## Tailored bandgaps: iterative algorithms of diffractive optics |

Optics Express, Vol. 21, Issue 11, pp. 13810-13817 (2013)

http://dx.doi.org/10.1364/OE.21.013810

Acrobat PDF (807 KB)

### Abstract

A diffractive optics design method based on a phase retrieval algorithm and carrier grating coding is modified to enable designing of photonic bandgap reflectances. Discrete and continuous signals are designed for a fiber grating to demonstrate the capability of the approach. The method is proved a versatile tool for synthesizing reflectance spectra of periodic structures.

© 2013 OSA

## 1. Introduction

1. K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photo-sensitivity in optical fiber waveguide: Application to reflection filter fabrication,” Appl. Phys. Lett. **32**, 647–649 (1978) [CrossRef] .

6. T. Aalto, S. Yliniemi, P. Heimala, P. Pekko, J. Simonen, and M. Kuittinen, “Integrated Bragg gratings in silicon-on-insulator waveguides,” in *Integrated Optics: Devices, Materials, and Technologies VI*, Y. S. Sidorin and A. Tervonen, eds., *Proc SPIE*4640, 117–124 (2002) [CrossRef] .

7. R. Kashyap, “Design of step-chirped fibre Bragg gratings,” Opt. Commun. **136**, 461–469 (1997) [CrossRef] .

8. D. Wiesmann, R. Germann, G.-L. Bona, C. David, D. Erni, and H. Jäckel, “Add-drop filter based on apodized surface-corrugated gratings,” J. Opt. Soc. Am. B **20**, 417–423 (2003) [CrossRef] .

9. K. A. Winick, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. **26**, 1918–1929 (1990) [CrossRef] .

10. M. Verbist, D. Van Thourhout, and W. Bogaerts, “Weak gratings in silicon-on-insulator for spectral filtering based on volume holography,” Opt. Lett. **38**, 386–388 (2013) [CrossRef] [PubMed] .

## 2. Presentation of the methods

### 2.1. Coupled mode equations and Fourier transform pair

12. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. **9**, 919–933 (1973) [CrossRef] .

13. H. Kogelnik and C. V. Shank, “Coupled-mode theory of distributed feedback lasers,” Appl. Phys. **43**, 2327–2335 (1972) [CrossRef] .

*z*-direction the equations for the pump mode amplitude

*A*(

*z*) and reflected mode amplitude

*B*(

*z*) can be written as where

*κ*(

*z*) is the coupling coefficient of the refractive index modulation [5] and Δ

*k*=

*β*−

_{a}*β*; the paramaters

_{b}*β*and

_{a}*β*are the propagation constants of the modes

_{b}*A*and

*B*, respectively. The coupling coefficient describes a carrier grating that induces coupling between the amplitudes

*A*and

*B*. For periodic structures the exact solution for the coupled Eqs (1)–(2) can be written as an elliptic integral [14

14. J. A. Armstrong, N. Bloembergen, J. Ducing, and P. S. Pershan, “Interactions between light waves in a nonlinear medium,” Phys. Rev. **127**, 1918–1939 (1962) [CrossRef] .

*z*, we can write the reflected mode amplitude in the form where

*L*denotes the length of the grating. Correspondingly, the coupling coefficient can be obtained with the inverse transform of the reflected amplitude Consequently,

*κ*and

*B*(Δ

*k*) in Eqs. (3)–(4) form a Fourier transform pair that can be treated with standard methods of wave optical engineering [11] since the methods of designing computer generated holograms are mainly based on Fourier transform pairs, as we shall see in Sec. 2.3.

### 2.2. Coding of the carrier grating

*κ*(

*z*) is shifted by an amount of Δ

*z*in Eq. (3), the phase of the coupling coefficient is changed by as stated by the classical Lohmann’s detour-phase principle [22

22. A. W. Lohmann and D. P. Paris, “Binary Franhofer holograms, generated by computer,” Appl. Opt. **6**, 1739–1748 (1967) [CrossRef] [PubMed] .

23. B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. **5**, 967–969 (1966) [CrossRef] [PubMed] .

*κ*(

*z*) and

*B*(Δ

*k*) form a Fourier transform pair, given by Eqs. (3)–(4), the required phase shift function

*ϕ*(

*z*) that yields the desired reflectance can be calculated from the reflectance spectrum. Now we only need to find out a mathematical way to obtain the phase function that defines the shift function Δ

*z*(

*z*).

### 2.3. Optimal phase retrieval

16. F. Wyrowski, “Diffractive optical elements: Iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A **7**, 961–969 (1990) [CrossRef] .

19. D. Prongu, H. P. Herzig, R. Dänliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. **31**, 5706–5711 (1992) [CrossRef] .

21. V. Kettunen, P. Vahimaa, J. Turunen, and E. Noponen, “Zeroth-order coding of complex amplitude in two dimensions,” J. Opt. Soc. Am. A **14**, 808–815 (1997) [CrossRef] .

18. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. **54**, 1481–1571 (1991) [CrossRef] .

*B*

_{t}(Δ

*k*)| and the initial

*κ*(

*z*) is set by using a parabolic phase function exp(

*iz*

^{2}/

*R*) because that resembles the phase function of the lens and the mathematically the problem is identical to the Fourier transform performed by lenses. Now the iteration goes along the following procedure.

*κ*(

*z*). However, designing components with phase and amplitude modulation, such as continuous signals from non-periodic structures, we allow amplitude modulation and apply the clipping method [21

21. V. Kettunen, P. Vahimaa, J. Turunen, and E. Noponen, “Zeroth-order coding of complex amplitude in two dimensions,” J. Opt. Soc. Am. A **14**, 808–815 (1997) [CrossRef] .

*C*× max[

*κ*(

*z*)] is set to

*C*× max[

*κ*(

*z*)]. This way we can decrease the modulation and increase the performance of the component.

18. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. **54**, 1481–1571 (1991) [CrossRef] .

## 3. Results

*λ*is in the infrared region. From now on, the following parameters will be used: central wavelength

*λ*

_{0}= 1.55

*μ*m, fiber core radius 5

*μ*m, core and cladding refractive indices 1.442 and 1.44. The effective index of the fiber is

*n*

_{eff}= 1.4405 and the V-number is

*V*= 1.5388. We assume that the grating has a binary modulation of

*n*(

*z*) =

*n*

_{eff}± Δ

*n*, where Δ

*n*= 0.0001, and the amplitudes

*A*(

*z*) and

*B*(

*z*) refer to the forward and backward propagating fundamental mode. Therefore the parameter Δ

*k*can be written Δ

*k*= 4

*πn*

_{eff}/

*λ*and at the central wavelength we denote Δ

*k*

_{0}= 4

*πn*

_{eff}/

*λ*

_{0}.

### 3.1. Periodic structures

*k*= 0. We use the initial phase

*R*= 30

*μ*m and 50 iterations for both phase (A) and (B). No amplitude modulation of

*κ*(

*z*) is allowed. The resulting phase function is seen in Fig. 1. The shape is familiar from Ref. [19

19. D. Prongu, H. P. Herzig, R. Dänliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. **31**, 5706–5711 (1992) [CrossRef] .

*z*for each period from Eq. (5) using Δ

*k*= Δ

*k*

_{0}. We use the carrier period

*d*= 0.538

*μ*m and the total grating period of

*D*= 300

*μ*m that yields the wavelength spacing of 2.8 nm or 3.48 GHz. Ten periods are used resulting in the total component length of 3 mm. Because no coupling exist between the fundamental and radiation modes, we can analyze the grating using only the mode amplitudes

*A*(

*z*) and

*B*(

*z*) with the pump depletion, which gives us the reflectance shown in Fig. 2. The response gives us nine distinct reflectance peaks with low noise level. The location and spacing of the peaks can be freely tuned by using different values for the carrier and total periods

*d*and

*D*; the widths of the peaks can be changed by using different number of periods. The reflectance is low because of the low number of grating periods and low refractive index contrast.

*d*= 0.538

*μ*m and the total period of

*D*= 300

*μ*m using this time 30 periods. The application of the coupled-mode analysis gives the reflectance in Fig. 4. Owing to the longer total component length the reflectance is now notably higher. Further, the designed weight factors of the different reflectance peaks are produced reliably implying that phases yielding even more complicated signals can be synthesized without major difficulties.

### 3.2. Continuous signals

*C*= 0.9 following the procedure presented in [21

21. V. Kettunen, P. Vahimaa, J. Turunen, and E. Noponen, “Zeroth-order coding of complex amplitude in two dimensions,” J. Opt. Soc. Am. A **14**, 808–815 (1997) [CrossRef] .

*κ*(

*z*) as a function of the fill factor

*f*, we obtain the following relation between the amplitude of the coupling coefficient and the fill factor We must bear in mind that changing the centre of the grating lines we also change the phase of the coupled field. Therefore the fill factor modulation must not affect the locations of the grating lines, only the widths.

*w*

_{1}= 2 nm and

*w*

_{2}= 1 nm. One hundred iteration loops are applied to the IFTA phases (A) and (B) using the carrier grating period of 0.538

*μ*m and the total grating length of 2000

*μ*m. The resulting phase and amplitude can be seen in Fig. 5. The phase distribution is even surprisingly smooth whereas the amplitude undergoes stronger modulation. The calculated response of the coded carrier structure is shown in Fig. 6. For comparison, we have also plotted the desired target response given in Eq. (7): the curves are nearly indistinguishable and overlap completely when considering the linear scale plot. The decibel scale image reveals that even though the signal window obediently follows the target function, the side lobes of the realized signal contains noise level 40 dB below the peak of the signal.

## 4. Discussion

*κ*(

*z*) in the IFTA procedure has a significant effect on the result and the algorithm can stagnate to a poor local maximum if the initial function is too far from the optimum. Therefore further research on obtaining the optimal initial

*κ*(

*z*) with an analytical method needs to be done. One potential approach would be the use of the optical map transform [24

24. H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, T. Beth, and F. Wyrowski, “Analytical beam shaping with application to laser diode arrays,” J. Opt. Soc. Am. A **14**, 1549–1553 (1997) [CrossRef] .

## 5. Conclusions

## Acknowledgments

## References and links

1. | K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photo-sensitivity in optical fiber waveguide: Application to reflection filter fabrication,” Appl. Phys. Lett. |

2. | G. Meltz, W. W. Morey, and W. H. Glen, “Formation of Bragg grating in optical fibers by transverse holographic method,” Opt. Lett. |

3. | K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fibre by UV exposure through a phase mask,” Appl. Phys. Lett. |

4. | R. Kashyap, |

5. | H. Nishihara, M. Haruna, and T. Suhara, |

6. | T. Aalto, S. Yliniemi, P. Heimala, P. Pekko, J. Simonen, and M. Kuittinen, “Integrated Bragg gratings in silicon-on-insulator waveguides,” in |

7. | R. Kashyap, “Design of step-chirped fibre Bragg gratings,” Opt. Commun. |

8. | D. Wiesmann, R. Germann, G.-L. Bona, C. David, D. Erni, and H. Jäckel, “Add-drop filter based on apodized surface-corrugated gratings,” J. Opt. Soc. Am. B |

9. | K. A. Winick, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. |

10. | M. Verbist, D. Van Thourhout, and W. Bogaerts, “Weak gratings in silicon-on-insulator for spectral filtering based on volume holography,” Opt. Lett. |

11. | T. Vallius, J. Konttinen, and P. Tuomisto, “An optical broadband filter and a device comprising the same,” EpiCrystals Inc., US Patent Application US61/491,007, (2011). |

12. | A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. |

13. | H. Kogelnik and C. V. Shank, “Coupled-mode theory of distributed feedback lasers,” Appl. Phys. |

14. | J. A. Armstrong, N. Bloembergen, J. Ducing, and P. S. Pershan, “Interactions between light waves in a nonlinear medium,” Phys. Rev. |

15. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik |

16. | F. Wyrowski, “Diffractive optical elements: Iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A |

17. | J. R. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt. |

18. | F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. |

19. | D. Prongu, H. P. Herzig, R. Dänliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. |

20. | J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, and E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. |

21. | V. Kettunen, P. Vahimaa, J. Turunen, and E. Noponen, “Zeroth-order coding of complex amplitude in two dimensions,” J. Opt. Soc. Am. A |

22. | A. W. Lohmann and D. P. Paris, “Binary Franhofer holograms, generated by computer,” Appl. Opt. |

23. | B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. |

24. | H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, T. Beth, and F. Wyrowski, “Analytical beam shaping with application to laser diode arrays,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.1590) Diffraction and gratings : Chirping

(050.1960) Diffraction and gratings : Diffraction theory

(060.2310) Fiber optics and optical communications : Fiber optics

(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 12, 2013

Revised Manuscript: May 4, 2013

Manuscript Accepted: May 9, 2013

Published: May 31, 2013

**Citation**

Tuomas Vallius, "Tailored bandgaps: iterative algorithms of diffractive optics," Opt. Express **21**, 13810-13817 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13810

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### References

- K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photo-sensitivity in optical fiber waveguide: Application to reflection filter fabrication,” Appl. Phys. Lett.32, 647–649 (1978). [CrossRef]
- G. Meltz, W. W. Morey, and W. H. Glen, “Formation of Bragg grating in optical fibers by transverse holographic method,” Opt. Lett.14, 823–825 (1989). [CrossRef] [PubMed]
- K. O. Hill, B. Malo, F. Bilodeau, D. C. Johnson, and J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fibre by UV exposure through a phase mask,” Appl. Phys. Lett.62, 1035–1037 (1993). [CrossRef]
- R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 2010).
- H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).
- T. Aalto, S. Yliniemi, P. Heimala, P. Pekko, J. Simonen, and M. Kuittinen, “Integrated Bragg gratings in silicon-on-insulator waveguides,” in Integrated Optics: Devices, Materials, and Technologies VI, Y. S. Sidorin and A. Tervonen, eds., Proc SPIE4640, 117–124 (2002). [CrossRef]
- R. Kashyap, “Design of step-chirped fibre Bragg gratings,” Opt. Commun.136, 461–469 (1997). [CrossRef]
- D. Wiesmann, R. Germann, G.-L. Bona, C. David, D. Erni, and H. Jäckel, “Add-drop filter based on apodized surface-corrugated gratings,” J. Opt. Soc. Am. B20, 417–423 (2003). [CrossRef]
- K. A. Winick, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron.26, 1918–1929 (1990). [CrossRef]
- M. Verbist, D. Van Thourhout, and W. Bogaerts, “Weak gratings in silicon-on-insulator for spectral filtering based on volume holography,” Opt. Lett.38, 386–388 (2013). [CrossRef] [PubMed]
- T. Vallius, J. Konttinen, and P. Tuomisto, “An optical broadband filter and a device comprising the same,” EpiCrystals Inc., US Patent Application US61/491,007, (2011).
- A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron.9, 919–933 (1973). [CrossRef]
- H. Kogelnik and C. V. Shank, “Coupled-mode theory of distributed feedback lasers,” Appl. Phys.43, 2327–2335 (1972). [CrossRef]
- J. A. Armstrong, N. Bloembergen, J. Ducing, and P. S. Pershan, “Interactions between light waves in a nonlinear medium,” Phys. Rev.127, 1918–1939 (1962). [CrossRef]
- R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).
- F. Wyrowski, “Diffractive optical elements: Iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A7, 961–969 (1990). [CrossRef]
- J. R. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt.52, 45–56 (2013). [CrossRef] [PubMed]
- F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys.54, 1481–1571 (1991). [CrossRef]
- D. Prongu, H. P. Herzig, R. Dänliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt.31, 5706–5711 (1992). [CrossRef]
- J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, and E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt.43, 1389–1398 (1996).
- V. Kettunen, P. Vahimaa, J. Turunen, and E. Noponen, “Zeroth-order coding of complex amplitude in two dimensions,” J. Opt. Soc. Am. A14, 808–815 (1997). [CrossRef]
- A. W. Lohmann and D. P. Paris, “Binary Franhofer holograms, generated by computer,” Appl. Opt.6, 1739–1748 (1967). [CrossRef] [PubMed]
- B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt.5, 967–969 (1966). [CrossRef] [PubMed]
- H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, T. Beth, and F. Wyrowski, “Analytical beam shaping with application to laser diode arrays,” J. Opt. Soc. Am. A14, 1549–1553 (1997). [CrossRef]

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