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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 13810–13817
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Tailored bandgaps: iterative algorithms of diffractive optics

Tuomas Vallius  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13810-13817 (2013)
http://dx.doi.org/10.1364/OE.21.013810


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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

Since the introduction of the waveguide gratings, myriad applications of both fiber gratings [1

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] .

4

4. R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 2010).

] and channel waveguide gratings [5

5. H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).

, 6

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 SPIE4640, 117–124 (2002) [CrossRef] .

] have arisen. As the research advances expanding the domain of potential applications, an insatiable demand increases for going beyond the traditional forbidden bandgap shapes. Therefore several approaches have been developed to modify the grating response. Chirping and apodization of the grating period [7

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

, 8

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] .

] and refractive index modulation methods based on Fourier-transform [9

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

,10

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] .

] are the most well known methods. However, limitations still exist for the spectrum and problems appear when arbitrary spectral shapes are targeted with high side mode suppression ratios.

Here we introduce a new approach based on finding the optimal phase function and coding that to the grating period variation [11

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).

]. The coupled wave equations are used as the framework because they allow the treatment of the waveguide grating function and reflectance as a Fourier transform pair. The design methods are borrowed from the discipline of diffractive optics thus enabling the use of the state-of-the-art design algorithms polished to perfection during the last two decades.

2. Presentation of the methods

2.1. Coupled mode equations and Fourier transform pair

The coupled-mode method for analyzing waveguide gratings [12

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

, 13

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

] is reliable when all the modes with notable coupling are taken into account. In the case of two modes propagating along the ±z-direction the equations for the pump mode amplitude A(z) and reflected mode amplitude B(z) can be written as
dA(z,Δk)dz=iκ(z)B(z,Δk)exp(iΔkz)
(1)
dB(z,Δk)dz=iκ(z)A(z,Δk)exp(iΔkz),
(2)
where κ(z) is the coupling coefficient of the refractive index modulation [5

5. H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).

] and Δk = βaβb; the paramaters βa and βb are the propagation constants of the modes 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] .

] and the reflectance/transmittance obtains a closed form algebraic expression [Ref. [5

5. H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).

], Eq. (4.13)].

Applying the undepleted pump approximation [5

5. H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).

] to Eq. (2) and integrating with respect to z, we can write the reflected mode amplitude in the form
B(Δk)=L0iκ(z)exp(iΔkz)dz,
(3)
where L denotes the length of the grating. Correspondingly, the coupling coefficient can be obtained with the inverse transform of the reflected amplitude
κ(z)=12πiB(Δk)exp(iΔkz)dΔk.
(4)
Consequently, κ and Bk) in Eqs. (3)(4) form a Fourier transform pair that can be treated with standard methods of wave optical engineering [11

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).

] 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

Let us then proceed to consider the tools of the phase modulation. If the location of the period of the carrier grating κ(z) is shifted by an amount of Δz in Eq. (3), the phase of the coupling coefficient is changed by
ϕ=ΔzΔk
(5)
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

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

]. Evidently, we can modulate the phase of the coupling coefficient by shifting the center of individual carrier grating periods. And because κ(z) and Bk) 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 Δz(z).

2.3. Optimal phase retrieval

The target reflected amplitude is denoted by |Btk)| and the initial κ(z) is set by using a parabolic phase function exp(iz2/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.
  1. Calculate Bk) from Eq. (3).
  2. Correct the amplitude |Bk)|.
    1. Set Bk) = |Btk)|exp{iarg[Bk)]} within the signal window. Remove the noise surrounding the signal window.
    2. Set Bk) = |Btk)|exp{iarg[Bk)]} within the signal window. Keep the distribution outside the signal window.
  3. Calculate κ(z) from Eq. (4).
  4. Correct the amplitude of κ(z) and retain the phase. Return to step 1.

For phase only structures the step 4 removes the amplitude modulation of 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] .

] where the amplitude exceeding 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.

Traditionally, the iteration is divided into two phases [18

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

]. The phase (A) aims at the maximal efficiency and applies the part (A) of step 2. The phase (B) enhances the uniformity and uses the part (B) of step 2. Several other formulations also exist but in this article we apply only this version.

3. Results

Even though the presented formalism can be applied to any kind of periodic structures from Bragg wire nanochannels to slab waveguides, we present the applicability of the coding approach by designing photonic bandgaps for a single mode fiber to illustrate the procedure and the wavelength λ is in the infrared region. From now on, the following parameters will be used: central wavelength λ0 = 1.55 μm, fiber core radius 5 μm, core and cladding refractive indices 1.442 and 1.44. The effective index of the fiber is neff = 1.4405 and the V-number is V = 1.5388. We assume that the grating has a binary modulation of n(z) = neff ± Δn, where Δn = 0.0001, and the amplitudes A(z) and B(z) refer to the forward and backward propagating fundamental mode. Therefore the parameter Δk can be written Δk = 4πneff/λ and at the central wavelength we denote Δk0 = 4πneff/λ0.

3.1. Periodic structures

Let us begin with designing a structure that reflects nine discrete wavelengths centered around Δk = 0. We use the initial phase R = 30 μm 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] .

] where a similar distribution was obtained for diffractive fan-out elements.

Fig. 1 Phase distribution that produces nine uniform reflectance peaks.

Fig. 2 Reflectance spectrum of the grating with uniform discrete wavelengths.

Then we use the same number of reflectance peaks but now the response is modulated so that the reflectance decreases linearly from the outermost wavelength to the center wavelength so that the center reflectance is 40 % lower. The resulting phase function can be seen in Fig. 3. Again, we code the phase to the carrier period of d = 0.538 μm and the total period of D = 300 μm 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.

Fig. 3 Phase distribution that produces nine modulated reflectance peaks.
Fig. 4 Reflectance spectrum of the grating with modulated discrete wavelengths.

3.2. Continuous signals

Continuous signals require nonperiodic structures posing extra challenge to the noise level of the signal. This phenomenan is mathematically identical to the boundary diffraction perturbation attributable to the aperture when designing, e.g., map transform elements of diffractive optics. Therefore we now resort to amplitude freedom of the grating in addition to the phase freedom. The IFTA procedure introduced in Sec. 2.3 uses amplitude clipping at the step 4 with the clipping parameter 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] .

]. This enables the further reduction of the noise level of the signal. The resulting amplitude distribution is coded to the fill factor modulation of the grating. Calculating the amplitude of the first Fourier coefficient of the coupling coefficient 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
f=arcsin[|κ(z)|]π.
(6)
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.

To test the algorithm with nonuniform continuous signals, we use a target function comprising a sum of a Gaussian and Supergaussian functions
S(λ)=|exp[(λλ0)4/w14]exp[(λλ0)2/w22]/2|,
(7)
where w1 = 2 nm and w2 = 1 nm. One hundred iteration loops are applied to the IFTA phases (A) and (B) using the carrier grating period of 0.538 μm and the total grating length of 2000 μm. 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.

Fig. 5 Phase (a) and amplitude (b) distribution that produces the continuous signal.
Fig. 6 Reflectance of the non-periodic structure. Both target signal (green) and the realized reflectance (blue) are given using linear (a) and logarithmic (b) scales.

4. Discussion

The undepleted pump approximation used in Eq. (3) suggest that the method is applicable to the low reflectance regime only. However, the beauty of the IFTA approach lies in its versatily; the decay of the pump mode can be taken into account in the iterative phase retrieval by using a suitable constraint for the amplitude correction step. Furthermore, also the data quantization for the fabrication process can easily be done during the iteration.

Neither the coupling is limited to the contradirectional modes but can take place between codirectional light fields, and even between the pump field and converted harmonic frequencies. Although the example cases were presented with a single mode fiber, the method is not limited to fiber optics but is equally applicable to any kind of periodic structures such as three dimensional photonic crystals, grating couplers, periodic poling and mode couplers.

The initial function for 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] .

] that has proved efficient for obtaining the initial guess for the computer generated holograms.

The computational efficiency of the proposed method is very high since it is based on the FFT algorithm. Therefore the time required to design the gratings takes only a few seconds with standard notebook computers.

5. Conclusions

A new approach for tailoring the bandgap properties of periodic structure has been presented. The coupled-mode theory of waveguide gratings yields a Fourier transform pair enabling a direct application of the Iterative Fourier Transform Algorithm for phase retrieval as well as carrier grating coding of the field distribution. The well established tools of diffractive optics proved functional also with the new application allowing designing of different types of discrete and continuous photonic bandgaps.

It would be of great interest to consider three dimensional periodic structures to tailor their optical properties. However, the further development of the method is still required before that is viable.

Acknowledgments

The author is grateful for helpful discussions with Dr. Juha Pietarinen, Prof. Seppo Honkanen, Dr. Antti Laakso, and Prof. Markku Kuittinen. The work of the author was funded by the Academy of Finland, project 14910.

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. 32, 647–649 (1978) [CrossRef] .

2.

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] .

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. 62, 1035–1037 (1993) [CrossRef] .

4.

R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 2010).

5.

H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).

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 SPIE4640, 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] .

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. 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] .

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] .

15.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

16.

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

17.

J. R. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt. 52, 45–56 (2013) [CrossRef] [PubMed] .

18.

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

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. 43, 1389–1398 (1996).

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] .

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] .

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] .

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

  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]
  2. 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]
  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.62, 1035–1037 (1993). [CrossRef]
  4. R. Kashyap, Fiber Bragg Gratings (Academic Press, San Diego, 2010).
  5. H. Nishihara, M. Haruna, and T. Suhara, Integrated Optical Circuits (McGraw-Hill, New York, 1989).
  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 SPIE4640, 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. B20, 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]
  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.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]
  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]
  15. 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).
  16. F. Wyrowski, “Diffractive optical elements: Iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A7, 961–969 (1990). [CrossRef]
  17. J. R. Fienup, “Phase retrieval algorithms: a personal tour,” Appl. Opt.52, 45–56 (2013). [CrossRef] [PubMed]
  18. F. Wyrowski and O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys.54, 1481–1571 (1991). [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]
  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.43, 1389–1398 (1996).
  21. 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]
  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]
  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. A14, 1549–1553 (1997). [CrossRef]

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