Perturbation Analysis with Approximate Integration for Propagation Mode in Two-Dimensional Two-Slab Waveguides

Abstract

On the basis of perturbation expansion from a gapless system, we calculate the propagation constant and propagation mode wave function in two-dimensional two-slab waveguides with a core gap small enough that there is only one propagation mode. We also perform calculations without the approximation for comparison. Our result shows that first-order perturbation contains the first-order Taylor expansion of (core gap)/(core width), and when the integration of the perturbation is suitably approximated, the result of the first-order perturbation is the same as that of the first-order Taylor expansion of (core gap)/(core width).

1. Introduction

When monochromatic light enters into one of the cores of a waveguide array in which each core has the same width and the same gap, the light propagates to adjacent cores in turn, and its trajectory becomes V-shaped. Such optical behavior has been theoretically and numerically analyzed. Theoretical analysis includes matrix method [1], the coupled mode equation, and the coupled power equation [2–9]. In the coupling mode equations, it is assumed that there is at least one propagation mode for each core and that all of the propagation modes are independent from each other.

This assumption is invalid in the case that the number of independent propagation modes is less than the number of cores. Such a situation can occur when the core gap becomes small enough that the index distribution is approximated by a gapless core. When such a situation occurs, analysis based on the coupled mode or the coupled power equation is invalid. Our interest is finding a simple method to analyze an optical behavior in such a situation.

An optical behavior for parallel slab waveguides with small core gap was theoretically analyzed using even and odd mode analysis, which was called supermode analysis later [3–5]. In this analysis, the exact solution for the Maxwell’s equations with the index distribution for two parallel slab waveguides, which becomes essentially one for Schrödinger equation for double finite wells in the field of quantum mechanics, is used to analyze the optical behavior. However, the method using supermodes is not useful when the number of cores increases, because the solution must be individually expressed in every core and clad.

In the field of quantum mechanics, perturbation theory is widely used as an approximation [10, 11]. In the field of electromagnetics and optics, it is also explained in [12], and applied to explain bend losses for a fiber [13, 14]. Coupled mode equation is also based on the perturbation theory [4–7]. The situation that we are interested in is suitable for using the perturbation theory.

In this paper, we calculate the supermode for a two-dimensional two-slab waveguide that has a small core gap so that the optical system has only one propagation mode, using first-order perturbation from gapless system. We also calculate the same without approximation for comparison. In this way, we show that first-order perturbation with suitable approximate integration and the first-order Taylor expansion of (core gap)/(core width) give the same result. Furthermore, we discuss higher-order relationships between perturbation with suitable approximate integration and Taylor expansion.

2. Model

For our optical model of two cores with a gap, the spatial configuration of the cores and clads are shown in the left panel of Figure 1. There exist two cores with the same index 𝑛₁ and the same core width 𝑑, and clads with index 𝑛₂, where 𝑛₁ and 𝑛₂ are constants, and 𝑛₁>𝑛₂. The core width 𝑑, core gap 𝑤, and indices 𝑛₁ and 𝑛₂ are chosen so that there is only one propagation mode. The core is configured in 𝑤/2<𝑥<𝑤/2+𝑑 and −𝑤/2−𝑑<𝑥<−𝑤/2. In other words, the index distribution 𝑛(𝑥) is set as 𝑛(𝑥)=𝑛₂,|𝑥|<𝑤2,𝑛₁,𝑤2<|𝑥|<𝑤2𝑛+𝑑,₂,𝑤2+𝑑<|𝑥|, (1) which is constant except for 𝑥=±𝑤/2 and 𝑥=±(𝑤/2+𝑑) and is illustrated in the right panel of Figure 1. The index distribution is constant in the 𝑦 and 𝑧 directions. The 𝑧 direction is defined as the direction of light propagation, and 𝜔 is used to denote the angular frequency of the monochromatic light.

(a)

(b)

Figure 1 

Optical system configuration (a) and its index distribution on 𝑧=const. line (b).

Since the index distribution has no variation in the 𝑦 direction, electromagnetic fields do not depend on 𝑦. For the electric field 𝐄(𝑡,𝑥,𝑧) and the magnetic field 𝐇(𝑡,𝑥,𝑧), the time variable 𝑡 can be separated from the spatial variables as 𝐄(𝑡,𝑥,𝑧)=exp(𝑖𝜔𝑡)𝐞(𝑥,𝑧) and 𝐇(𝑡,𝑥,𝑧)=exp(𝑖𝜔𝑡)𝐡(𝑥,𝑧), because we are interested in monochromatic light with angular frequency 𝜔. Then, from Faraday’s and Ampére-Maxwell’s laws, components of the electric and magnetic fields have the following relationships: ℎ_𝑥()𝑖𝑥,𝑧=−𝜔𝜇₀𝜕_𝑧𝑒_𝑦(),ℎ𝑥,𝑧_𝑦()=𝑖𝑥,𝑧𝜔𝜇₀𝜕_𝑧𝑒_𝑥(𝑥,𝑧)−𝜕_𝑥𝑒_𝑧(,ℎ𝑥,𝑧)_𝑧()=𝑖𝑥,𝑧𝜔𝜇₀𝜕_𝑥𝑒_𝑦,𝑒_𝑥()=𝑖𝑥,𝑧𝜔𝜀₀𝑛(𝑥)²𝜕_𝑧ℎ_𝑦(),𝑒𝑥,𝑧_𝑦()𝑖𝑥,𝑧=−𝜔𝜀₀𝑛(𝑥)²𝜕_𝑧ℎ_𝑥(𝑥,𝑧)−𝜕_𝑥ℎ_𝑧(,𝑒𝑥,𝑧)_𝑧()𝑖𝑥,𝑧=−𝜔𝜀₀𝑛(𝑥)²𝜕_𝑥ℎ_𝑦(𝑥,𝑧). (2)(3)(4)(5)(6)(7) It is straightforward to see that (2) to (7) are also solutions to the chargeless Gauss’s law for electric and magnetic fields, which are expressed by the rest of Maxwell’s equations. Equations (3) and (6) can be written as the second-order differential equations 𝜕²_𝑥+𝜕²_𝑧+𝑛(𝑥)²₋𝜆²ℎ_𝑦()𝜕𝑥,𝑧=0,_2𝑥+𝜕²_𝑧+𝑛(𝑥)²⁻𝜆²𝑒_𝑦(𝑥,𝑧)=0, (8)(9) by using (5) and (7) and (2) and (4), respectively, where the derivative of 𝑛(𝑥) is neglected in (9) because 𝑛(𝑥) is constant except at some discrete points. Since (8) and (9) are basically the same differential equation, we introduce the field Ψ(𝑥,𝑧) instead of ℎ_𝑦(𝑥,𝑧) and 𝑒_𝑦(𝑥,𝑧). Equations (8) and (9) are usually solved by separation of variables: Ψ(𝑥,𝑧)=𝜓(𝑥)𝜙(𝑧). (10) So 𝜕²_𝑧𝜙(𝑧)=₋𝑒𝜆²𝜙(𝑧),−𝜕²_𝑥−𝑛(𝑥)²⁻𝜆²𝜓(𝑥)=⁻𝑒𝜆²𝜓(𝑥), (11)(12) where 𝑒 is a constant. In (11), 𝑒 obviously cannot be positive if the region of 𝑧 is −∞<𝑧<∞, because positive 𝑒 makes 𝜙(𝑧) diverge in 𝑧→∞ or 𝑧→−∞. For any negative 𝑒, there exists a positive 𝛽 such that 𝑒=−𝛽², (13) where 𝛽 is the propagation constant, and the solution of (11) represents propagation in the positive or negative 𝑧 direction with 𝛽=[−𝑒]^1/2.

Equation (12) is essentially the one-dimensional Schrödinger equation [10, 11], which is an eigenfunction problem, where 𝑒 is the eigenvalue and 𝜓 is the eigenfunction of the operator: −𝜕²_𝑥−𝑛(𝑥)²₋𝜆². (14) Equation (12) has nontrivial solutions for special values 𝑒 in −𝑛²₁<𝑒<−𝑛²₂ (propagation mode solution), and two nontrivial solutions exist for each 𝑒 in −𝑛²₂<𝑒 (radiation mode solution).

In the following section, we solve (12) by using first-order perturbation with a suitable approximation integration in order to calculate the difference of its propagation constant and propagation mode solution from that of a gapless system. In Section 4, we solve (12) without approximation in order to obtain the first-order Taylor expansion of 𝑤/𝑑, and then we compared the result with those obtained in Section 3. The final section is devoted to summary and discussion.

3. Solution with First-Order Perturbation

To solve (12), we decompose the square of the index distribution −𝑛(𝑥)² into the square of the index distribution for 𝑤=0 (−𝑛₀(𝑥)²) and the rest (−𝑛_𝑝(𝑥)²) as −𝑛(𝑥)²=−𝑛₀(𝑥)²−𝑛_𝑝(𝑥)², (15) where 𝑛₀(𝑥)=𝑛₁,𝑤𝑥−2𝑛<𝑑,₂𝑤,𝑑<𝑥−2,𝑛_𝑝(𝑥)=𝑛²₁−𝑛²₂^1/2𝑤,−2𝑤−𝑑<𝑥<2−𝑛−𝑑,₂₁−𝑛²₂^1/2𝑤,−2𝑤<𝑥<2,0,otherwise. (16) Figure 2 shows the distribution of 𝑛₀(𝑥) (solid line) and 𝑛_𝑝(𝑥) (dotted line) on the 𝑧=const. line. The index distribution of 𝑛₀ represents the core that is laid on 𝑤/2−𝑑<𝑥<𝑤/2+𝑑. Instead of directly solving (12), we use the complete set of solutions for −𝜕²_𝑥−𝑛₀(𝑥)²₋𝜆²𝑣(𝑥)=₋𝑒𝜆²𝑣(𝑥), (17) in order to solve (12) perturbatively. All independent solutions for (17) are 𝑉₀(𝑥)=𝑣₀𝑃cos⁰(2𝑥−𝑤)𝑑,𝑤𝑥−2𝑣<𝑑,₀cos2𝑃₀−𝑄×exp⁰(|2𝑥−𝑤|−2𝑑)𝑑𝑤,𝑑<𝑥−2,𝑉^𝑠_𝑒(𝑥)=𝑣^𝑠_𝑒𝑃cos^𝑒(2𝑥−𝑤)𝑑,𝑤𝑥−2𝑣<𝑑,_𝑠𝑒cos2𝑃_𝑒𝑄×cos^𝑒(|2𝑥−𝑤|)−2𝑑𝑑−𝑃_𝑒𝑄_𝑒sin2𝑃_𝑒𝑄×sin^𝑒(|2𝑥−𝑤|)−2𝑑𝑑𝑤,𝑑<𝑥−2,𝑉^𝑎_𝑒(𝑥)=𝑣^𝑎_𝑒𝑃sin^𝑒(2𝑥−𝑤)𝑑,𝑤𝑥−2𝑣<𝑑,_𝑎𝑒(𝑥)×signsin2𝑃_𝑒𝑄×cos^𝑒(|2𝑥−𝑤|−2𝑑)𝑑+𝑃_𝑒𝑄_𝑒cos2𝑃_𝑒𝑄×sin^𝑒(|2𝑥−𝑤|−2𝑑)𝑑𝑤,𝑑<𝑥−2, (18)(19)(20) where 𝑃₀=𝑛²₁+𝑒₀^1/2𝑑2𝜆,𝑄₀=−𝑛²₂+𝑒₀^1/2𝑑⁻2𝜆,𝑃_𝑒=𝑛²₁+𝑒^1/2𝑑⁻2𝜆,𝑄_𝑒=𝑛²₂+𝑒^1/2𝑑⁻2𝜆, (21)(22) and 𝑒₀ (or the pair of 𝑃₀ and 𝑄₀) in (21) satisfy 𝑄₀=𝑃₀tan2𝑃₀,𝑃²₀+𝑄²₀=𝑉², (23)(24) where 𝑛𝑉=²¹−𝑛²₂^1/2𝑑⁻2𝜆. (25) In (22), 𝑒 is in the region −𝑛²₂<𝑒<0. (26)

Figure 2 

Decomposed index distributions. Solid line represents 𝑛0(𝑥), and dotted line represents 𝑛𝑝(𝑥).

𝑉₀ represents the propagation mode, and 𝑉^𝑠_𝑒 and 𝑉^𝑎_𝑒 represent symmetric and antisymmetric radiation modes according to 𝑒, respectively. For the reasons stated in the introduction, we are interested in the case that the propagation mode of the optical system has only one independent mode. Therefore, we examine the case where the normalized propagation mode solution of (18) is 𝑉₀ only.

Here, let us define the inner product of fields 𝑓⋅𝑔 as 1𝑓⋅𝑔=𝑑^∞_−∞𝑑𝑥𝑓(𝑥)𝑔(𝑥). (27) Then, 𝑣₀, 𝑣^𝑠_𝑒, and 𝑣^𝑎_𝑒 are defined so that 𝑉₀, 𝑉^𝑠_𝑒, and 𝑉^𝑎_𝑒 are normalized to 𝑉₀⋅𝑉₀𝑉=1,_𝑝𝑒⋅𝑉^𝑞_𝑓=𝛿^𝑝𝑞𝛿𝑒−𝑓, (28) where 𝑝,𝑞∈{𝑠,𝑎} (note that 𝑉₀,𝑉^𝑠_𝑒 and 𝑉^𝑎_𝑒 can be regarded as eigenfunctions of the eigenequation (17), and their eigenvalues are 𝑒₀, 𝑒, and 𝑒, respectively. In addition to the relationships in (28), 𝑉₀⋅𝑉^𝑠_𝑒=0 and 𝑉₀⋅𝑉^𝑎_𝑒=0 [10, 11]). For instance, 𝑣₀ is 𝑣₀=2𝑄₀2𝑄₀+1^1/2. (29) In perturbation theory, the propagation mode solution for (12) is determined by using (18) to (20), which results in −𝜕²_𝑥−𝑛₀(𝑥)²₋𝜆²−𝑛_𝑝(𝑥)²₋𝜆²𝑉₀(𝑥)(𝑥=𝑒+Δ𝑉)₀+Δ𝑒⁻𝜆²𝑉₀(𝑥)+Δ𝑉(𝑥). (30) The results of the first-order perturbation for Δ𝑒 and Δ𝑉 are as follows [10, 11]: Δ𝑒=−𝑉₀⋅𝑛²_𝑝𝑉₀𝑉₀,𝑉⋅Δ𝑉=0,_𝑝𝑒𝑉⋅Δ𝑉=^𝑝𝑒⋅𝑛²_𝑝𝑉₀𝑒−𝑒₀, (31)(32)(33) where 𝑛²_𝑝𝑉₀(𝑥)=𝑛_𝑝(𝑥)²𝑉₀(𝑥). (34) In (33), 𝑉^𝑠_𝑒⋅Δ𝑉 and 𝑉^𝑎_𝑒⋅Δ𝑉 mean the 𝑉^𝑠_𝑒 and 𝑉^𝑎_𝑒 components of Δ𝑉, respectively. Note that for any field 𝑓(𝑥), the inner product with 𝑛²_𝑝𝑉₀ is 𝑓⋅𝑛²_𝑝𝑉₀=𝑛²₁−𝑛²₂𝑑₋₍₎^{(𝑤/2)−𝑑}_{𝑤/2−𝑑}𝑑𝑥𝑓(𝑥)𝑉₀(𝑥)−₋₍₎^𝑤/2_𝑤/2𝑑𝑥𝑓(𝑥)𝑉₀(𝑥,=𝑛₂₁)−𝑛²₂𝑓𝑤2−𝑑𝑉₀𝑤2𝑤−𝑑−𝑓2𝑉₀𝑤2_𝑤=0𝑤𝑑𝑤+𝑜𝑑², (35)(36) which means that it contains the first-order Taylor expansion of 𝑤/𝑑. In the deformation from (35) to (36), we use the Euler method for the integration. Using (18) through (20), we obtain the first-order Taylor expansion of 𝑤/𝑑 for Δ𝑒, 𝑉^𝑠_𝑒⋅Δ𝑉, and 𝑉^𝑎_𝑒⋅Δ𝑉 from (31) and (33), respectively, as follows: Δ𝑒=𝑣²₀𝑛²₁−𝑛²₂1−cos²2𝑃₀=2𝑄₀𝑛²₁−𝑛²₂sin²2𝑃₀2𝑄₀𝑤+1𝑑,𝑉^𝑠_𝑒⋅Δ𝑉=−𝑣₀𝑣^𝑒_𝑠𝑛²₁−𝑛²₂1−cos2𝑃₀cos2𝑃_𝑒𝑒−𝑒₀𝑤𝑑,𝑉^𝑎_𝑒⋅Δ𝑉=−𝑣₀𝑣^𝑒_𝑠𝑛²₁−𝑛²₂sin2𝑃₀cos2𝑃_𝑒𝑒−𝑒₀𝑤𝑑. (37)(38)(39)

4. Solution without Approximation

When the exact eigenvalue of the propagation mode of (12) is 𝑒_𝑤, its eigenfunction is 𝑊(𝑥)=𝑤₀cosh2𝑄_𝑤𝑥𝑑,|𝑥|<𝑤2,𝑤₀𝑄cosh^𝑤𝑤𝑑𝑃×cos^𝑤(2|𝑥|−𝑤)𝑑+𝑄_𝑤𝑃_𝑤𝑄sinh^𝑤𝑤𝑑𝑃×sin^𝑤(2|𝑥|−𝑤)𝑑,𝑤2<|𝑥|<𝑤2𝑤+𝑑,₀𝑄cosh^𝑤𝑤𝑑cos2𝑃_𝑤+𝑄_𝑤𝑃_𝑤𝑄sinh^𝑤𝑤𝑑×sin2𝑃_𝑤−𝑄×exp^𝑤(2|𝑥|−2𝑑−𝑤)𝑑,𝑤2+𝑑<|𝑥|, (40) where 𝑃_𝑤=𝑛²₁+𝑒_𝑤^1/2𝑑⁻2𝜆,𝑄_𝑤=−𝑛²₂+𝑒_𝑤^1/2𝑑⁻2𝜆. (41)𝑃_𝑤 and 𝑄_𝑤 satisfy the relationships 𝑃_𝑤𝑃_𝑤sin2𝑃_𝑤−𝑄_𝑤cos2𝑃_𝑤=𝑉²−𝑄exp^𝑤𝑤𝑑𝑄sinh^𝑤𝑤𝑑sin2𝑃_𝑤,𝑃²_𝑤+𝑄²_𝑤=𝑉², (42) such that 𝑊(𝑥) and its derivative are continuous for any 𝑥. Note that 𝑊₀(𝑥) agrees with 𝑉₀(𝑥) in the limit of 𝑤/𝑑=0, and 𝑤₀ becomes 𝑣₀ when 𝑤/𝑑=0. When 𝑤/𝑑≠0, the relationships between 𝑊(𝑥) and Δ𝑉(𝑥) and between 𝑒_𝑤 and Δ𝑒 become the following: 𝑊(𝑥)=𝑉₀(𝑥)𝑒+Δ𝑉,_𝑤=𝑒₀+Δ𝑒. (43)(44) Substituting (44) into (42) by using (41), and applying (23), we obtain 2𝑃₀sin2𝑃₀+cos2𝑃₀𝑉²Δ𝑒2𝑛²₁−𝑛²₂sin2𝑃₀cos2𝑃₀[1+𝑂(Δ𝑒)]=𝑉²𝑄₀sin2𝑃₀𝑤𝑑𝑤1+𝑂𝑑,Δ𝑒. (45) Comparing the lowest order of Δ𝑒 and 𝑤/𝑑 in (45) using (23), we obtain (37) from (45). Because of (13), the first-order perturbation and the first-order Taylor expansion of 𝑤/𝑑 give the same propagation constant for a two-core optical system with gap 𝑤.

𝑉^𝑠_𝑒⋅𝑊 and 𝑉^𝑎_𝑒⋅𝑊 are directly derived, which results in 𝑉^𝑠_𝑒⋅𝑊₀=𝑣₀𝑣^𝑠_𝑒𝑄_𝑤1+cosh2𝑃_𝑒𝑤/𝑑sin𝑄_𝑤𝑤/𝑑+𝑃_𝑒sinh2𝑃_𝑒𝑤/𝑑cos𝑄_𝑤𝑤/𝑑2𝑃²_𝑒+𝑄²_𝑤+𝑃_𝑤cosh𝑄_𝑤𝑤/𝑑cos2𝑃_𝑤+𝑄_𝑤sinh𝑄_𝑤𝑤/𝑑sin2𝑃_𝑤𝑃_𝑤𝑄²_𝑒+𝑄²_𝑤×1+2cos2𝑄_𝑒𝑤/𝑑−exp2𝑄_𝑤𝑤/𝑑𝑄_𝑤cos2𝑃_𝑒−𝑃_𝑒sin2𝑃_𝑒2+𝑄²_𝑒cos2𝑃_𝑒+𝑃_𝑒𝑄_𝑤sin2𝑃_𝑒sin2𝑄_𝑒𝑤/𝑑𝑄_𝑒−1𝑃²_𝑒−𝑃²_𝑤𝑃_𝑤𝑄cosh^𝑤𝑤𝑑1+cos2𝑃_𝑤𝑤/𝑑sin2𝑃_𝑤cos2𝑃_𝑒2−sin2𝑃_𝑤𝑤/𝑑cos2𝑃_𝑤cos2𝑃_𝑒2−𝑃_𝑒𝑃_𝑤1+cos2𝑃_𝑤𝑤/𝑑cos2𝑃_𝑤cos2𝑃_𝑒2+sin2𝑃_𝑤𝑤/𝑑sin2𝑃_𝑤sin2𝑃_𝑒2−sin2𝑃_𝑒𝑤𝑑+𝑄_𝑤𝑄sinh^𝑤𝑤𝑑1+cos2𝑃_𝑒𝑤/𝑑1−cos2𝑃_𝑤cos2𝑃_𝑒2−sin2𝑃_𝑤𝑤/𝑑sin2𝑃_𝑤sin2𝑃_𝑒2+cos2𝑃_𝑒𝑤𝑑−cos2𝑃_𝑤𝑤𝑑−1+cos2𝑃_𝑤𝑤/𝑑sin2𝑃_𝑤sin2𝑃_𝑒2+sin2𝑃_𝑤𝑤/𝑑cos2𝑃_𝑤cos2𝑃_𝑒2−𝑃_𝑤sin2𝑃_𝑤𝑤𝑑−𝑄_𝑒sin2𝑄_𝑒𝑤𝑑𝑄cosh^𝑤𝑤𝑑−𝑄_𝑤cos2𝑄_𝑒𝑤𝑑−cos2𝑃_𝑤𝑤𝑑𝑄sinh^𝑤𝑤𝑑cos2𝑃_𝑤cos2𝑃_𝑒×2𝑄²_𝑒−𝑃²_𝑤⁻¹−𝑃_𝑒sin2𝑃_𝑤𝑤𝑑+𝑃_𝑤𝑄_𝑒sin2𝑄_𝑒𝑤𝑑𝑄cosh^𝑤𝑤𝑑−𝑄_𝑤𝑃_𝑒𝑃_𝑤cos2𝑄_𝑒𝑤𝑑−cos2𝑃_𝑤𝑤𝑑𝑄sinh^𝑤𝑤𝑑sin2𝑃_𝑤sin2𝑃_𝑒×2𝑄²_𝑒−𝑃²_𝑤⁻¹−𝑄_𝑤sin2𝑃_𝑤𝑤𝑑+𝑄_𝑒𝑃_𝑤sin2𝑄_𝑒𝑤𝑑𝑄sinh^𝑤𝑤𝑑+cos2𝑄_𝑒𝑤𝑑−cos2𝑃_𝑤𝑤𝑑𝑄cosh^𝑤𝑤𝑑sin2𝑃_𝑤cos2𝑃_𝑒×2𝑄²_𝑒−𝑃²_𝑤⁻¹+𝑃_𝑒𝑃_𝑤1𝑃_𝑤sin2𝑃_𝑤𝑤𝑑+1𝑄_𝑒sin2𝑄_𝑒𝑤𝑑𝑄sinh^𝑤𝑤𝑑+𝑃_𝑒cos2𝑄_𝑒𝑤𝑑−cos2𝑃_𝑤𝑤𝑑𝑄cosh^𝑤𝑤𝑑cos2𝑃_𝑤sin2𝑃_𝑒×2𝑄²_𝑒−𝑃²_𝑤⁻¹,𝑉^𝑎_𝑒⋅𝑊₀=𝑣^𝑎_𝑒𝑣₀−𝑃_𝑒1−cos2𝑃_𝑒𝑤/𝑑cosh𝑄_𝑤𝑤/𝑑+𝑄_𝑤sinh𝑄_𝑤𝑤/𝑑sin2𝑃_𝑒𝑤/𝑑2𝑃²_𝑒+𝑄²_𝑤+𝑃_𝑤cosh𝑄_𝑤𝑤/𝑑cos2𝑃_𝑤+𝑄_𝑤sinh𝑄_𝑤𝑤/𝑑sin2𝑃_𝑤𝑃_𝑤𝑄²_𝑒+𝑄²_𝑤×1−cos2𝑄_𝑒𝑤/𝑑𝑄_𝑤sin2𝑃_𝑒+𝑃_𝑒cos2𝑃_𝑒2+𝑄²_𝑒sin2𝑃_𝑒−𝑃_𝑒𝑄_𝑤cos2𝑃_𝑒sin2𝑃_𝑒𝑤/𝑑4𝑄_𝑒+1−cos2𝑃_𝑒𝑤/𝑑2𝑃_𝑤cosh𝑄_𝑤𝑤/𝑑𝑃_𝑒1−cos2𝑃_𝑤cos2𝑃_𝑒−𝑃_𝑤sin2𝑃_𝑤sin2𝑃_𝑒𝑃_𝑤𝑃²_𝑒−𝑃²_𝑤+1−cos2𝑃_𝑒𝑤/𝑑2𝑄_𝑤sinh𝑄_𝑤𝑤/𝑑𝑃_𝑒cos2𝑃_𝑤sin2𝑃_𝑒−𝑃_𝑤sin2𝑃_𝑤cos2𝑃_𝑒𝑃_𝑤𝑃²_𝑒−𝑃²_𝑤+sin2𝑃_𝑒𝑤/𝑑2𝑃_𝑤cosh𝑄_𝑤𝑤/𝑑𝑃_𝑤sin2𝑃_𝑤cos2𝑃_𝑒−𝑃_𝑒cos2𝑃_𝑤sin2𝑃_𝑒𝑃_𝑤𝑃²_𝑒−𝑃²_𝑤+sin2𝑃_𝑒𝑤/𝑑2𝑄_𝑤sinh𝑄_𝑤𝑤/𝑑𝑃_𝑤1−cos2𝑃_𝑤cos2𝑃_𝑒−𝑃_𝑒sin2𝑃_𝑤sin2𝑃_𝑒𝑃_𝑤𝑃²_𝑒−𝑃²_𝑤−𝑄cosh^𝑤𝑤𝑑𝑃_𝑒cos2𝑃_𝑤cos2𝑃_𝑒−cos2𝑃_𝑤(𝑑−𝑤)𝑑cos2𝑃_𝑒(𝑑−𝑤)𝑑+𝑃_𝑤sin2𝑃_𝑤sin2𝑃_𝑒−sin2𝑃_𝑤(𝑑−𝑤)𝑑sin2𝑃_𝑒(𝑑−𝑤)𝑑×2𝑃²_𝑒−𝑃²_𝑤⁻¹+𝑄_𝑤𝑄sinh^𝑤𝑤𝑑𝑃_𝑒cos2𝑃_𝑤sin2𝑃_𝑒−cos2𝑃_𝑤(𝑑−𝑤)𝑑sin2𝑃_𝑒(𝑑−𝑤)𝑑−𝑃_𝑤sin2𝑃_𝑤cos2𝑃_𝑒−sin2𝑃_𝑤(𝑑−𝑤)𝑑cos2𝑃₀(𝑑−𝑤)𝑑×2𝑃_𝑤𝑃²_𝑒−𝑃²_𝑤⁻¹+𝑄sinh^𝑤𝑤𝑑𝑃_𝑒sin2𝑃_𝑤𝑤𝑑+𝑃_𝑤𝑄_𝑒sin2𝑄_𝑒𝑤𝑑cos2𝑃_𝑤cos2𝑃_𝑒−𝑄_𝑤𝑃_𝑤𝑃_𝑤sin2𝑃_𝑤𝑤𝑑+𝑄_𝑒sin2𝑄_𝑒𝑤𝑑sin2𝑃_𝑤sin2𝑃_𝑒×2𝑄²_𝑒−𝑃²_𝑤⁻¹+𝑄cosh^𝑤𝑤𝑑𝑄_𝑒𝑃_𝑤sin2𝑃_𝑤𝑤𝑑+𝑄_𝑒sin2𝑄_𝑒𝑤𝑑cos2𝑃_𝑤sin2𝑃_𝑒−𝑃_𝑒𝑄_𝑒sin2𝑃_𝑤𝑤𝑑+𝑃_𝑤sin2𝑄_𝑒𝑤𝑑sin2𝑃_𝑤cos2𝑃_𝑒×2𝑄_𝑒𝑄²_𝑒−𝑃²_𝑤⁻¹. (46)

From (46), we obtain the first-order Taylor expansion of 𝑤/𝑑 of 𝑉^𝑠_𝑒⋅𝑊 and 𝑉^𝑎_𝑒⋅𝑊 after a straightforward but lengthy calculation:𝑉^𝑠_𝑒⋅𝑊=−𝑣₀𝑣^𝑠_𝑒𝑛²₁−𝑛²₂1−cos2𝑃₀cos2𝑃_𝑒𝑒−𝑒₀𝑤𝑑,𝑉^𝑎_𝑒⋅𝑊=−𝑣₀𝑣^𝑎_𝑒𝑛²₁−𝑛²₂sin2𝑃₀cos2𝑃_𝑒𝑒−𝑒₀𝑤𝑑. (47)(48)𝑉^𝑠_𝑒⋅Δ𝑉 and 𝑉^𝑎_𝑒⋅Δ𝑉 are equal to 𝑉^𝑠_𝑒⋅𝑊 and 𝑉^𝑎_𝑒⋅𝑊, respectively, because of (32) and (43). As shown above, (38) and (39) completely coincide with (47) and (48), respectively.

5. Conclusion and Discussion

We have shown that in a two-dimensional two-slab waveguide system that has a small core gap so that the optical system has only one propagation mode, the supermode and its propagation constant calculated by first-order perturbation by the Euler method and the first-order Taylor expansion of 𝑤/𝑑 for the exact solution give the same result up to (𝑤/𝑑)¹ order.

As for higher-order perturbation, the correction of 𝑒_𝑤 by second-order perturbation is given by [10] _{𝑝∈{𝑠,𝑎}}⁰_−𝑛22(𝑒)𝑉𝑑𝑒𝜌^𝑝𝑒⋅𝑛²₂𝑉₀²𝑒−𝑒₀, (49) where 𝜌(𝑒) represents mode density function, and the correction of 𝑉^𝑝_𝑒⋅𝑊 is _{{}𝑞∈𝑠,𝑎}⁰_−𝑛22𝑉𝑑𝑓^𝑝𝑒⋅𝑛²₂𝑉^𝑞_𝑓𝑉^𝑞_𝑓⋅𝑛²₂𝑉₀𝑒−𝑒₀𝑓−𝑒₀−𝑉^𝑝_𝑒⋅𝑛²₂𝑉₀𝑉₀⋅𝑛²₂𝑉₀𝑒−𝑒₀². (50) The lowest order of 𝑤/𝑑 for (49) and (50) is 𝑂(𝑤/𝑑)², because of the argument given in Section 3. Therefore, second-order perturbation with the Euler method is a part of the second-order Taylor expansion of 𝑤/𝑑. As shown in (36), the first-order perturbation can include the second order of 𝑤/𝑑. To obtain the second order of 𝑤/𝑑 from the first-order perturbation, integration over −𝑤/2<𝑥<𝑤/2−𝑑 and −𝑤/2<𝑥<𝑤/2 in the perturbation must be approximated by the trapezoidal rule instead of the Euler method. Thus, first-order perturbation by the trapezoidal rule in combination with a second-order perturbation by the Euler method gives the same result as the exact calculation given by the second-order Taylor expansion of 𝑤/𝑑. In a similar way, the 𝑛th-order coefficient of the Taylor expansion of 𝑤/𝑑 is obtained from all orders of perturbation less than the 𝑛th order. In each order of perturbation, exact integration over −𝑤/2−𝑑<𝑥<𝑤/2−𝑑 and −𝑤/2<𝑥<𝑤/2 is not necessary; instead, a suitable approximation of integration including the (𝑤/𝑑)^𝑛 order term is necessary. Thus, when an optical system has two cores with a small gap, it can be analyzed by perturbation using modes of an optical system with a gapless core with twice the width.

Obtaining the exact solution becomes time consuming as the number of cores increases. Even in such situations, perturbation analysis can be executed easily. Hence, perturbation analysis using suitable approximate integration is a promising candidate for approximation in the region 𝑤/𝑑≪1 when the coupled mode or power mode equation is invalid because 𝑤/𝑑 is very small.

Acknowledgment

The authors thank all members of the Tobe Laboratory in the Department of Integrated Information Technology, College of Science and Engineering, Aoyama Gakuin University, for comments and discussion.