1. Introduction

Bragg gratings have been used extensively over many decades as photonic bandgap structures in optical fibers or in photonic integrated circuits, as filtering elements, or in distributed-feedback lasers [1–5]. They have been theoretically investigated using analytical methods such as coupled-mode theory (CMT), which is accurate for weak gratings and simple perturbation profiles that can be described by analytical functions [6]. In CMT a Fourier representation is used to describe the permittivity perturbation along the grating. The non-uniform permittivity along the waveguide generally consists of real and imaginary index perturbations, which induce coupling between different modes. In Bragg gratings, such coupling is contra-directional, and the strength is typically defined by coupling coefficients [7]. More recently, numerical techniques such as finite-difference time-domain (FDTD) and eigenmode expansion (EME) methods were employed to model complex Bragg gratings [8,9]. The common drawback of such numerical methods is computational expense.

Electromagnetic inverse scattering techniques provide a wide range of design possibilities for gratings. These mathematical methods allow the definition of a structure, such as its refractive index or shape, by analyzing the scattering of waves. Multiple approaches are used to solve inverse scattering problems, including layer-peeling and layer-adding [10–13], genetic algorithms [14], simulated annealing [15,16], spectral techniques [17,18], and other optimization approaches. These methods typically solve equations iteratively that link the computed scattering data to the unknown properties of the medium or structure. The ultimate objective is to identify the design that best corresponds to the measured scattering data.

In this paper we exploit the discrete Fourier transform (DFT) representation of a permittivity perturbation and find a direct relationship between its Fourier coefficients and the Bragg grating’s reflectance spectrum. We then show that by applying the inverse discrete Fourier transform (IDFT) to a desired or prescribed spectral response, the permittivity profile can be determined. This method offers a simple and efficient approach to Bragg grating design, provided that the phase and amplitude of the grating's spectral response are known.

Using the IDFT method, we then arrive at a novel type of grating, which we refer to as Dirac gratings. These gratings are characterized by sharp periodic perturbations in the permittivity along the propagation axis, and they have a near-zero duty cycle. Notably, Dirac gratings may exhibit order-independent performance, meaning that the spectral response is identical for higher-order gratings compared to first-order gratings of the same length. Other types of Dirac gratings can be designed, including odd-equal Diracs where only odd order gratings reflect light (inspired by quarter-wavelength gratings), and exceptional point Dirac gratings with unidirectional reflectance. These findings have significant implications for the design and fabrication of optical gratings for various applications.

This paper is structured as follows: In Section II, we derive analytically the coupling coefficients of coupled-mode theory using the DFT amplitudes representing a spatial permittivity perturbation. We introduce the concept of Dirac gratings by demonstrating that under specific conditions the reflectivity is independent of the order of the grating. In Section III, we employ the IDFT to arrive at perturbation profiles for various designs of Dirac gratings. In Section IV, we evaluate the performance of these gratings through the analytical Transfer Matrix Method (TMM) and the numerical Eigenmode Expansion (EME) method. Finally, we provide concluding remarks in Section 5.

2. Fourier and coupling coefficients

A Bragg grating with a rectangular profile can be mathematically modeled as a rectangular pulse train, where the relative permittivity changes between high and low values within each period. The rectangular pulse train is described by (1) and sketched in Fig. 1.

 

Fig. 1. Relative permittivity distribution described by (1).

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2.1 Spectral representation

Generally, we can apply the DFT to derive the spectral-domain representation of the discretized form of Eq. (1) [19]:

The Fourier coefficients ${F_k}$ which have the largest magnitude are denoted ${\mathrm{{\cal F}}_M}$ and they occur for $k = MN$ where M is an integer. They are identified directly from the last equality of Eq. (3), after resubstituting H by ${e^{ - 2\pi j/\xi }}$:

We can re-write (4) in terms of trigonometric functions:

When the number of samples $\xi $ is sufficiently large to ensure $\xi \gg \pi MN$, the expression for $|{{\mathrm{{\cal F}}_M}} |$ reduces to:

It is observed that $|{{\mathrm{{\cal F}}_M}} |$ is directly proportional to $\xi $. To remove this dependence, we normalize $|{{\mathrm{{\cal F}}_M}} |$ by dividing by $\xi $:

2.2 Coupling coefficients

The expression for the coupling coefficient, ${\kappa _{ {\pm} k}}$, that results from coupled mode theory (CMT) applied to a weak waveguide grating, is given by [21]:

The strongest coupling coefficient also occurs at the Bragg wavelength ${\lambda _B}$ [22]. So, by substituting $\lambda_0 = \lambda_B$ and using the Bragg equation, ${\lambda _B} = 2\bar{n}\mathrm{\Lambda }/M$, Eq. (12) becomes:

This expression for the coupling coefficient (Eq. (13)) is consistent with the literature [21–24], indicating that the integer M, first introduced here in Eq. (4) to identify the largest Fourier coefficients, also corresponds to the grating order in the Bragg equation. We also note that $\bar{n}$ is sometimes taken as the effective refractive index of the waveguide mode averaged over a period, or for a dielectric stack as the refractive index averaged over a period.

To operate at the same Bragg wavelength at order M, it is necessary to increase the pitch by a factor of M, resulting in a reduction of $|{{\kappa_M}} |$ (Eq. (13)). The reflectance of an $M$-th order Bragg grating following CMT is given by [21]:

Thus, conventionally, increasing M decreases the reflectance if the length of the grating, L, remains constant.

2.3 Limit of small duty cycle: Dirac gratings

If the duty cycle, d, is small enough for $\pi Md\;\to 0$, then Eqs. (8), (11) and (12) simplify, respectively, to:

Equation (15) shows that for the same material system, i.e., fixed $\Delta \varepsilon $, Dirac gratings yield a weaker $|{{\kappa_M}} |$ compared to a conventional first order grating due to the small d. However, a high reflectance can be maintained by increasing L to compensate for the small ${\kappa _{ - M}}$ (Eq. (14)). For example, for a first order quarter-wavelength grating, the coupling coefficient is $|{{\kappa_1}} |\cong |{\Delta \varepsilon {/\lambda_0}\bar{n}} |$ according to Eq. (12), which is $1/\pi d$ times larger than that of a Dirac grating (Eq. (15)). Thus, for a Dirac grating to produce the same reflectance as the corresponding quarter-wavelength grating, L should be $1/\pi d$ times longer.

We observed that the reflectance at ${\lambda _B}$, given by Eq. (14), depends on ${\kappa _{ - M}}$ and $\overline {{\mathrm{{\cal F}}_M}} $, as revealed by (11). Therefore, the behavior of ${R_M}$ is dependent on the Fourier amplitudes ${F_M}$. The Fourier amplitudes can thus be prescribed, and the IDFT applied to determine the required permittivity modulation in the spatial domain. These observations motivate the development of a design technique based on the IDFT. In the next section, we will use this technique to explore intricate Dirac grating designs.

3. IDFT design and Dirac gratings

Initially we use the IDFT to verify that a Dirac grating has a reflectance response independent of grating order, $M = 1,\; 2,\; 3, \ldots $,. To verify this, we set the magnitude of the spectral components $|{{F_k}} |$ to the same arbitrary constant, e.g., unity, for $k = MN$, and zero elsewhere, as shown in Fig. 2(a), and their phase to zero so that the incident and reflected waves are in phase at the Bragg wavelength.

 

Fig. 2. (a) Desired distribution of $|{{\mathrm{{\cal F}}_k}} |$ in the spectral domain for order independent reflectance. (b) Spatial permittivity profile of the corresponding Dirac grating over ten periods.

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Applying the IDFT [19] to a discrete spectral distribution, ${F_k}$, leads to the required permittivity perturbation in the spatial domain, described by the samples ${f_m}$:

Note that the magnitude of the resulting samples (Eq. (16)) depends on the magnitude chosen for $|{{F_k}} |$ and on $\xi $.

The permittivity modulation resulting from the spectral distribution of Fig. 2(a) is shown in Fig. 2(b), where the horizontal axis is normalized so that a period is equal to unity. We observe that the real part of the permittivity modulation, $\mathrm{\Delta }{\varepsilon _{re}}$, required to implement the grating consists of Dirac-like peaks, whereas the imaginary part of the permittivity modulation is zero ($\mathrm{\Delta }{\varepsilon _{im}} = 0$), which follows our definition of Dirac gratings and Eq. (15).

Other types of Dirac gratings can be envisioned and designed using the IDFT. The conventional quarter-wavelength grating is of particular interest in semiconductor laser design because it can act as an efficient and high-reflectance mirror while also being relatively easy to fabricate. For the case of Bragg gratings formed as a quarter-wavelength stack ($d = 0.5$), from Eq. (8) it follows that $|{\overline {{\mathrm{{\cal F}}_M}} } |$ vanishes for $d = 0.5$ and even M, while for odd $M$:

This implies that the coupling coefficient (12) and the reflectance (14) are zero for even orders and non-zero for odd orders. Inspired by the conventional quarter-wavelength stack, we set as an alternative design in Fig. 3(a) (left panel) the odd spectral components $|{{\mathrm{{\cal F}}_M}} |$ to unity and the even ones to zero. We then find the permittivity profile by applying the IDFT, yielding a Dirac-like profile involving sharp positive and negative perturbations occurring at specific points along the grating length, as shown in Fig. 3(a) (right panel). We term this design an odd equal-order grating.

 

Fig. 3. Left panels: Desired distribution of $|{{F_k}} |$ in the spectral domain. Right panels: Spatial permittivity profile of the corresponding Dirac gratings over ten periods. (a) Odd equal-order grating, and (b) exceptional point grating.

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In Fig. 3(b), a Dirac grating is designed to produce unidirectional reflection, which means that light is reflected from one grating port but not from the other. This is achieved by ensuring that the spectral representation of the permittivity perturbation has odd orders occurring at only one spectral location such that the spectrum is single-sided, as shown in the left panel of Fig. 3(b). Such gratings are known as exceptional point gratings where parity-time symmetry breaks down [25–27]. The required permittivity modulation shown in the right panel of Fig. 3(b) exhibits even and odd symmetries for the real and imaginary parts of the refractive index, respectively, in agreement with the literature [28].

The IDFT approach can be extended to create gratings with arbitrary reflectance responses, resulting in a complex permittivity profile. Our findings are consistent with the findings of [29], which showed that the reflection spectrum of weak gratings is the Fourier transform of the refractive index profile.

4. Modelling of Dirac gratings

Dirac gratings theoretically require infinitesimally narrow perturbations, which could be practically implemented as thin layers in a dielectric stack, or as short discontinuities in waveguide geometry along the propagation direction, such that the effective index perturbation resembles, e.g., the profile of Fig. 2(b). In this section, we investigate analytically the limit up to which the length of the perturbations can be increased while retaining the properties of a Dirac grating. Additionally, we further validate the concept of Dirac gratings by modelling the performance of two example structures using the TMM and EME method.

4.1 Upper limit of duty cycle

As described in Section 2, a Dirac grating is formed when the duty cycle d is small, such that the small-angle approximation for the sine function in Eq. (12) holds. This sine function can be expressed using the Maclaurin series expansion as follows [30]:

It is evident from the above that the second and subsequent terms can be disregarded for sufficiently small arguments. To obtain Fourier coefficients (coupling coefficients and a reflectance) that are equal for all grating orders to within a specified error $\zeta $, then using Eq. (18), the upper limit for the duty cycle, ${d_{max}}$, should respect:

It can be observed that ${d_{max}}$ depends solely on $\zeta $ and M. For instance, if the designer aims to achieve order-independency for orders up to 100 with an error less than 2%, the maximum allowed duty cycle is ${d_{max}} \approx 0.001$. So, the designer faces a trade-off between the error ($\zeta $) and fabrication-related considerations including the grating period ($M\mathrm{\Lambda }$) and the smallest feature size (${d_{max}}\mathrm{\Lambda }$). The other grating parameters such as index contrast and grating length do not play a defining role in realizing a Dirac grating.

4.2 Dirac dielectric stack and TMM analysis

We employ the TMM to investigate a dielectric stack composed of Si/SiO₂ multilayers on a Si substrate, as an order-independent Dirac grating (following Fig. 2). The simulation setup, along with the multilayer stack considered, are illustrated in Fig. 4. In this figure, $M\mathrm{\Lambda }$ is the pitch and ${L_D}$ is the length of each perturbation approximating Dirac peaks (Dirac length). The duty cycle $d = {L_D}/M\mathrm{\Lambda }$ is the ratio of the Dirac length to the pitch. The refractive indices of Si and SiO₂ were taken as ${n_0} = 3.47$ and ${n_1} = 1.44$, respectively, [31] for operation near $\lambda_0 = $ 1550 nm. To ensure that the first order $({M = 1} )$ Bragg reflection occurs near 1550 nm, we set $\mathrm{\Lambda }$ to 225 nm.

 

Fig. 4. Schematic of the periodic multilayer stack of pitch $M\mathrm{\Lambda }$, used in TMM calculations. The grating is bounded by two semi-infinite layers - a Si substrate of index ${n_0}$ and air of index 1.

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The reflectance response R was computed for several grating orders and different values of d while maintaining the total stack thickness fixed to 10.8 µm, which holds N = 48 periods for M = 1. Figure 5(a) plots the reflectance at ${\lambda _B}$ as a function of duty cycle d for four different grating structures, corresponding to $M = 1$, $2$, $4$, and $8$. We fixed d when comparing different orders to verify the order independent performance predicted by (14) and (15). Figure 5(a) reveals two key observations. First, the Dirac stack indeed exhibits order-independent performance, as the reflectance response remains constant for different orders at the same d. Second, the order-independent behavior is not significantly affected by ${L_D}$ values up to 2.5% of the pitch. Thus, in a dielectric stack, thin slabs of finite thickness are good approximations to a Dirac function.

 

Fig. 5. (a) Reflectance near the Bragg wavelength as a function of d for several grating orders. (b) Comparison of reflectance responses for several grating orders (shown in legend) for $d = 0.02$, calculated via the TMM. The grating length is fixed to 10.8 µm for all orders.

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Figure 5(b) compares the reflectance responses of different gratings for $d = 0.02$. For the case where $d = 0.02$, the SiO₂ and Si thicknesses are 4.5 and 220.5 nm, respectively, for 1^st order, and 36 and 1764nm, respectively for 8^th order. Both designs produce the same reflectance for the same total stack thickness (10.8 µm). Furthermore, the reflectance responses of Fig. 5(b) exhibit good agreement with one another, not only in terms of the reflectance values at ${\lambda _B}$ but also in terms of the width of the reflection band. The TMM simulations thus provide confirmation of the effectiveness of Dirac gratings.

The order-independence of the spectral responses can be explained phenomenologically as follows. The Dirac-like perturbations have a short length ${L_D}$ so they may be viewed as scatterers as long as they remain shorter than the wavelength. As the grating pitch is increased from $\Lambda $ to $M\Lambda $, the perturbation length must also be increased from ${L_D}$ to $M{L_D}$ in order to maintain the duty cycle d constant. The $M$-times longer perturbation produces $M$-times stronger scattering, such that for gratings of the same length but different orders, the reflectance response remains identical even though the $M$-th order grating has $M$-times fewer scatterers compared to the first-order grating. So, the integrated effect of the scatterers in both cases remains identical.

The fabrication of a Dirac grating as a dielectric stack requires precise control of the refractive index and thickness of individual layers, which must also be uniform and of high quality (low roughness, particles and defects). Semiconductor deposition tools readily offer the required control and quality in material, via physical or chemical vapour deposition [32].

4.3 Dirac waveguide grating and EME analysis

The example waveguide assumed in this section is a Silicon-on-Insulator (SOI) strip-loaded waveguide as shown in Fig. 6(a). The details of each layer (thickness, refractive index) are given in Table 1. The refractive indices of Si and SiO₂ were extracted from [31]. The waveguide is taken as lossless and passive, i.e., the imaginary parts of the refractive indices are zero. The top view of the parameterized grating structure is shown in Fig. 6(b), where $M\mathrm{\Lambda }$ is the pitch and ${L_D}$ is the length of the Dirac peaks. $\mathrm{\Lambda }$ is assumed to be 280 nm such that the first order ${\lambda _B}$ occurs at ∼1550 nm. Figure 6(c) shows the electric field profile of the fundamental transverse-electric (TE) mode for the two ridge widths used within each period.

 

Fig. 6. (a) 1D cross section of the assumed strip-loaded waveguide. (b) Schematic of the Dirac waveguide grating. (c) Electric field profiles of the fundamental mode for the two ridge widths used in (b).

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Table 1. Properties of the multilayer grating structure at 1550 nm

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We now use the EME method to model the grating structure. The EME method is rigorous, modelling the response of the 3D structure, including any scattering losses due to mode mismatch along a grating. The simulations were carried out using the EME solver in Lumerical Mode [33]. The fundamental TE mode was selected as the input source.

Similarly to the previous section, we vary the parameter $d = {L_D}/M\mathrm{\Lambda }$ and the grating order M, and we compute the reflectance response for a constant grating length set to $L = 560$ µm, i.e., $N = 2000$ for $M = 1$. Figure 7(a) plots the reflectance at ${\lambda _B}$ vs. d for four different orders. It is apparent that for $d > 0.01$, the performance is not independent of order as different reflectance values are produced for the same grating length. This contrasts with Fig. 5(a), where a stronger grating maintained order-independence for larger d. Additionally, the EME method considers radiation loss, which means that the reflectance might diverge for duty cycles smaller than expected based on Eq. (20). In this example, the upper limit for the duty cycle should be $0.013$ according to Eq. (20) for $\zeta = 10\%$, whereas the EME simulations show $d < 0.01$. The structural parameters for $d = 0.005$ for the 20^th order grating are: $M\mathrm{\Lambda } = 5.6$ µm and ${L_D} = 28$ nm, whereas the structural parameters for the first order grating are: $M\Lambda = 0.28$ µm and ${L_D} = 1.4$ nm. The former which has only 100 periods produces almost the same performance as the latter which has 2000 periods.

 

Fig. 7. (a) Reflectance near the Bragg wavelength as a function of d for several grating orders. (b) Comparison of reflectance responses for several grating orders (shown in legend) for $d = 0.005$, calculated via the EME method. The grating length is fixed to 560 µm for all cases.

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Figure 7(b) shows the wavelength response of the reflectance R computed using the EME of the four grating designs for the case $d = 0.005$. It is clear that despite very different orders, the gratings share almost identical wavelength responses, thereby confirming order-independent behavior for Dirac waveguide gratings.

The fabrication of a Dirac waveguide grating shares similar considerations as a dielectric stack, in that precise control of material properties and quality are required. Additionally, high-resolution lithography is required to define the waveguides and the Dirac perturbations of length ${L_D}$, as well as an etching process that produces vertical and smooth sidewalls. High-resolution features can be accessed via electron beam or helium ion beam lithography [34], and the latter by reactive ion etching.

5. Conclusion

We demonstrated theoretically that the reflectance of a grating, which depends on the coupling coefficient ${\kappa _{ - M}}$, is proportional to the Fourier coefficients of the largest magnitude in the discrete spectral representation of a permittivity perturbation. Based on this observation, we introduced a design technique whereby a desired spectral response is inverse Fourier transformed to produce the corresponding permittivity perturbation in space. The approach is general and can be used to design gratings of arbitrary reflectance response. The outcome is a complex permittivity profile of the desired Bragg grating. This method provides an efficient way to design devices for a variety of purposes, such as generating frequency combs, single-frequency lasers, unidirectional lasers (e.g., exceptional point gratings), of arbitrary permittivity profile.

Using this technique, we arrived at the concept of a Dirac grating, where (strong or weak) periodic perturbations of vanishing duty cycle approach a train of Dirac functions. Such gratings produce interesting spectral responses, such as order-independent performance for the same grating length. We provided practical design examples based on a dielectric stack and on an SOI waveguide. The analytical TMM and numerical EME methods were used to verify the performance on these examples and verify the concept of Dirac gratings. Apart from their interesting behavior, Dirac gratings have practical significance in that higher order gratings can be used in place of lower order gratings while maintaining the same performance for the same grating length.