Focusing of partially coherent light into planar waveguides

Toni Saastamoinen; Markku Kuittinen; Pasi Vahimaa; Jari Turunen; Jani Tervo

doi:10.1364/OPEX.12.004511

1 Introduction

Waveguide discontinuities have been a subject of major interest of research during the past few decades because they play an important part in the design of many optical components. Problems involving such discontinuities are encountered in various situations such as waveguide ends, couplers and, as a consequence of misaligments, in component interconnections.

A guided mode incident on any waveguide discontinuity excites, in general, all (discrete) propagating guided modes supported by the structure, as well as continuous angular spectra associated with (leaky) radiation modes and evanescent modes that have appreciable amplitudes only in the immediate vicinity of the abrupt discontinuity. In a rigorous solution of the scattering problem all of these field modes have to be taken into account, although the radiation and evanescent modes make the solution difficult. Despite of this, a wide range of numerical methods to solve waveguide discontinuity problems have been presented: see, e.g., Refs. [1–10].

If we replace the real waveguide by a stack of identical waveguides separated by layers of homogenous material, the continuous angular spectra of the radiation and evanescent fields become discrete. This geometry allows the exploitation of the well-established rigorous techniques developed for grating problems. The effect of the artificial periodicity can be in principle reduced to any desired degree by choosing a sufficiently large separation layer. This approach was applied to various problems involving discontinuities in Ref. [11] but it did not gain wider attention until recently [12–16].

In the aforementioned method, however, the incident light is always assumed to be spatially fully coherent, although many common light sources used in connection of integrated optics have poor spatial coherence properties. In this paper we extend the analysis of discontinuities in planar waveguides into the domain of partial coherence. As an example, we examine the edge coupling of a Gaussian Schell-model beam into planar waveguides by employing the coherent-mode representation of partially coherent fields [17, 18]. The coupling efficiencies obtained by a grating model are compared to the predictions of the overlap integral method.

2 Coherent mode representation of spatially partially coherent fields

It was shown by Wolf that under very general conditions the cross-spectral density function, which characterizes the coherence properties of the field in the space–frequency domain [17], may be represented as an incoherent sum of completely coherent modes [18]

W (r_{1}, r_{2}) = \sum_{n = 1}^{\infty} λ_{n} ϕ_{n}^{*} (r_{1}) ϕ_{n} (r_{2}) .

Here λ and ϕ(r) are the eigenvalues and the eigenfunctions, respectively, of the Fredholm integral equation of the second kind:

\int_{D} W (r_{1}, r_{2}) ϕ_{n} (r_{1}) d^{3} r_{1} = λ_{n} ϕ_{n} (r_{2}),

where D is the domain of integration. We note that the dependence of quantities such as the modal eigenfunctions and eigenvalues on the angular frequency ω is left implicit here and in what follows.

Since the cross-spectral density function is Hermitian and non-negative definite [17,18], the eigenvalues λ_n are real and non-negative, and they may be understood to represent the energy distribution between the modes. The eigenfunctions ϕ(r) are orthonormal, i.e.,

\int_{D} ϕ_{m}^{*} (r) ϕ_{n} (r) d^{3} r = δ_{mn},

where δ_mn denotes the Kronecker delta symbol.

The decomposition (1) is important since it enables the treatment of partially coherent fields by applying methods already developed for coherent fields. Specifically, one needs to solve the Fredholm equation (2), propagate each coherent mode individually, and finally combine the results using Eq. (1).

3 Field representation in the waveguide

Let us consider an infinite planar waveguide with refractive index distribution

n (x) = {\begin{matrix} n_{c} & when & x > d_{g} \\ n_{l} & when & x_{l} \leq x < x_{l + 1}, \\ n_{s} & when & x < 0 \end{matrix}

where l = 1,…,L, x ₀ = 0, and x_L = d_g . We replace this real structure with an artificially periodic structure illustrated in Fig. 1, with a period d = d _s + d _g + d _c, where the subscripts s, g, and c denote substrate, guide, and cover respectively. In addition, we demand that

n_{max} = max {n_{l}}

is larger than the refractive indices n _s and n _c of the substrate and cover. If we let d _c → ∞ and d _s → ∞, the periodic structure reduces to the real non-periodic waveguide described by Eq. (4). Graded-index waveguides fabricated, for example, by ion-exchange techniques can be modelled to any degree of accuracy by choosing a sufficient large value for L.

Fig. 1. The periodic waveguide structure of period d used to model the real non-periodic structure of Eq. (4).

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Let us denote a single scalar component of the electromagnetic field by U (which is E_y in TE polarization and H_y in TM polarization). To find the solutions of the Maxwell equations in the waveguide structure separate the variables as

U (x, z) = X (x) [a \exp (iγ z) + b \exp (- iγ z)] .

Here a, b and γ are constants and X(x) is a Bloch wave function that satisfies the condition

X (x + d) = X (x) \exp (i α_{0} d),

where α ₀ is a pseudoperiodicity constant. The periodicity leads to a discrete set of possible values of γ and the associated functions X(x). The solutions may be divided in two categories: The homogenous eigenmodes for which 0 < γ ² < k ² $n_{max}^{2}$ and the inhomogenous eigenmodes for which γ ² < 0, where k = 2π/λ and λ is the wavelength of light in vacuum. Obviously, the latter class of modes, whose field amplitudes grow exponentially in the z direction, are non-physical. The modes with exponentially decaying field amplitude have no contribution in a waveguide with an infinite length but they play an important role in the vicinity of the structural discontinuities.

In grating theory a wide range of efficient numerical methods have been presented to solve the propagation constants γ_m and the corresponding eigenfunctions X_m (x) for all homogenous modes as well as for a sufficient number of inhomogenous modes. The field in the waveguide structure (4) illustrated in Fig. 1 must satisfy the Helmholtz equation in each homogenous region, the appropriate electromagnetic boundary conditions at all interfaces, and the pseudo-periodicity condition (7). An algebraic eigenvalue equation can be obtained [19], which can be solved numerically even if some of the refractive indices are complex. If the eigenvalues in each interval of the structure are presented on a basis formed by Legendre polynomials, the algebraic eigenvalue problem can be replaced with a matrix eigenvalue problem [20]. Furthermore, it is possible to express the refractive index profile in the form of a Fourier series, which leads to a matrix eigenvalue problem that can be solved by standard numerical algorithms (see e.g., Refs. [21–25]).

The structure defined in Eq. (4) in the non periodic limit d → ∞ supports the following types of modes [26–30]

{\begin{matrix} guided modes : & {[k max (n_{s}, n_{c})]}^{2} < γ^{2} < {(k n_{max})}^{2} \\ evanescent modes : & γ^{2} < 0 \\ radiation modes : & otherwise \end{matrix} .

The eigenvalues γ of the guided modes form a discrete spectrum, but the spectra of the all other types of modes are continuous, which makes the treatment of non-periodic waveguide discontinuity problems more difficult than the solution of the grating diffraction problem.

4 Boundary value problem

After the solution for the modes X_m (x) and the eigenvalues γ_m using one of the methods discussed in the previous section, it remains to solve the amplitudes of the reflected and transmitted fields.

The interaction of the incident field with the waveguide structure excites all guided modes, a radiation field, a reflected field, and an evanescent field that is appreciable only in the vicinity of the plane z = 0. The periodicity of the geometry discretizes the known angular spectrum A_m (α) of the incident field denoted by $E_{y}^{in}$ (x,z) and the unknown angular spectrum R(α) of the reflected field $E_{y}^{r}$ (x,z) in the half-space z ≤ 0. This leads to the use of Rayleigh expansions, which give the incident and reflected fields in the half-space z ≤ 0 in the form

E_{y}^{in} (x, z) = \sum_{m = - M}^{M} A_{m} \exp [i (α_{m} x + r_{m} z)],

E_{y}^{r} (x, z) = \sum_{m = - \infty}^{\infty} R_{m} \exp [i (α_{m} x + r_{m} z)],

where A_m and R_m represent the sampled values of the angular spectrum of the incident and the reflected fields, respectively,

α_{m} = α_{0} + \frac{2 π m}{d},

r_{m} = {\begin{matrix} {(k^{2} n^{2} - α_{m}^{2})}^{\frac{1}{2}} & if ∣ α_{m} ∣ \leq k n \\ i {(α_{m}^{2} - k^{2} n^{2})}^{\frac{1}{2}} & otherwise \end{matrix},

and n is the refractive index of the medium in the half-space z < 0. In Eq. (9) the summation of the incident field is restricted to the index M to take account only the homogenous plane waves with real-valued r_m , since we assume that sources are located sufficiently far away from the examined boundary.

The field expression in the waveguide in TE polarization is of the form of Eq. (6) with b = 0 because we assume that there are no sources in the positive half-space that would give rise to modal fields propagating in the negative z direction:

E_{y}^{t} (x, z) = \sum_{m = 1}^{\infty} a_{m} \exp (i γ_{m} z) X_{m} (x),

where a_m are the unknown amplitudes of the mth mode propagating in the positive z direction and γ_m are the propagation constants, which can be determined by methods presented in Sect. 3.

To solve the modal amplitudes a_m and the coefficients R_m of the angular spectrum of the reflected field, we require the continuity of the electric field and its z derivative at the boundary between the half-space z < 0 and the waveguide:

E_{y}^{in} (x, z) + E_{y}^{r} (x, z) = E_{y}^{t} (x, z),

\frac{\partial}{\partial z} E_{y}^{in} (x, z) + \frac{\partial}{\partial z} E_{y}^{r} (x, z) = \frac{\partial}{\partial z} E_{y}^{t} (x, z) .

The substitution of the field expressions $E_{y}^{in}$ , $E_{y}^{t}$ , and $E_{y}^{r}$ into Eqs. (14)–(15) yields two equations that include summations, which are more conveniently expressed in the matrix forms

A + R = Pa,

r (A - R) = P Γ a,

where the elements of A, R, and a are A_m , R_m , and a_m , respectively, and Γ and r are diagonal matrices with the elements γ_m and r_m , respectively. The elements of the matrix P are obtained from the equation

P_{m q} = \frac{1}{d} \int_{o}^{d} X_{m} (x) \exp (\frac{- i 2 π q x}{d}) d x .

The coefficients P_mq are the solved eigenvector coefficients of the eigenvalue problem discussed in Section 3.

The amplitudes R_m and a_m may be solved from Eqs. (16) and (17). For example, the amplitudes a_m of the waveguide modes are obtained by eliminating R, which yields

a = 2 {(PΓ + rP)}^{- 1} rA .

This is easily solved by truncation of the matrices. Finally, the treatment in TM polarization is analogous, with only a slight difference in the boundary conditions.

The transmitted and reflected amplitudes of a random partially coherent field are now achieved simply by solving the eigenvalues and eigenfunctions of the Fredholm integral equation (2) and replacing the incident field $E_{y}^{in}$ (x,z) by the coherent eigenfunctions ϕ_n (x,z). Repeating this procedure for each eigenfunction separately we obtain the unknown amplitudes a_m and R_m corresponding to each incident coherent mode. We note that the waveguide modes X_m (x) and the eigenvalues γ_m need to be solved only once regardless of the number of coherent modes.

As an example of the procedure that normally would have to be carried out numerically, we consider a normally incident Gaussian Schell-model (GSM) beam with the cross-spectral density function of the form

W_{y}^{in} (x_{1}, x_{2}) = \exp [- \frac{{(x_{1} - \bar{x})}^{2} + {(x_{2} - \bar{x})}^{2}}{w_{0}^{2}}] \exp [- \frac{{(x_{1} - x_{2})}^{2}}{2 σ_{0}^{2}}],

where w ₀ and σ ₀ denote the beam 1/e² width and coherence width at the beam-waist plane, respectively, and x̅ is the beam center position. The coherent modes of the incident beam (20) can be solved analytically, which gives the eigenfunctions and eigenvalues of Eq. (2) in the form [31]

ϕ_{n} (x, z) = {(\frac{2 c}{π})}^{\frac{1}{4}} \frac{1}{\sqrt{2^{n} n!}} H_{n} [(x - \bar{x}) \sqrt{2 c}] \exp [- c {(x - \bar{x})}^{2}],

and

λ_{n} = {(\frac{π}{a + b + c})}^{\frac{1}{2}} {(\frac{b}{a + b + c})}^{n},

respectively, where

{\begin{matrix} a = w_{0}^{- 2} \\ b = \frac{σ_{0}^{- 2}}{2} \\ c = {(a^{2} + 2 a b)}^{\frac{1}{2}} \end{matrix},

and H_n is a Hermite polynomial of the order n. The geometry of the waveguide and the incident beam are illustrated in Fig. 2.

Fig. 2. Coupling of a set of Gaussian Schell-model beams from a uniform medium into a periodically modulated waveguide used to model single-beam coupling into non-periodic waveguide.

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5 Input coupling efficiency

In beam coupling problems one is usually interested in the coupling efficiency, defined as the fraction of the incident power coupled into guided modes of the waveguide:

η_{m} = \frac{P_{m}}{P_{in}} .

Here the power P is given by the z component of the time-averaged Poynting vector as

P = \frac{1}{2} \int_{0}^{d} ℜ {E_{y} (x, z) H_{x}^{*} (x, z)} d x,

where ℜ denotes the real part.

To obtain the coupling efficiency of the incident partially coherent field into the waveguide, each set of the angular spectrum components $A_{m}^{n}$ and the modal amplitudes $a_{m}^{n}$ of the waveguide, excited by the corresponding coherent mode with index n, has to be weighted by the modal coefficients λ_n ) in Eq. (22). In view of Eq. (25), the power carried by the incoming partially coherent field is

P_{in} = \frac{d}{2 ω μ_{0}} \sum_{n = 1}^{\infty} λ_{n} \sum_{m = - M}^{M} r_{m} {∣ A_{m}^{n} ∣}^{2},

and the total power carried by the mth coherent mode is given by

P_{m} = \frac{d}{2 ω μ_{0}} ℜ {γ_{m}} \sum_{n = 1}^{\infty} λ_{n} {∣ a_{m}^{n} ∣}^{2} \sum_{q = - \infty}^{\infty} {∣ P_{m q} ∣}^{2} .

Similarly, for the reflected field, we obtain

P_{r} = \frac{d}{2 ω μ_{0}} ℜ {γ_{m}} \sum_{n = 1}^{\infty} λ_{n} \sum_{q = - \infty}^{\infty} {∣ R_{m}^{n} ∣}^{2} .

The coherent beam coupling efficiency into waveguides is usually determined by using the overlap integral method [32]. In the case of partially coherent illumination each coherent mode, in general, provides a contribution to all waveguide modes as well as to evanescent and radiation modes. Since the coherent modes are mutually uncorrelated, the coupling efficiency into a waveguide mode with index m may be defined as an incoherent sum of the coupling efficiencies from different coherent modes:

η_{m} = \frac{\sum_{n = 1}^{\infty} λ_{n} {∣ \int_{- \infty}^{\infty} X_{m}^{*} (x) ϕ_{n} (x) d x ∣}^{2}}{\int_{- \infty}^{\infty} {∣ X_{m} (x) ∣}^{2} d x \sum_{n = 1}^{\infty} λ_{n} \int_{- \infty}^{\infty} {∣ ϕ_{n} (x) ∣}^{2} d x} .

Because the modes X_m are defined piecewise as sinusoidal, exponential, and hyperbolic functions, the integrals can be evaluated analytically for Gaussian illumination.

6 Optimization of coupling efficiency

The best possible coupling efficiency into a non-symmetric waveguide depends on three parameters: the beam half-width w, the coherence width σ, and the beam center position x̅. The determination of the efficiency is a rather simple task by using the overlap integral method [32] but rather time consuming with our quasi-rigorous method, which however is more accurate.

In the following examples the parameters for the best possible coupling efficiency are determined by scanning over the different values of the beam center position x̅ and the half-width w. In practice the latter scan can be performed with a positive lens. The beam half-width at the focal plane of the lens is

W_{F} = \frac{λ f}{π w_{0} β},

where β = (1 +1/δ ²)^-1/2 and δ = w ₀/σ ₀ is the global degree of spatial coherence of the GSM beam [17]. The coherence width at the focal plane is

σ_{F} = \frac{σ_{0} w_{F}}{w_{0}} .

In our analysis a restriction w_F ≥ λ/2 is made, which is rather optimistic for practical measurements. The coupling efficiencies are calculated by the overlap integral method (29) and the quasi-rigorous method for two values of the global degree of coherence, namely δ = 1 and δ = 1/2, in TE polarization. The number of significant coherent modes retained in the calculations depends on the value of δ. In the following examples the ratio of the normalized modal weights is chosen to be λ_N /λ ₁ < 10^-3, where N is the order of the highest mode. For comparison, results are also provided for the fully coherent Gaussian beam that is obtained in the limit δ → ∞.

The convergence of the coupling efficiencies is verified by increasing the period of the structure and the number of eigenmodes retained in calculations. This naturally implies that the eigenvalues, which in our examples are calculated by using the Fourier modal method [21–25], are also converged to sufficient accuracy.

6.1 Single mode waveguide

Let us first analyze the single mode planar waveguide with the following parameters: n _s = 1.55, n_g = 1.97, and n _c = 1 with d _g = 0.21 μm. The wavelength of the light in all calculations is λ = 0.6328 μm.

The coupling efficiencies with the optimum values of the beam half-width w and the offset x̅ in the cases δ → ∞, δ = 1, and δ = 1/2 are presented for both methods in Table 1. As an example, the results in the case δ = 1 are illustrated in Figs. 3(a) and (b), and the convergence of the coupling efficiency for the optimum parameters is shown in Fig. 4 with several values of the period d as a function of the number of retained eigenmodes. We emphasize that the convergence of the results must be confirmed for every configuration.

Table 1. Optimum coupling efficiencies of the GSM beam into a single mode waveguide with guiding layer thickness d_g = 0.21 μm.

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6.2 Three-mode waveguide

Next we consider a three-mode waveguide with the same refractive index distribution as in the case of single-mode waveguide, but with the guiding layer thickness d_g = 0.7μm. The coupling efficiencies are presented in Table 2.

The optimum values for w are no longer correctly predicted by the overlap integral method when the value of δ is decreased. Interestingly, the coupling efficiencies into the fundamental mode of the three-mode waveguide are higher than in the case of single-mode waveguide.

Table 2. Optimum coupling efficiencies of the GSM beam into a three-mode waveguide with guiding layer thickness d_g = 0.7 μm.

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6.3 Graded index waveguide

A similar analysis as presented above is given for a graded-index planar waveguide with n _s = 1.5232, n _c = 1 and the guiding layer thickness d _g = 5.0 μm. The refractive index distribution of the structure is shown in Fig. 5 where the number of layers [Eq. (4)] is chosen to be L = 50. The optimum coupling efficiencies into waveguide modes m = 1 and m = 3 are presented in Tables 3 and 4, respectively.

In view of the results presented above, the overlap integral method is surprisingly accurate for weakly modulated refractive index structures. The minor differences in the coupling efficiencies are mostly explained by the absence of reflection losses at the edge of the waveguide in the overlap integral method.

Fig. 3. Coupling efficiencies η ₁ into the fundamental waveguide mode as a function of beam center position x̅ and beam width w when the global degree of coherence δ = 1. (a) Overlap integral method. (b) Quasi-rigorous method.

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Fig. 4. Convergence of the coupling efficiency into the fundamental mode of the single mode waveguide in the case δ = 1 as a function of the number of eigenmodes retained in the calculation. The curves represent the periods d = 80d _g, d = 125d _g, d = 150d _g, and d = 175d _g respectively, starting from the lowermost.

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As the analysis shows, the best coupling efficiency for every type of waveguide is obtained in the coherent limit δ → ∞, i.e., when the incident field reduces to a conventional fully coherent Gaussian beam. This result is easily understood if we consider the transversal mode profiles of the GSM beam and the overlap integral method. The best coupling efficiency is obtained for the lowest-order coherent mode, n = 1, as it resembles most the fundamental waveguide mode m = 1. The fully coherent Gaussian beam is obtained when the coherence length σ → ∞ and thus only one modal coefficient is non-zero. The more the value of δ decreases the greater number of coherent modes have to be retained to model the incident beam. The higher order coherent modes, however, are Hermite-Gaussian modes, which have weaker coupling efficiencies.

Table 3. Optimum coupling efficiencies of the GSM beam into a graded index waveguide mode m = 1 with guiding layer thickness d_g = 5.0 μm.

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Table 4. Optimum coupling efficiencies of the GSM beam into a graded index waveguide mode m = 3 with guiding layer thickness d_g = 5.0 μm.

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Fig. 5. The refractive index distribution of the graded index waveguide as a function of the guiding layer thickness d _g.

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7 Conclusions

We have examined the problem of focusing partially coherent Gaussian Schell-model beams into planar waveguides as an example of the use of the periodic model of the waveguide structure in discontinuity problems. The main advantage of the periodic model is the discretization of the angular spectra of the radiation and evanescent fields, which enables the application of numerically efficient rigorous diffraction models for gratings. The widely used overlap integral method was found to be inadequate to predict the coupling efficiencies into strongly waveguiding structures, but sufficiently accurate in the case of weakly modulated structures. The optimum coupling parameters were, however, predicted rather accurately in all cases by the overlap integral method. The coupling efficiencies were found to decrease along with the decrease of the global degree of coherence of the incident beam.

Acknowledgments

The work of T. Saastamoinen was funded by The Finnish Academy of Science and Letters. J. Tervo is a research fellow with the Alexander von Humboldt foundation.

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	overlap integral method			quasi-rigorous method
δ	η ₁(%)	w(μm)	x̅(μm)	η ₁(%)	w(μm)	x̅(μm)
∞	83.3	0.32	0.08	70.9	0.32	0.07
1	74.7	0.32	0.08	63.8	0.32	0.07
1/2	59.8	0.32	0.08	52.3	0.32	0.07

	overlap integral method			quasi-rigorous method
δ	η ₁(%)	w(μm)	x̅(μm)	η ₁(%)	w(μm)	x̅(μm)
∞	99.5	0.32	0.34	86.9	0.32	0.34
1	82.4	0.37	0.34	74.7	0.32	0.34
1/2	61.3	0.42	0.34	58.7	0.32	0.34

	overlap integral method			quasi-rigorous method
δ	η ₁(%)	w(μm)	x̅(μm)	η ₁(%)	w(μm)	x̅(μm)
∞	97.6	0.81	4.13	92.8	0.80	4.11
1	80.9	0.95	4.13	77.1	0.94	4.12
1/2	60.4	1.20	4.13	58.0	1.16	4.12

	overlap integral method			quasi-rigorous method
δ	η ₃(%)	w(μm)	x̅(μm)	η ₃(%)	w(μm)	x̅(μm)
∞	67.8	0.92	1.77	64.7	0.92	1.75
1	56.2	1.10	1.77	53.8	1.09	1.77
1/2	42.3	1.40	1.77	40.7	1.37	1.77

	overlap integral method			quasi-rigorous method
δ	η ₁(%)	w(μm)	x̅(μm)	η ₁(%)	w(μm)	x̅(μm)
∞	83.3	0.32	0.08	70.9	0.32	0.07
1	74.7	0.32	0.08	63.8	0.32	0.07
1/2	59.8	0.32	0.08	52.3	0.32	0.07

Focusing of partially coherent light into planar waveguides

Abstract

1 Introduction

2 Coherent mode representation of spatially partially coherent fields

3 Field representation in the waveguide

4 Boundary value problem

5 Input coupling efficiency

6 Optimization of coupling efficiency

6.1 Single mode waveguide

6.2 Three-mode waveguide

6.3 Graded index waveguide

7 Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (4)

Equations (31)

Optics Express