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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 Optical Society of America
1. Introduction
Since the introduction of the waveguide gratings, myriad applications of both fiber gratings [1–4] and channel waveguide gratings [5, 6] 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, 8] and refractive index modulation methods based on Fourier-transform [9,10] 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]. 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, 13] 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
where κ(z) is the coupling coefficient of the refractive index modulation [5] and Δk = βa − βb; 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] and the reflectance/transmittance obtains a closed form algebraic expression [Ref. [5], Eq. (4.13)].Applying the undepleted pump approximation [5] to Eq. (2) and integrating with respect to 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(Δk) 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
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
as stated by the classical Lohmann’s detour-phase principle [22, 23]. Evidently, we can modulate the phase of the coupling coefficient by shifting the center of individual carrier grating periods. And because κ(z) and B(Δk) 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
We have already showed that the coupling coefficient and reflected amplitude form a Fourier transform pair and next we shall utilize this property. Fortunately, an efficient iterative Gerchberg-Saxton phase retrieval algorithm [15] exists for such cases enabling the calculation of the optimal phase distribution of the coupling coefficient. This method was further developed to the Iterative Fourier Transform Algorithm (IFTA) by Fienup and Wyrowski to design diffractive optical elements [16–19] known as computer generated holograms. It finds the phase distribution that produces the desired signal. The most sophisticated approach is to use IFTA for calculating the optimal field and then code the field information into a carrier grating to obtain excellent signal-to-noise ratios as shown in [20, 21]. Next we briefly introduce the basic form of IFTA for designing waveguide gratings. An excellent review of the method for diffractive elements can be found in [18].
The target reflected amplitude is denoted by |Bt(Δk)| 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.
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] 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]. 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] where a similar distribution was obtained for diffractive fan-out elements.
Next we code this phase function to the carrier grating periods by solving Δz for each period from Eq. (5) using Δk = Δk0. We use the carrier period d = 0.538 μm and the total grating period of D = 300 μm 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.
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.
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]. 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
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
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. 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] 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.
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