and J. Wang , “Diffraction of phase mask with stitching errors in fabrication of fiber Bragg gratings”, Opt. Eng., Special section on Diffractive Optics 43, 2570–2574, (2004)
  10. R. Kashyyap, “Fiber Bragg gratings” Chap. 6.1 (Academic, San Diego, 1999)
  11. V. Jayaraman, Z. Chuang, and L. Coldren, “Theory, Design, and Performance of Extended tuning Range Semiconductor Lasers with Sampled Gratings” IEEE J. Quantum Electron. 29, 1824–1834 (1993)
    [CrossRef]
  12. J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
    [CrossRef]
  13. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High Channel Counts Chromatic Dispersion Compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
    [CrossRef]
  14. Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).
  15. L. Poladian, B. Ashton, and W. Padden, “Interactive design and fabrication of complex FBGs,” OFC paper WL1, Tech. Digest vol.1, 378–79 (2003)
  16. B. J. Lin, “Electromagnetic Near-Field Diffraction of a medium Slit,” J. Opt. Soc. Am. 62, 976–981 (1972)
    [CrossRef]

2004 (2)

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

Y. Sheng, Y. Qiu, and J. Wang , “Diffraction of phase mask with stitching errors in fabrication of fiber Bragg gratings”, Opt. Eng., Special section on Diffractive Optics 43, 2570–2574, (2004)

2003 (3)

2002 (1)

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

2000 (1)

J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
[CrossRef]

1999 (2)

1997 (2)

Z. S. Hegedus, “Contact printing of Bragg gratings in optical fibers: rigorous diffraction analysis,” Appl. Opt. 36, 247–252 (1997)
[CrossRef] [PubMed]

J. A. R. Williams et al. “The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near-field holography,” ECOC 97, 187–190 (1997)

1995 (1)

P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. 115, 327–334 (1995)
[CrossRef]

1993 (1)

V. Jayaraman, Z. Chuang, and L. Coldren, “Theory, Design, and Performance of Extended tuning Range Semiconductor Lasers with Sampled Gratings” IEEE J. Quantum Electron. 29, 1824–1834 (1993)
[CrossRef]

1972 (1)

Ashton, B.

L. Poladian, B. Ashton, and W. Padden, “Interactive design and fabrication of complex FBGs,” OFC paper WL1, Tech. Digest vol.1, 378–79 (2003)

Baxter, G. W.

Beaulieu, C.

Blott, B. H.

J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
[CrossRef]

Brocklesby, W. S.

J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
[CrossRef]

Chuang, Z.

V. Jayaraman, Z. Chuang, and L. Coldren, “Theory, Design, and Performance of Extended tuning Range Semiconductor Lasers with Sampled Gratings” IEEE J. Quantum Electron. 29, 1824–1834 (1993)
[CrossRef]

Coldren, L.

V. Jayaraman, Z. Chuang, and L. Coldren, “Theory, Design, and Performance of Extended tuning Range Semiconductor Lasers with Sampled Gratings” IEEE J. Quantum Electron. 29, 1824–1834 (1993)
[CrossRef]

Collins, S. F.

Dragomir, N. M.

Dyer, P. E.

P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. 115, 327–334 (1995)
[CrossRef]

Farley, R. J.

P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. 115, 327–334 (1995)
[CrossRef]

Giedl, R.

P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. 115, 327–334 (1995)
[CrossRef]

Hegedus, Z. S.

Hillman, C. W. J.

J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
[CrossRef]

Jayaraman, V.

V. Jayaraman, Z. Chuang, and L. Coldren, “Theory, Design, and Performance of Extended tuning Range Semiconductor Lasers with Sampled Gratings” IEEE J. Quantum Electron. 29, 1824–1834 (1993)
[CrossRef]

Kashyyap, R.

R. Kashyyap, “Fiber Bragg gratings” Chap. 6.1 (Academic, San Diego, 1999)

Li, H.

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High Channel Counts Chromatic Dispersion Compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
[CrossRef]

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).

Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).

Li, Y.

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High Channel Counts Chromatic Dispersion Compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
[CrossRef]

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Lin, B. J.

Mills, J. D.

J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
[CrossRef]

Padden, W.

L. Poladian, B. Ashton, and W. Padden, “Interactive design and fabrication of complex FBGs,” OFC paper WL1, Tech. Digest vol.1, 378–79 (2003)

Poladian, L.

L. Poladian, B. Ashton, and W. Padden, “Interactive design and fabrication of complex FBGs,” OFC paper WL1, Tech. Digest vol.1, 378–79 (2003)

Popelek, J.

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Qiu, Y.

Y. Sheng, Y. Qiu, and J. Wang , “Diffraction of phase mask with stitching errors in fabrication of fiber Bragg gratings”, Opt. Eng., Special section on Diffractive Optics 43, 2570–2574, (2004)

Y. Qiu, Y. Sheng, and C. Beaulieu, “Optimal phase masks for fiber Bragg grating fabrication,” J. Lightwave Technol. 17, 2366–2370 (1999).
[CrossRef]

Rollinson, C.

Rothenberg, J. E

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

Rothenberg, J. E.

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High Channel Counts Chromatic Dispersion Compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
[CrossRef]

Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).

J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).

Rothenberg, J.E.

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Sheng, Y.

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

Y. Sheng, Y. Qiu, and J. Wang , “Diffraction of phase mask with stitching errors in fabrication of fiber Bragg gratings”, Opt. Eng., Special section on Diffractive Optics 43, 2570–2574, (2004)

H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High Channel Counts Chromatic Dispersion Compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
[CrossRef]

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Y. Qiu, Y. Sheng, and C. Beaulieu, “Optimal phase masks for fiber Bragg grating fabrication,” J. Lightwave Technol. 17, 2366–2370 (1999).
[CrossRef]

J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).

Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).

Stevenson, A. J.

Wade, S.

Wang, J.

Y. Sheng, Y. Qiu, and J. Wang , “Diffraction of phase mask with stitching errors in fabrication of fiber Bragg gratings”, Opt. Eng., Special section on Diffractive Optics 43, 2570–2574, (2004)

Wang, Y.

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Wilcox, R. B.

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Williams, J. A. R.

J. A. R. Williams et al. “The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near-field holography,” ECOC 97, 187–190 (1997)

Ying, W.

J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).

Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).

Zweiback, J.

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).

Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).

Appl. Opt. (2)

Z. S. Hegedus, “Contact printing of Bragg gratings in optical fibers: rigorous diffraction analysis,” Appl. Opt. 36, 247–252 (1997)
[CrossRef] [PubMed]

J. D. Mills, C. W. J. Hillman, B. H. Blott, and W. S. Brocklesby, “Imaging of free-space interference patterns used to manufacture fiber Bragg gratings,” Appl. Opt. 39, 6129–6135 (2000)
[CrossRef]

ECOC (1)

J. A. R. Williams et al. “The effects of phase steps in e-beam written phase masks used for fiber grating fabrication by near-field holography,” ECOC 97, 187–190 (1997)

IEEE J. Quantum Electron. (1)

V. Jayaraman, Z. Chuang, and L. Coldren, “Theory, Design, and Performance of Extended tuning Range Semiconductor Lasers with Sampled Gratings” IEEE J. Quantum Electron. 29, 1824–1834 (1993)
[CrossRef]

IEEE Photonics Technol. Lett. (2)

J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann fiber Bragg gratings and phase-only sampling for high channel counts,” IEEE Photonics Technol. Lett. 14, 1309–1311, (2002).
[CrossRef]

Y. Sheng, J. E Rothenberg, H. Li, Y. Wang, and J. Zweiback, “Split of phase-shift in a phase mask for fiber Bragg gratings,” IEEE Photonics Technol. Lett. 16, 1316–1318 (2004)
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (1)

OFC paper WL1, Tech. Digest (1)

L. Poladian, B. Ashton, and W. Padden, “Interactive design and fabrication of complex FBGs,” OFC paper WL1, Tech. Digest vol.1, 378–79 (2003)

Opt. Commun. (1)

P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. 115, 327–334 (1995)
[CrossRef]

Opt. Eng., Special section on Diffractive Optics (1)

Y. Sheng, Y. Qiu, and J. Wang , “Diffraction of phase mask with stitching errors in fabrication of fiber Bragg gratings”, Opt. Eng., Special section on Diffractive Optics 43, 2570–2574, (2004)

Opt. Lett. (1)

Other (3)

R. Kashyyap, “Fiber Bragg gratings” Chap. 6.1 (Academic, San Diego, 1999)

J. E. Rothenberg, Y. Sheng, H. Li, W. Ying, and J. Zweiback, “Diffraction compensation of masks for high channel-count phase-only sampled fiber Bragg gratings,” OSA Topical meeting on Bragg gratings, Photosensitivity and Poling in Glass Waveguides, Post deadline paper, PDP-2, September (2003).

Y. Sheng, J. E. Rothenberg, H. Li, W. Ying, and J. Zweiback, “Phase mask design and phase mask for writing optical fiber Bragg gratings,” International Patent PCT, WO 03/062880 (2003).

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Figures (3)

Fig. 1.
Fig. 1.

Near-field diffraction of a phase mask with one phase step δ ; (a) regions of superposition of the 4 diffracted beams; (b) a small central part of the calculated interference pattern of the 4 beams, illustrating the phase shift split.

Fig. 2.
Fig. 2.

Near-field diffraction of a phase mask with two phase shifts; (a): regions of superposition of the 6 diffracted beams; (b): part of the interference pattern calculated by superposing the beam 1–6, described in the text, in the respective regions.

Fig. 3.
Fig. 3.

Near-field intensity distribution of a phase mask with one phase-shift.; Left-Top: from 0 to 5 μm computed by the FDTD; Left-Bottom from 5 to 15 μm computed by Fourier free space propagation filter; Right-Top: Plot of the amplitude in arbitrary scale of the near field at y=15 μm ; Right-Bottom: Plot of the fringe periods at y=15 μm .

Equations (3)

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

e j 2 πy cos θ λ [ e j 2 π ( x x 01 ) Λ + e j 2 π ( x x 02 ) Λ ] =
= 2 e j 2 πy cos θ λ e j 2 π ( x 02 x 01 ) 2 Λ cos ( 2 π Λ ( x x 01 δ 2 ) )
1 + cos ( 2 π ( x x 01 γ + δ 2 ) ( Λ 2 )

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