## Abstract

A unified approach to obtain the characteristics of almost-periodic grating slab waveguides including gain in the waveguide is reported. In this approach the waveguides are divided into short segments, and in each segment the gratings are assumed to be periodic, that is, parameters such as coupling coefficient, grating phase, deviations from the Bragg frequency, and gain in the waveguide are independent of a propagation direction *z*. Then characteristics of almost-periodic grating slab waveguides can be obtained by multiplying each F matrix of a short segment with the proper grating phase conditions at the interface between two adjacent segments. The appropriateness of this approach is shown for typical aperiodic grating waveguides such as tapered, chirped, and phase-shifted gratings. The results obtained by this method are compared with others and prove to be in good agreement with the results obtained by other methods. In addition to these characteristics, it is shown that the F matrix can be used to obtain the threshold conditions for distributed feedback laser oscillations including reflections from cleaved edges.

© 1987 Optical Society of America

## I. Introduction

Almost-periodic distributed feedback waveguides such as tapered, chirped, or phase-shifted gratings play an important role in improving the characteristics of a grating filter[1]–[5] and stabilizing a longitudinal mode oscillation for a DFB (distributed feedback) laser,[6]–[10] etc. It is important to derive a unified approach for almost-periodic distributed feedback waveguides with or without gain in the waveguides. So far two different methods of approximation to solve these problems have been reported, namely, utilizing Riccati differential equations for a contradirectional coupled wave interaction near the Bragg frequency[1] and numerical repetitions by integral forms of contradirectional coupled mode equations.[2],[3] In addition to these, one can derive an approximate form by dividing the waveguide into short segments in which the grating is assumed to be constant. If the fundamental matrices for each short segment are determined, one can obtain the characteristics of almost-periodic distributed feedback waveguides by multiplying these fundamental matrices in certain phase conditions of the gratings at the interface between two adjacent segments.

A similar method has been reported for analyzing hologram gratings,[11] but in this case the grating consists of multilayered dielectrics, so it is not suitable for waveguide analysis. Since the grating phase is not included in this analysis, there is a limitation for dividing nonuniform grating structures into small segments, that is, dividing locations of the gratings is limited to where the grating phase is equal to zero. Because of explicit grating phase relationships at the interfaces of each segment, it is easy to analyze phase-shifted grating characteristics; in particular it is useful to study the characteristics of a phase-controlled DFB laser which indicates fantastic dynamic single-mode characteristics under pulse modulations. On the other hand, the methods used in [Refs. 2]–[4] are difficult to handle this kind of problem.

In this paper we derive an F matrix (fundamental matrix) for nonuniform but almost-periodic distributed feedback waveguides and compare the results obtained by this method with others to justufy it.

## II. Derivation of F Matrix for a Periodic Distributed Feedback Waveguide

The basic coupled wave equations for a uniform grating with gain are

*β*is the difference between a propagation constant in the

*z*direction

*β*and the

*M*th Bragg frequency

*Mπ*/Λ of a grating period Λ,

*κ*is a coupling coefficient between forward and backward waves,

*g*is the gain per unit length, and

*ϕ*is a grating phase as shown in Fig. 1.

From the coupled wave equations given by Eq. (1), the F matrix for a periodic distributed feedback slab waveguide is obtained. In the case of periodic waveguides, *κ, ϕ*, Δ*β*, and *g* are independent of *z*; letting *A* = *E _{A}* amd

*B*=

*E*for complex amplitudes of forward and backward propagating waves, respectively, one gets the following relations:

_{B}*β*is a propagation constant in the

*z*direction.

Considering the relation given by Eq. (3), the solutions of Eq. (1) are as follows:

*c*

_{1}and

*c*

_{2}are arbitrary constants and Δ

*β′*and Γ

_{1,2}, are defined as follows:

*γ*

^{2}=

*κ*

^{2}− (Δ

*β′*)

^{2}.

Assuming the continuity conditions of forward and backward waves at the interfaces of *z* = 0 and *z* = *L* as shown in Fig. 2, the F matrix for *E _{A}*(0),

*E*(0), and

_{B}*E*(

_{A}*L*),

*E*(

_{B}*L*) is given by

*M*= 1, the grating phase

*ϕ*is equal to the phase at

*z*= 0.

The F matrix satisfies the reciprocity, that is, the determinant of the F matrix is unity, namely,

## III. F-Matrix Representation for a Nonuniform but Almost-Periodic Distributed Feedback Slab Waveguide

To begin, let us consider the conditions for dividing almost-periodic waveguides into *N* segments. To obtain the coupled wave equations in Eq. (1), the first order of perturbed term Δ*n′*^{2}(*x,z*) for distributed feedback regions is given by

*z*is

**F**is the Fourier transform of Δ

*n′*

^{2}(

*x,z*),

*S*is a spatial frequency, and

*δ*is the delta function. To get the above relation, one tacitly assumes that the distributed feedback region is infinite in the

*z*direction, but it is truncated by a finite length

*L*as shown in Fig. 3. Therefore Δ

*n*

^{2}(

*x,z*) is given by

*n′*

^{2}≒ Δ

*n*

^{2}is given by Λ ≪

*L*, and the F matrix in Eq. (7) includes this condition tacitly, namely, the condition of dividing the waveguide must satisfy

*k*indicates a

*k*th segment. The condition given by Eq. (9) is usually satisfied for slowly varying almost-periodic gratings. An almost-periodic distributed feedback waveguide can be divided into

*N*small segments provided the above condition is satisfied. In each segment corrugations are assumed to be periodic and the F matrix for the

*k*th segment is represented by (see Fig. 4)

*] as follows:*

^{k}#### A. Tapered Grating

A tapered grating can be characterized by the coupling coefficient *κ* which varies along the propagation direction *z* as shown in Fig. 5(a) and is generally given by

*κ*

_{0}is a reference coupling coefficient and

*T*(

_{a}*z*) is the tapered function, namely, linear, quadratic, etc. The grating phase at the interface between two adjacent small segments must satisfy

*ϕ*

^{k}^{−1}+ 2

*β*

_{B}L^{k}^{−1}in Eq. (13) represents the grating phase at the end of the (

*k*− 1)th segment, Eq. (13) gives the continuity condition of the grating phase between the (

*k*− 1)th and

*k*th segments.

#### B. Chirped Grating

A chirped grating can be characterized by changing the grating period Λ in the *z* direction. In other words, the Bragg frequency *β _{B}* changes in the

*z*direction as shown in Fig. 5(b). The general expression for a chirped grating is given by

*β*is the reference first-order Bragg frequency

_{BO}*π*/Λ

_{0}; here Λ

_{0}is a reference grating period and

*β*is the deviation from the reference Bragg frequency

_{Bs}*β*.

_{BO}The relations of deviation quantities from the reference, namely, between *β _{Bs}*(

*z*) and Λ

*(*

_{s}*z*) and the phase deviation

*ϕ*

_{s}_{(}

_{z}_{)}are

*k*− 1)th and

*k*th segments, and this also represents the approximation of the phase deviation

*ϕ*(

_{s}*z*).

#### C. Phase-Shifted Grating

A phase-shifted grating can be characterized by the collections of several periodic grating segments which are assembled in such a way that the grating phases between two adjacent segments are not continuous as shown in Fig. 5(c). In this example it consists of two segments and the condition given by Eq. (13) does not hold at the interface of each segment. The phase shift Δ*ϕ ^{k}* at the interface is given by

To justify the F-matrix approach, reflection and transmission characteristics for three different types of grating filter, such as linear tapered, linear chirped, and phase-shifted gratings are obtained. The coefficients of reflection **R** and transmission **T** are given by the F-matrix representation as follows:

*κ*is given by

*T*is a tapered coefficient and

*L*is a waveguide length. Figure 6(b) shows the results for linear chirped gratings when the Bragg frequency

*β*(

_{B}*z*) is given by

*V*is a chirped coefficient and

*L*is a waveguide length. Figure 6(c) shows the characteristics of the reflection coefficient

**| R|**vs the normalized frequency Δ

*βL*for a phase-shifted grating which consists of two grating segments of equal length and period, that is,

*L*

_{1}=

*L*

_{2}=

*L/*2 for various phase shift parameters Δ

*ϕ*. In this analysis the number of divisions

*N*is chosen as

*N*= 50 except for the phase-shifted grating.

In the above examples the following parameters are used for three types of grating, namely, a tapered coefficient *T* for linear tapered gratings, a chirped coefficient *V* for a linear chirped grating, and a phase shift Δ*ϕ* for phase-shifted gratings.

In addition to this filter analysis the threshold conditions of distributed feedback laser oscillations[6],[7] are investigated by the F-matrix approach. Let *r*_{1} and *r*_{2} be the reflection coefficients at the cleaved facets of a DFB laser and let the F matrix for the distributed feedback region of a DFB laser be [F]; then the characteristic F matrix for the DFB laser [F* _{R}*] is given as follows:

*g*and the normalized frequency Δ

_{th}L*βL*is given by

*κL*= 3, the reflection coefficients

*r*

_{1}= 0.533

^{2}and

*r*

_{2}= 0 at each facet, respectively, and three different grating phases such as

*ϕ*= 0,

*ϕ*= −

*π*/2, and

*ϕ*=

*π*/2. These results obtained by the F-matrix method are in good agreement with the results shown in [Ref. 6]. The definition of

*ϕ*in this paper, however, is different from that in [Ref. 6]. When the grating phase

*ϕ*is not integer multiplications of

*π*, single longitudinal mode oscillation conditions are obtained.[6]

## IV. Conclusion

A unified approach in terms of the F matrix for a nonuniform but almost-periodic distributed feedback slab waveguide is obtained, and this method is used to investigate the characteristics of an active device, such as a DFB laser, and passive devices, such as tapered, chirped, and phase-shifted gratings. The results obtained by this method are in good agreement with previously reported results obtained by other methods. Now w.e are in a situation to utilize this method to design various types of almost-periodic corrugated waveguide; in particular we are interested in designing a wideband light amplifier whose bandwidth is adjustable by virtue of corrugations. This will be reported elsewhere.[12]

The authors wish to thank S. Iida and T. Kambayashi of the Technological University of Nagaoka for their stimulating discussions and comments.

## Figures

## References

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