## Near-field diffraction of irregular phase gratings with multiple phase-shifts

Optics Express, Vol. 13, Issue 16, pp. 6111-6116 (2005)

http://dx.doi.org/10.1364/OPEX.13.006111

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### 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 [

© 2005 Optical Society of America

## 1. Introduction

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]

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]

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]

*μm*from the phase mask to the fiber core for a fiber diameter of 125

*μm*. The numerical finite difference in time domain (FDTD) calculation showed the split of the phase-shift into two half-magnitude phase shifts that are propagated at the angle of the ±1 diffracted orders. The split of the phase shifts caused large roll-off errors in the multi-channel spectrum that can be corrected by new designs of the phase-shifted phase masks [1

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]

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]

*mm*for 100

*GHz*channel spacing) [1

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]

**16**, 1316–1318 (2004) [CrossRef]

## 2. Ideal diffraction near field of phase-shifted phase mask

**16**, 1316–1318 (2004) [CrossRef]

**16**, 1316–1318 (2004) [CrossRef]

*62.5 μm*, the evanescent waves vanish. Only homogeneous waves should be considered. We use the scalar diffraction theory and we neglect the diffraction at the ends of the finite length gratings. When the gap width δ >

*λ*/2 , the interaction between the two gratings separated by δ can be neglected [16

16. B. J. Lin, “Electromagnetic Near-Field Diffraction of a medium Slit,” J. Opt. Soc. Am. **62**, 976–981 (1972) [CrossRef]

*L*and

_{1}*L*. Without loss of generality, we assume

_{2}*L*and

_{1}*L*to be integer multiples of the grating period Λ . We define the phase reference points

_{2}*x*

_{01}and

*x*

_{02}that are at the same location within the grating period profile of the two gratings respectively. The values of

*x*and

_{01}*x*can change by any integer multiples of Λ. Thus the phase-step takes place from (

_{02}*x*

_{01}+

*L*

_{1}) to

*x*

_{02}with δ = (

*x*

_{02}-(

*x*

_{01}+

*L*

_{1}))

_{mod Λ}as shown in Fig. 1(a).

*y*=

*0*, every point is a source emitting ±1 diffracted orders resulting from interference of the Huygens wavelets. We write down the four diffracted beams: beam (1) exp[

*j2π*((

*x*-

*x*

_{01})/Λ

*y*cos

*θ*/

*λ*)] and beam (2) exp[

*j2π*((

*x*-

*x*

_{01})/Λ

*y*cos

*θ*/

*λ*)] are diffracted from the grating

*L*, beam (3) exp[

_{1}*j2π*((

*x*-

*x*

_{02})/Λ

*y*cos

*θ*/

*λ*)] and beam (4) exp[

*j2π*(-(

*x*-

*x*

_{02})/Λ

*y*cos

*θ*/

*λ*)] are diffracted from the grating

*L*. In the enlarged groove of width Λ/2 + δ , the wavelets emitted from the segment from (

_{2}*x*

_{02}- δ) to

*x*

_{02}should be a part of the diffracted beams of grating

*L*. The wavelets emitted from the segment from (

_{2}*x*

_{01}+

*L*

_{1}- Λ/2) to (

*x*

_{01}+

*L*

_{1}- Λ/2 + δ) should be that of grating

*L*. In the middle segment of width Λ/2 - δ of the enlarged groove, the ±1 diffracted orders are degenerate. They can belong either to grating

_{1}*L*or to grating

_{1}*L*.

_{2}*y*=

*0*, the widths of the four beams are equal to their respective grating section lengths. Then, each beam propagates at the diffracted angle ±

*θ*determined by the grating equation Λ sin

*θ*=

*λ*. Hence, we define the specific regions in the near field where the four beams are superposed respectively, as shown in Fig. 1(a). In addition to the ordinary interference fringe patterns formed by the superposition of beams (1) and (2) both diffracted from grating

*L*in region

_{1}*A*, and that of beams (3) and (4) both diffracted from grating

*L*in region

_{2}*C*, there is a triangular region

*B*, where beams (2) and (3) diffracted from the two gratings

*L*and

_{1}*L*overlap. The vertex of region B is at the location of the phase step. The boundaries of region B are at the diffraction angle ±

_{2}*θ*with respect to the

*y*-axis. There is an uncertainty of Λ/2 - δ on the location of the vertex of the region B, because of the degeneracy of the diffracted beams in the middle segment of the enlarged groove. We calculated the superposition of beams (1)–(4) in the respective regions with a phase mask period Λ = 1

*μm*and the wavelength

*λ*= 250

*nm*with a phase-step δ = 250

*nm*in the middle of the phase mask at

*x*= 10

*μm*, corresponding to a phase-shift of 2

*πδ*/(Λ/2) =

*π*in the FBG of period Λ/2 . A part of the near field intensity distribution is shown in Fig. 1(b). A shift of a quarter of the fringe period between the fringes in regions B and A, and another

*π*/2 shift between the fringes in regions C and B can be seen clearly at the boundaries of region B.

*π*(

*x*-

*x*

_{01})/(Λ/2)). In region B, the superposition of beams (2) and (3), diffracted by gratings

*L*and

_{1}*L*respectively, is

_{2}*x*

_{02}-

*x*

_{01})

_{mod Λ}and the intensity distribution is

*π*(

*x*-

*x*

_{01}- δ)/(Λ/2)). If the fiber core was in contact with the phase mask, the width of region B would be zero at

*y*=0 and the signal propagating along the FBG would experience a phase-shift 2

*πδ*/(Λ/2) when passing from region A to C, as expected for a phase-shifted FBG. However, at a given distance

*y*>0 from the phase mask, the interference of beams (2) and (3) produces an additional fringe system of width ∆

*x*= 2

*ytan*θ in region B. The two half-amplitude phase shifts

*πδ*/(Λ/2) occur at the boundaries between regions A, B and B, C, respectively.

*δ*.

*δ*=

*x*

_{02}-(

*x*

_{01}+

*L*

_{1}) and

*γ*=

*x*

_{03}-(

*x*

_{01}+

*L*

_{1}+

*δ*+

*L*

_{2}) =

*x*

_{03}-(

*x*

_{02}+

*L*

_{2}) are separated by

*L*

_{2}, where

*x*

_{03}is the phase reference point in the grating section

*L*

_{3}, as shown in Fig. 2(a). The beams (1)–(4) are described in the same manner as previously. We define two new beams diffracted by the grating

*L*

_{3}, as beam (5):exp[

*j*2

*π*:((

*x*-

*x*

_{03})/Λ Λ

*y*cosθ/

*λ*,)] and beam (6): exp[

*j*2

*π*(-(

*x*-

*x*

_{03})/Λ +

*y*cos

*θ*/

*λ*)]. When the separation between δ and

*γ*is large such that

*L*

_{2}/2tan

*θ*>

*y*, the signal propagating to the fiber core at distance

*y*will “see” the split of phase-shift

*δ*in regions A, B and C, and then the split of phase-shifts

*γ*in the regions C, D and E. The two phase-shifts are split independently. When the adjacent phase-steps

*δ*and

*γ*are close such that

*L*

_{2}/2tan

*θ*<

*y*, the two fringe systems in regions B and D can overlap in region F , where beam (2) diffracted from grating

*L*and beam (5) diffracted from grating

_{1}*L*are superposed, as shown in Fig. 2(a). One can readily calculate the intensity distribution in region F by the interference between beams (2) and (5) as

_{3}*y*>

*L*

_{2}/2tan

*θ*the signal in the fiber core “sees” the fringes in the regions, A, B, F, D and E as: in region A, 1 + cos(2

*π*(

*x*-

*x*

_{01})/(Λ/2)); region B

*π*(

*x*-

*x*

_{01}-(

*γ*+

*δ*))/(Λ/2)). A phase shift

*δ*/2 occurs when crossing region A to B, and an additional phase shift

*γ*/2 occurs when crossing region B to F, the third phase shift

*δ*/2 occurs when crossing region F to D and the last phase shift

*γ*/2 will be added when crossing region D to E. Each half-amplitude phase shift appears as being “propagated” from their respective original phase steps in the phase mask at the angle of first order diffraction to the

*y*-axis. Thus, the multiple phase-shifts in the phase mask are split and individually and independently. All the split phase shifts occur at the boundaries between regions of different interference patterns. The split phase-shifts can then “propagate” and crossover during the beam propagation without interfere and change of their values. This model has been used without proving it in Ref [2] for designing the diffraction compensation phase mask for high channel count phase-only sampled FBGs with a high number of phase-shifts introduced in each sampling period.

## 3. Numerical solution with FDTD

*1 μm*. A phase shift of

*250 nm*is located at

*x*=

*8 μm*. The structure was modeled by a fine meshing made of

*λ*/20 square cells. A Gaussian pulse excitation was launched on the phase mask at normal incidence. The field amplitude and phase distributions were computed by a recursive temporal Fourier transform of the instantaneous electrical field intensity at an arbitrary chosen distance

*y*=

*5μm*from the phase mask. Then, we continued the free space propagation computation using the Fourier optics spatial filter for another 10

*μm*as shown in Fig. 3.

*500 nm*. However, the boundaries between the middle fringe system and the right and left fringe systems are blurred. At distance

*y*=

*15 μm*there is a set of 5 periods of

*525 nm*and 2 periods of

*487.5 nm*and

*512.5 nm*, in the left and right boundaries respectively as shown in Right-Bottom of Fig. 3. The variations of those 7 periods make a total phase shift of

*125 nm*, which is equal to half of the phase step of

*250 nm*in the phase mask. This is a clear indication that the phase shift of

*250 nm*in the phase mask is split into two half-magnitude phase-shifts of

*125 nm*each, which appear to propagate from the original phase step in the phase mask at the angle

*θ*of ±1 diffracted orders. Their separation along the fiber core at distance

*y*from the mask is equal to ∆

*x*= 2

*y*tan

*θ*.

## 4. Conclusion

## Acknowledgments

## References

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. |

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, |

3. | P. E. Dyer, R. J. Farley, and R. Giedl, ”Analysis of grating formation with excimer laser irradiated phase mask,” Opt. Commun. |

4. | Z. S. Hegedus, “Contact printing of Bragg gratings in optical fibers: rigorous diffraction analysis,” Appl. Opt. |

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 |

6. | Y. Qiu, Y. Sheng, and C. Beaulieu, “Optimal phase masks for fiber Bragg grating fabrication,” J. Lightwave Technol. |

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. |

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. |

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 |

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. |

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. |

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. |

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 |

16. | B. J. Lin, “Electromagnetic Near-Field Diffraction of a medium Slit,” J. Opt. Soc. Am. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(050.1970) Diffraction and gratings : Diffractive optics

(050.5080) Diffraction and gratings : Phase shift

(060.5060) Fiber optics and optical communications : Phase modulation

(220.3740) Optical design and fabrication : Lithography

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 29, 2005

Revised Manuscript: July 27, 2005

Published: August 8, 2005

**Citation**

Yunlong Sheng and Li Sun, "Near-field diffraction of irregular phase gratings with multiple phase-shifts," Opt. Express **13**, 6111-6116 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6111

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### References

- 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]
- 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).
- 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] [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)
- Y. Qiu, Y. Sheng, and C. Beaulieu, "Optimal phase masks for fiber Bragg grating fabrication,�?? J. Lightwave Technol. 17, 2366-2370 (1999). [CrossRef]
- 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]
- 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]
- 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)
- R. Kashyyap, �??Fiber Bragg gratings�?? Chap. 6.1 (Academic, San Diego, 1999)
- 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]
- 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]
- 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).
- L. Poladian, B. Ashton and W. Padden, �??Interactive design and fabrication of complex FBGs,�?? OFC paper WL1, Tech. Digest vol.1, 378-79 (2003)
- B. J. Lin, �??Electromagnetic Near-Field Diffraction of a medium Slit,�?? J. Opt. Soc. Am. 62, 976-981 (1972) [CrossRef]
- Z. S. Hegedus, �??Contact printing of Bragg gratings in optical fibers: rigorous diffraction analysis,�?? Appl. Opt. 36, 247-252 (1997)

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