Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach

Makoto Yamada; Kyohei Sakuda

doi:10.1364/AO.26.003474

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

\begin{matrix} dA / dz & = κ exp [i (2 Δ β z - ϕ)] B + gA, \\ dB / dz & = κ exp [- i (2 Δ β z - ϕ)] A - gB, \end{matrix}

Δ β = β - β_{B} = β - M π / Λ,

where Δβ is the difference between a propagation constant in the z direction β and the Mth 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, Δβ, and g are independent of z; letting A = E_A amd B = E_B for complex amplitudes of forward and backward propagating waves, respectively, one gets the following relations:

\begin{matrix} E_{A} (z) & = A (z) exp (- i β z), \\ E_{B} (z) & = B (z) exp (+ i β z), \end{matrix}

where β is a propagation constant in the z direction.

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

\begin{matrix} E_{A} (z) & = [c_{1} exp (Γ_{1} z) + c_{2} exp (Γ_{2} z)] exp [(g - i β) z], \\ E_{B} (z) & = {exp [- i (2 Δ β^{'} z - ϕ)] / κ} [c_{1} Γ_{1} exp (Γ_{1} z) + c_{2} Γ_{2} \\ \times exp (Γ_{2} z)] exp [- (g - i β) z], \end{matrix}

where c₁ and c₂ are arbitrary constants and Δβ′ and Γ_1,2, are defined as follows:

\begin{matrix} Δ β^{'} & = Δ β + ig, \\ Γ_{1} & \begin{matrix} = i Δ β - γ, & Γ_{2} = i Δ β + γ, \end{matrix} \end{matrix}

where γ² = κ² − (Δβ′)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_B(0), and E_A(L), E_B(L) is given by

(\begin{matrix} E_{A} (0) \\ E_{B} (0) \end{matrix}) = [F] (\begin{matrix} E_{A} (L) \\ E_{B} (L) \end{matrix}) .

The elements of the F matrix in Eq. (6) are given as follows:

\begin{matrix} F_{11} & = [cosh (γ L) + i Δ β^{'} L sinh (γ L) / (γ L)] exp (i β_{B} L), \\ F_{12} & = - κ L sinh (γ L) exp [- i (β_{B} L + ϕ)] / (γ L), \\ F_{21} & = - κ L sinh (γ L) exp [i (β_{B} L + ϕ)] / (γ L), \\ F_{22} & = [cosh (γ L) - i Δ β^{'} L sinh (γ L) / (γ L)] exp [- i (β_{B} L)] . \end{matrix}

Since we are concerned with the first order of the grating, namely, 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,

| F | = F_{11} F_{22} - F_{12} F_{21} = 1 .

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 Δn′²(x,z) for distributed feedback regions is given by

Δ n^{' 2} (x, y) = 2 a_{1} (x) cos (2 π z / Λ),

and its Fourier transform with respect to z is

F [Δ n^{' 2} (x, z)] = 4 π a_{1} (x) [δ (S - 2 π / Λ) + δ (S - 2 π / Λ)],

where F is the Fourier transform of Δn′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 Δn²(x,z) is given by

Δ n^{2} (x, z) = {\begin{matrix} 2 a_{1} (x) cos (2 π z / Λ) & | z | \leq L / 2, \\ 0 & | z | > L / 2, \end{matrix}

and its Fourier transform is given by

\begin{matrix} F [Δ n^{2} (x, z)] & = 2 {sin [(S + 2 π / Λ) L / 2] / [S + 2 π / Λ] \\ + sin [(S - 2 π / Λ) L / 2] / [S - 2 π / Λ]} a_{1} (x) . \end{matrix}

Therefore the validity of the approximation condition, that is, Δn′² ≒ Δn² 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

Λ^{k} ≪ L^{k},

where the superscript k indicates a kth 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 kth segment is represented by (see Fig. 4)

[F^{k}] = [F (κ^{k}, Δ β^{k}, L^{k}, g^{k}, ϕ^{k})] .

Then the charcteristics of the F matrix for an almost-periodic waveguide can be given by multiplication of [F^k] as follows:

[F] = Π_{k = 1}^{N} [F^{k}] .

In the following paragraphs we. show the results for tapered, chirped, and phase-shifted gratings as typical examples of almost-periodic distributed feedback waveguides. These gratings are shown schematically in Figs. 5(a), (b), and (c), respectively.

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 (z) = κ_{0} [1 + T_{a} (z)],

where κ₀ 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} = ϕ^{k - 1} + 2 β_{B} L^{k - 1} .

Since ϕ^k⁻¹ + 2β_BL^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 kth 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_{B} (z) = β_{B O} + β_{Bs} (z),

where β_BO is the reference first-order Bragg frequency π/Λ₀; here Λ₀ is a reference grating period and β_Bs is the deviation from the reference Bragg frequency 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

\frac{Λ_{s} (z)}{Λ_{0}} = - \frac{β_{Bs} (z)}{β_{BO}},

ϕ_{s} = 2 \int β_{Bs} (ζ) d ζ .

In this case the piecewise F-matrix approximation also requires the following phase relation at the interface between the two adjacent segments, namely,

ϕ^{k} = ϕ^{k - 1} + 2 β_{B} L^{k - 1} .

Equation (17) shows the continuity condition of the grating at the interface between the (k − 1)th and kth 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 Δϕ^k at the interface is given by

Δ ϕ^{k} = ϕ^{k} - ϕ^{k - 1} - 2 β_{B} L^{k - 1} .

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:

T = 1 / F_{11},

R = F_{21} / F_{11} .

Figure 6(a) shows the results for linear tapered gratings when the coupling coefficient κ is given by

κ (z) = κ_{0} [1 + T (z - L / 2) / L],

where 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

β_{B} (z) = β_{BO} + 2 V (z - L / 2) / L^{2},

where 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 ΔβL for a phase-shifted grating which consists of two grating segments of equal length and period, that is, L₁ = L₂ = L/2 for various phase shift parameters Δϕ. 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 Δϕ 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 r₁ and r₂ 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:

[F_{R}] = 1 / {(1 - r_{1}) (1 - r_{2})}^{1 / 2} (\begin{array}{l} 1, & - {(r_{1})}^{1 / 2} \\ - {(r_{1})}^{1 / 2}, & 1 \end{array}) [F] (\begin{array}{l} 1, & {(r_{2})}^{1 / 2} \\ {(r_{1})}^{1 / 2}, & 1 \end{array}) .

The relation between the threshold gain g_thL and the normalized frequency ΔβL is given by

F_{R 11} = 0 .

Figure 7 shows the results of Eq. (24) for the following parameters: the normalized length of a uniform periodic distributed feedback waveguide κL = 3, the reflection coefficients r₁ = 0.533² and r₂ = 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

Fig. 1 Definition of grating phase for the first-order grating.

Download Full Size | PDF

Fig. 2 Schematic diagram of a periodic distributed feedback waveguide.

Download Full Size | PDF

Fig. 3 Corrugations of finite length L.

Download Full Size | PDF

Fig. 4 Schematic diagram of N divisions of an almost-periodic distributed feedback waveguide and parameters in the kth segment: k = 1,2,…, N, i.e., κ^k is a coupling coefficient, β_Bk is the Bragg frequency, ϕ^k is a grating phase, g^k is the gain, and $E_{A}^{k}$ and $E_{B}^{k}$ are forward and backward electric field amplitudes, respectively.

Download Full Size | PDF

Fig. 5 Some examples of aperiodic distributed feedback waveguides, i.e., (a) tapered grating, (b) chirped grating, and (c) phase-shifted grating.

Download Full Size | PDF

Fig. 6 Characteristics of reflection coefficient |R|² vs normalized frequency ΔβL for three different types of aperiodic distributed feedback waveguide: (a) linear tapered grating, (b) linear chirped grating, and (c) phase-shifted grating.

Download Full Size | PDF

Fig. 7 Threshold characteristics for the DFB laser, threshold gain g_thL vs normalized frequency ΔβL.

Download Full Size | PDF

References

1. H. Kogelnik, “Filter Response of Non-Uniform Almost-Periodic Structure,” Bell Syst. Tech. J. 55, 109 (1976).

2. K. O. Hill, “Aperiodic Distributed-Parameter Waveguides for Integrated Optics,” Appl. Opt. 13, 1853 (1974). [CrossRef] [PubMed]

3. M. Matsuhara and K. O. Hill, “Optical-Waveguide Band-Rejection Filters: Design,” Appl. Opt. 13, 2886 (1974). [CrossRef] [PubMed]

4. M. Matsuhara, K. O. Hill, and A. Watanabe, “Optical-Waveguide Filters: Synthesis,” J. Opt. Soc. Am. 65, 804 (1975). [CrossRef]

5. S. H. Kim and C. G. Fostand, “Tunable Narrow-Band Thin Film Waveguide Grating Filters,” IEEE J. Quantum Electron. QE-15, 1405 (1979).

6. Y. Itaya, T. Matuoka, K. Kuroiwa, and T. Ikegami, “Longitudinal Mode Behaviors of 1.5 μm Range GalnAsP/InP Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-20, 230 (1984). [CrossRef]

7. K. Ukata, S. Akiba, K. Sakai, and Y. Matsushima, “Effect of Mirror Facets on Lasing Characteristics of Distributed Feedback InGaAsP/InP Laser Diode at 1.5 μm Range,” IEEE J. Quantum Electron. QE-20, 236 (1984).

8. W. Streifer, R. D. Burham, and D. R. Scifres, “Effect of External Reflectors on Longitudinal Modes of Distributed Feedback Lasers,” IEEE J. Quantum Electron. QE-11, 154 (1975). [CrossRef]

9. A. Suzuki and T. Tada, “Theory and Experiment on Distributed Feedback Lasers with Chirped Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 236, 532 (1981).

10. F. Koyama and Y. Suematsu, “Phase-Controlled Active-Distributed-Reflector Laser,” IECE Jpn. OQE84, 67 (1984), in Japanese.

11. D. Kermisch, “Nonuniform Sinusoidally Modulated Dielectric Gratings,” J. Opt. Soc. of Am. 59, 1409 (1969). [CrossRef]

12. M. Yamada and K. Sakuda, “Adjustable Gain and Bandwidth Light Amplifiers in Terms of Distributed Feedback Structures,” J. Opt. Soc. Am. A4, 69 (1987), to be published. [CrossRef]

Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach

Abstract

I. Introduction

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

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

A. Tapered Grating

B. Chirped Grating

C. Phase-Shifted Grating

IV. Conclusion

Figures

References

Cited By

Figures (7)

Equations (28)

Applied Optics