Theory of SNAP devices: basic equations and comparison with the experiment

M. Sumetsky

doi:10.1364/OE.20.022537

1. Introduction

Surface nanoscale axial photonics (SNAP), the record accurate and low loss platform for fabrication of complex miniature photonic circuits, was introduced and described in the series of recent publications [1–6]. A SNAP device is illustrated in Fig. 1 . It consists of an optical fiber with specially introduced nanometer-scale radius variation, called the SNAP fiber (SF), and transverse input/output waveguides coupled to this fiber. The waveguides are usually fabricated of a biconical fiber with micron-scale diameter waist, or, alternatively, can be planar waveguides fabricated lithographically. The input waveguide launches whispering gallery modes (WGMs), which circulate near the SF surface and experience slow propagation along the fiber axis. In SNAP, the WGMs have very small propagation constant and, for this reason, are sensitive to dramatically small nanoscale variation of the fiber radius and similar variation of the refractive index [1,6,7]. This enables fabrication of complex SNAP circuits (consisting, e.g., of long series of coupled microresonators) with record high accuracy and record small optical losses [3–6]. The performance of SNAP devices depends only on the effective radius variation of the fiber, which combines the contributions of variations of the fiber physical radius and its refractive index. For nanoscale variations, these contributions usually have the same order of magnitude [8,9].

Fig. 1 Illustration of a SNAP device. Red: WGM microresonators. WG1 and WG2 are the waveguide 1 and waveguide 2 coupled to the SF. Inset: magnified nanometer-scale fiber radius variation.

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The goal of this paper is to develop the theory which can be directly applied both to modeling of SNAP devices and to their evaluation from the experimental data. The results obtained are essentially based on the previous work [1,2] where the one-dimensional Schrödinger equation governing the slow propagation of WGMs along the fiber with nanoscale effective radius variation was derived. This equation is described in Section 2. In [1,2], the transmission amplitude of a SNAP device was expressed through the Green’s function of the Schrödinger equation which is supposed to be renormalized to take into account the losses due to coupling to the waveguide. However, the actual application of the theory [1,2] was limited because the problem of renormalization has not been addressed. The theory developed in Section 3 solves this problem. In addition, Section 3 includes the case of an SF coupled to several waveguides. It is shown that the transmission amplitudes of a SNAP device coupled to N waveguides, one of which is the input/output waveguide and others are the output waveguides, can be expressed through the solution of the one-dimensional Schrödinger equation of Refs [1,2]. and 3N complex constants, which determine the lossy SF/waveguide coupling. In the case of lossless coupling, the number of constants is reduced to N. In Section 4, the developed theory is applied to the investigation of transmission amplitudes through a localized state and also through a uniform SF coupled to one waveguide, which exhibits a variety of asymmetric Fano resonances. In Section 5, it is shown that sharp transmission resonances can characterize a lossy SNAP device coupled to two waveguides. In Section 6, the experimental characterization of the SNAP bottle microresonator and of a chain of 10 coupled microresonators is demonstrated to be in the excellent agreement with the developed theory. It is shown that the effective radius variation of a SNAP fiber and all of the transmission amplitude parameters can be accurately determined from the experiment. The results of this paper are discussed and summarized in Section 7.

2. WGMs in a SNAP fiber in the absence of input/output waveguides

The field distribution of a WGM adiabatically propagating along the axis z of an SF with nanoscale smooth radius variation (and/or equivalent refractive index variation) in the absence of input/output waveguides is defined as

U_{m, p, q} (r) = Ψ_{m, p, q} (z) Ξ_{m, p} (ρ) \exp (i m φ)

where

(z, ρ, φ)

are the cylindrical coordinates, m is the discrete azimuthal quantum number, p is the discrete radial quantum number, and q is the discrete or continuous axial quantum number. Function

Ψ_{m, p, q} (z)

determines the axial distribution of the WGM and satisfies the one-dimensional Schrödinger equation [1,2]:

\frac{d^{2} Ψ}{d z^{2}} + β^{2} (λ, z) Ψ = 0, β^{2} (λ, z) = E (λ) - V (z) .

In this equation, the energy is proportional to the variation of radiation wavelength

λ

and the potential is proportional to the fiber radius variation

Δ r (z) = r (z) - r_{0}

and refractive index variation

Δ n_{f} (z) = n_{f} (z) - n_{f 0}

:

\begin{array}{l} E (λ) = - 2 k^{2} (λ_{r e s}) \frac{λ - λ_{r e s} - i γ_{r e s}}{λ_{r e s}}, V (z) = - 2 k^{2} (λ_{r e s}) \frac{Δ r_{e f f} (z)}{r_{0}}, \\ \frac{Δ r_{e f f} (z)}{r_{0}} = \frac{Δ r (z)}{r_{0}} + \frac{Δ n_{f} (z)}{n_{f 0}}, k (λ_{r e s}) = \frac{2 π n_{f 0}}{λ_{r e s}} . \end{array}

The joint contribution of the fiber radius and refractive index variation is expressed in Eqs. (2) and (3) only through the effective radius variation

Δ r_{e f f} (z)

. In silica, the attenuation parameter

γ_{r e s}

is very small (typically,

γ_{r e s} < 0.1 pm

). The resonance wavelength

λ_{r e s}

coincides with the transverse eigenvalue

λ_{m p}

corresponding to the zero propagation constant, i.e., to the zero axial speed of light. Equation (2) is valid for wavelengths

λ

in a small neighborhood of

λ_{r e s} = λ_{m p}

with the fixed quantum numbers m and p. These numbers are omitted below for brevity.

3. Theory of a SNAP device

It is instructive to recall the general theory of transmission through waveguides coupled to a complex-shaped cavity. The latter has been of significant interest in the last few decades both in quantum mechanics and optics [10–16]. The important problem that has been addressed in this theory was the elucidation of the relation between the transmission amplitudes through the waveguides and the Green’s function of the isolated cavity. To solve this problem, the renormalized Green’s function of the cavity was introduced. This function modifies the Green’s function of the isolated cavity (i.e., the cavity uncoupled from the waveguides) called the bare Green’s function, and takes into account the losses due to coupling to the waveguides. It was found that the transmission amplitudes through the waveguides can be expressed through the overlap integrals between the renormalized Green’s function and the travelling waves propagating along the waveguides.

The major challenge of the theory of non-one-dimensional cavities coupled to waveguides is the actual determination of the renormalized Green’s function for a given cavity shape [10–15]. This problem has been addressed in several special cases only. For example, for quantum mechanical and optical billiards, this problem was approached in the semiclassical approximation [11]. For a cavity assembled of several weakly coupling elementary cavities, this problem was considered in the tight-binding approximation [16]. Besides, the behavior of non-one-dimensional cavities can be qualitatively analyzed with the quantum graph theory [17].

Crucially, it is shown in this paper that in many cases the relation between the bare Green’s function and renormalized Green’s function of a SNAP device can be found analytically. For the considered slow axial propagation of light, i.e., when the radiation wavelength is close to the resonance $λ_{r e s}$ , the bare 3D Green’s function of the SNAP device is expressed through the 1D Green’s function $G (λ, z_{1}, z_{2})$ of Eq. (2) (see Appendix 1). Similarly, the renormalized 3D Green’s function is expressed through the 1D renormalized Green’s function $\bar{G} (λ, z_{1}, z_{2})$ defined below. Finally, the transmission amplitudes through the waveguides are simply expressed through the Green’s function $\bar{G} (λ, z_{1}, z_{2})$ .

Consider an SF coupled to N transverse waveguides WG₁, WG₂, …, WG_N. Assume that WG₁ serves as the input and output waveguide while all other waveguides are the output waveguides only (in Fig. 1, $N = 2$ ). Then, the renormalized Green’s function of the SF, $\bar{G} (λ, z, z_{1})$ , is determined as the Green’s function of equation:

\frac{d^{2} Ψ (z)}{d z^{2}} + [β^{2} (λ, z) + \sum_{n = 1}^{N} D_{n} δ (z - z_{n})] Ψ (z) = 0.

This equation coincides with Eq. (2) for

D_{n} = 0

. The real part of

D_{n}

determines the phase shift due to coupling to WG_n while its imaginary part takes into account the radiation loss through WG_n, In Eq. (4), coupling to waveguides WG_n is modeled with zero-range potentials

D_{n} δ (z - z_{n})

[17,18], which is justified if the waveguide width is much smaller than the WGM axial wavelength. The latter assumption is in agreement with the experiment, where, typically, the microfiber diameter is ~1 μm and the axial wavelength is greater than 10 μm. Under the same assumptions, the transmission amplitude through WG₁ is simply found as (Appendix 1)

S_{11} (λ) = S_{11}^{(0)} - i | C_{1} |^{2} \bar{G} (λ, z_{1}, z_{1}),

and the transmission amplitude from WG₁ to WG_n is

S_{1 n} (λ) = S_{1 n}^{(0)} - i C_{1} C_{n}^{*} \bar{G} (λ, z_{1}, z_{n}),

where constants

C_{n}

are the SF/WG_n coupling parameters and

S_{1 n}^{(0)}

are the non-resonant components of the transmission amplitudes

S_{1 n}

. These parameters are slow functions of wavelength and can be set to constants in the considered neighborhood of resonance

λ_{r e s}

.

Equation (4) for the renormalized Green’s function together with Eqs. (5) and (6) express the transmission amplitudes through the effective radius variation of the SF and 3N complex constants $S_{1 n}^{(0)}$ , $C_{n}$ , and $D_{n}$ . These equations are applied below to the analysis of basic theoretical models and to the treatment of experimental data.

4. Transmission amplitude of a SNAP device coupled to a single waveguide

For a single waveguide WG₁ coupled to an SF ( $N = 1$ ) the renormalized Green’s function $\bar{G} (λ, z_{1}, z_{2})$ of Eq. (4) is expressed through the bare Green’s function $G (λ, z_{1}, z_{2})$ of Eq. (2) as (see Appendix 2):

\bar{G} (λ, z_{1}, z_{2}) = \frac{G (λ, z_{1}, z_{2})}{1 + D_{1} G (λ, z_{1}, z_{1})} .

Substitution of Eq. (7) into Eq. (5) yields:

S_{11} (λ) = S_{11}^{(0)} - \frac{i | C_{1} |^{2} G (λ, z_{1}, z_{1})}{1 + D_{1} G (λ, z_{1}, z_{1})} .

In the case of lossless resonant transmission,

G (λ, z_{1}, z_{2})

is real (e.g., as in a lossless bottle resonator [19]), and, due to energy conservation, the absolute value of the transmission amplitude defined by Eq. (5) equals unity,

| S_{11} (λ) | \equiv 1

for all real

G (λ, z_{1}, z_{2})

. The latter equation immediately leads to

S_{11}^{(0)} = 1

and

Im D_{1} = | C_{1} |^{2} / 2

. As a result, for lossless resonant coupling and any SF profile, which not necessarily corresponds to the real-valued

G (λ, z_{1}, z_{2})

(i.e., the WGM propagation can be lossy and delocalized),

S_{11} (λ) = \frac{1 + (Re D_{1} - \frac{i}{2} | C_{1} |^{2}) G (λ, z_{1}, z_{1})}{1 + (Re D_{1} + \frac{i}{2} | C_{1} |^{2}) G (λ, z_{1}, z_{1})} .

From the energy conservation law, the absolute value of the transmission amplitude defined by Eq. (9) is always less than unity. This can be verified using the known inequality

Im G (λ, z_{1}, z_{1}) < 0

[12]. The case of lossless resonant coupling described by Eq. (9) is often referred to as ideal coupling which is confirmed experimentally for weak coupling and a single mode waveguide [20]. For an SF ideally coupled to N waveguides,

S_{11}^{(0)} = 1

,

S_{1 n}^{(0)} = 0

, and

Im D_{n} = | C_{n} |^{2} / 2

.

For lossy coupling, the energy conservation law implies $| S_{11} (λ) | < 1$ , which restricts the possible values of $S_{11}^{(0)}$ , $C_{1}$ , and $D_{1}$ by the inequalities (Appendix 3):

| S_{11}^{(0)} | < 1,

Im (D_{1}) > | C_{1} |^{2} \frac{1 - Re (S_{11}^{(0)})}{1 - | S_{11}^{(0)} |^{2}} .

These inequalities are useful in the modeling of SNAP devices as well as in the numerical treatment of experimental data.

4.1. Transmission through localized states of an SF

Consider a fully localized SNAP bottle microresonator [19] with a real-valued axial distribution of eigenmode $Ψ_{n} (z)$ having the energy eigenvalue $E_{n}$ . The Green’s function of Eq. (2) near $E_{n}$ is [12]

G (λ, z_{1}, z_{2}) = \frac{Ψ_{n} (z_{1}) Ψ_{n} (z_{2})}{E (λ) - E_{n} + i Γ_{0}},

where

Γ_{0} = (8 π^{2} n_{f 0}^{2} / λ_{r e s}^{3}) γ_{r e s}

is the width of resonance due to the propagation losses in the SF. Substitution of Eq. (12) into Eq. (8) yields the Fano formula [21] for the transmission amplitude:

\begin{array}{l} S_{11} (λ) = S_{11}^{(0)} - \frac{i Λ_{n}}{(E (λ) - E_{n} - Δ_{n}) + i (Γ_{0} + Σ_{n})}, \\ Λ_{n} = | C_{1} |^{2} Ψ_{n}^{2} (z_{1}), Δ_{n} = - Re (D_{1}) Ψ_{n}^{2} (z_{1}), Σ_{n} = Im (D_{1}) Ψ_{n}^{2} (z_{1}) . \end{array}

Thus, coupling to the transverse WG₁ replaces the resonance width

Γ_{0}

with

Γ_{n} = Γ_{0} + Σ_{n}

, where the self-energy

Σ_{n}

is proportional to

Im (D_{1})

and describes additional losses due to leakage into the waveguide [12], and also shifts the resonance position by Δ_n, which is proportional to

Re (D_{1}) .

The self-energy term

Σ_{n}

is also proportional to the intensity of WGM

Ψ_{n} (z)

at position

z_{1}

of WG₁ [2]. In particular,

Σ_{n}

vanishes at the WGM nodes (here

Λ_{n} = 0

and the resonant WGM is uncoupled from the waveguide and, hence, is invisible or dark). Alternatively, the largest coupling to the microfiber is achieved at the WGM antinodes.

In the case of lossless coupling Eqs. (9) and (12) yield:

S_{11} (λ) = \frac{(E (λ) - E_{n} - Δ_{n}) + i (Γ_{0} - \frac{1}{2} Λ_{n})}{(E (λ) - E_{n} - Δ_{n}) + i (Γ_{0} + \frac{1}{2} Λ_{n})},

Equation (14) is the well-known Breit-Wigner resonance representation of the scattering amplitude [12].

The wavelength/spatial dependence in the neighborhood of a localized state can be elucidated by considering the characteristic fundamental state of a harmonic oscillator with Gaussian spatial mode distribution $Ψ_{0} (z) = {(2 / π)}^{1 / 4} z_{w}^{- 1 / 2} \exp [- {(z / z_{w})}^{2}]$ . For example, in the case of lossless coupling Eq. (14) yields the following wavelength/spatial dependence of the transmission amplitude:

S_{11} (λ) = \frac{(λ - λ_{r e s}) - i (γ_{r e s} - γ_{C} \exp [- {(z / z_{w})}^{2}])}{(λ - λ_{r e s}) - i (γ_{r e s} + γ_{C} \exp [- {(z / z_{w})}^{2}])},

where

γ_{C} = {(2 π)}^{- 5 / 2} λ_{r e s}^{3} n_{f 0}^{- 2} z_{w}^{- 1} | C_{1} |^{2}

.

Figure 2 shows the surface plots of the transmission amplitude $| S_{11} (λ) |$ as a function of distance along the fiber and wavelength deviation for $λ_{r e s} = 1.5 μ m$ , $γ_{r e s} = 0.1 pm$ , and $n_{f 0} = 1.5$ . The axial FWHM of the mode is set to to 20 μm, which corresponds to $z_{w} = 12 μ m .$ In Fig. 2(a)-2(e) the plots of transmission amplitude for ideal lossless coupling ( $S_{11}^{(0)} = 1, D_{1} = i | C_{1} |^{2} / 2$ ) are shown. It is seen that the resonance width grows approaching the WGM maxima. For relatively small coupling, the surface plot has a single maximum. For larger coupling, the spatial behavior of the transmission amplitude splits into two peaks indicating the condition of critical coupling. Next, the parameters $| C_{1} |^{2} = 0.026 μ m^{- 1},$ $D_{1} = 0.022 + 0.02 i μ m^{- 1},$ and $S_{11}^{(0)} = 0.88 - 0.08 i$ of Fig. 2(f) correspond to the lossy transmission amplitudes of the SNAP bottle microresonator considered in Section 6. In this case, the value of internal losses in the SF, $γ_{r e s}$ , is no longer important because its contribution is much smaller than the contribution of losses due to scattering into other WGMs and into continuum. It is seen that the scattering losses result in dramatic growth of the resonance widths compared to those in Fig. 2(a)-2(e). In addition, the lossy coupling leads to the asymmetric resonances indicating the Fano effect [21]. For comparison, Fig. 2(g) shows the situation with scattering parameters similar to those in Fig. 2(f) except for the increased value of output losses (determined by the imaginary part of $D_{1}$ ) $D_{1} = 0.022 + 0.06 i μ m^{- 1} .$ Finally, the surface plot of Fig. 2(h) corresponds to the same parameters as those in Fig. 2(f) except for the opposite coupling phase shift (determined by the real part of $D_{1}$ ), $D_{1} = - 0.022 + 0.02 i μ m^{- 1} .$ In the latter case the asymmetric incline of spectral resonances is opposite to that in Fig. 2(f).

Fig. 2 Surface plots of the transmission amplitude as a function of distance along the fiber and wavelength deviation for λ = 1.5 μm, γ_res = 0.1 pm, n_f₀ = 1.5, and different coupling parameters shown on the plots.

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For a larger number of discrete SF resonances,

G (λ, z_{1}, z_{1}) = \sum_{n} \frac{| Ψ_{n} (z_{1}) |^{2}}{E (λ) - E_{n} + i Γ_{n}}

and the generalization of Eqs. (13) and (16) is straightforward.

4.2. Transmission through a uniform SF

In the case of a uniform SF, the bare Green’s function is

G (λ, z_{1}, z_{2}) = \frac{\exp [i {(E (λ) - E_{0} + i Γ_{0})}^{1 / 2} | z_{2} - z_{1} |]}{2 i {(E (λ) - E_{0} + i Γ_{0})}^{1 / 2}} .

The surface plots of the transmission amplitude

| S_{11} (λ) |

in Fig. 3 as a function of wavelength deviation and coupling parameters are found from Eqs. (8) and (17) for

λ_{r e s} = 1.5 μ m

,

γ_{r e s} = 0.1 pm

,

n_{f 0} = 1.5

. The surface plots of the lossless transmission amplitude are shown in Fig. 3(a) and 3(b). In Fig. 3(a),

Re (D_{1}) = 0

, i.e., coupling between the SF and the WG₁ does not cause a phase shift. In the presence of a phase shift, the behavior of the transmission amplitude can be more complex. This is illustrated in Fig. 3(b) for

Re (D_{1}) = 2 Im (D_{1}) .

Fig. 3(c) shows the surface plot of the lossy transmission amplitude calculated for

S_{11}^{(0)} = 0.88 - 0.08 i

taken from the experiment of Section 6 and the ratio of coupling parameters

D_{1}

and

| C_{1} |^{2}

from the same experiment. For comparison, Fig. 3(d) shows the behavior of the transmission amplitude with the same parameters as in Fig. 3(c) except for the twice greater values of

Re (D_{1}) .

Fig. 3 Surface plot of the transmission amplitude through a uniform SF as a function of wavelength deviation and coupling parameters. (a), (b) – lossless transmission (b),(c) – lossy transmission. The SF parameters are λ = 1.5 μm, γ_res = 0.1 pm, and n_f₀ = 1.5. Other parameters are given in this Fig. 3.

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5. Transmission amplitudes of a SNAP device coupled to two waveguides

Equation (8) shows that the transmission amplitude can be expressed as a rational function of the coupling parameters $C_{1}$ , $D_{1}$ , and the bare Green’s function of the SF $G (λ, z_{1}, z_{1})$ . Similarly, it would appear reasonable that for a SNAP device coupled to several waveguides, WG₁, WG₂, …, WG_N, i.e., the transmission amplitudes $S_{1 n} (λ)$ are the rational functions of the coupling parameters $C_{n}$ , $D_{n}$ , and the values of Green’s function $G (λ, z_{n_{1}}, z_{n_{2}})$ calculated at the waveguide positions $z_{n_{1}}$ and $z_{n_{2}},$ $n_{1}, n_{2} = 1, 2, ..., N$ . In this Section, this result is demonstrated for a SNAP device coupled to two waveguides (Fig. 1). The transmission amplitude of light entering and exiting WG₁, $S_{11} (λ)$ , and the transmission amplitude of light entering WG₁ and exiting WG₂, $S_{12} (λ)$ , are determined from Eq. (5) and Eq. (6) for $N = 2$ . The values of renormalized Green’s function $\bar{G} (λ, z_{1}, z_{1})$ and $\bar{G} (λ, z_{1}, z_{2})$ in these equations are expressed through the coupling parameters $C_{1}, C_{2}, D_{1}, D_{2}$ , and the values of the bare Green’s function $G (λ, z_{1}, z_{1})$ and $G (λ, z_{1}, z_{2})$ as (Appendix 2):

\bar{G} (λ, z_{1}, z_{1}) = \frac{G (λ, z_{1}, z_{1}) + D_{2} (G (λ, z_{1}, z_{1}) G (λ, z_{2}, z_{2}) - G^{2} (λ, z_{1}, z_{2}))}{(1 + D_{1} G (λ, z_{1}, z_{1})) (1 + D_{2} G (λ, z_{2}, z_{2})) - D_{1} D_{2} G^{2} (λ, z_{1}, z_{2})},

\bar{G} (λ, z_{1}, z_{2}) = \frac{G (λ, z_{1}, z_{2})}{(1 + D_{1} G (λ, z_{1}, z_{1})) (1 + D_{2} G (λ, z_{2}, z_{2})) - D_{1} D_{2} G^{2} (λ, z_{1}, z_{2})},

where, as noted in Section 4, in the case of lossless coupling,

Im D_{n} = \frac{1}{2} | C_{n} |^{2} .

5.1. Transmission through the localized states of an SF

Near a separated resonance, the bare Green’s function is defined by Eq. (12). Then Eqs. (5), (6), (18) and (19) for the lossless coupling are simplified:

S_{11} (λ) = \frac{E (λ) - E_{n} + Δ_{n} + i (Γ_{0} + Σ_{n}^{(2)} - Σ_{n}^{(1)})}{E (λ) - E_{n} + Δ_{n} + i (Γ_{0} + Σ_{n}^{(2)} + Σ_{n}^{(1)})},

S_{12} (λ) = \frac{- 2 i Ξ_{n}}{E (λ) - E_{n} + Δ_{n} + i (Γ_{0} + Σ_{n}^{(1)} + Σ_{n}^{(2)})},

Σ_{n}^{(j)} = \frac{1}{2} | C_{j} |^{2} Ψ_{n}^{2} (z_{j}), Ξ_{n} = C_{1} C_{2}^{*} Ψ_{n} (z_{1}) Ψ_{n} (z_{2}), Δ_{n} = Re (D_{1}) Ψ_{n}^{2} (z_{1}) + Re (D_{2}) Ψ_{n}^{2} (z_{2}) .

Equation (21) is similar to Eq. (14) found for a SNAP device coupled to a single waveguide where the internal losses

Γ_{0}

are replaced with

Γ_{0} + Σ_{n}^{(2)}

, taking into account the leakage through WG₂. Thus, the presence of the second waveguide modifies the widths and shifts of the individual resonances (e.g., those in a bottle microresonator); however, it does introduce qualitative changes in the spectral behavior.

5.2. Transmission through a uniform SF

If the coupling to WG₂ is negligible then the renormalized Green’s function defined by Eqs. (18) and (19) coincides with that of a single waveguide device defined by Eq. (7). Here we consider the opposite case of large coupling to WG₂. Then, for the lossless coupling to WG₁, i.e., for $S_{11}^{(0)} = 1$ , $D_{2} = \infty$ , and $D_{1} = Re (D_{1}) + \frac{i}{2} | C_{n} |^{2}$ , Eqs. (5), (18), and (20) yield

S_{11} (λ) = \frac{G (λ, z_{2}, z_{2}) - (Re D_{1} - \frac{i}{2} | C_{1} |^{2}) (G (λ, z_{1}, z_{1}) G (λ, z_{2}, z_{2}) - G^{2} (λ, z_{1}, z_{2}))}{G (λ, z_{2}, z_{2}) - (Re D_{1} + \frac{i}{2} | C_{1} |^{2}) (G (λ, z_{1}, z_{1}) G (λ, z_{2}, z_{2}) - G^{2} (λ, z_{1}, z_{2}))} .

Figure 4 shows the surface plots of the absolute value of the transmission amplitude found from this equation for the uniform SF, when the bare Green’s function is defined by Eq. (17). In Fig. 4(a), it is assumed that

Re (D_{1}) = 0

, i.e., that coupling to WG₁ does not shift the resonances. To understand the effect of shifting, Fig. 4(b) shows the case when

Re (D_{1}) = 2 Im (D_{1}) .

Fig. 4(a) shows that the zero phase shift corresponds to symmetric resonance peaks, while, according to Fig. 4(b), the shapes of resonances become asymmetric when the real and imaginary parts of

D_{1}

are comparable. The interesting feature of spectra in Fig. 4 is the narrow transmission resonances appearing due to the interference between WGMs reflected from WG₁ and WG₂. The widths of these resonances are proportional to

1 / | C_{1} |^{2}

and become very small with the growth of SF/MF₁ coupling.

Fig. 4 Surface plots of the transmission amplitude S₁₁(λ) through a uniform SF coupled to two waveguides, WG₁ and WG₂, as a function of wavelength deviation and the coupling parameter |C₁|² to WG₁. Coupling to WG₂ is assumed to be large compared to coupling to WG₁. The waveguides are spaced by 150 µm. (a) – D₁ = i|C₁|²/2 (b) – D₁ = (1 + i/2)|C₁|². The SF parameters are λ = 1.5 μm, γ_res = 0.1 pm, and n_f₀ = 1.5.

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6. A SNAP bottle microresonator and a chain of 10 coupled microresonators: theory vs. experiment

Experimentally, SNAP bottle microresonators are fabricated along the surface of a 19 μm radius fiber following [3,4] by the local nanoscale variation of the effective fiber radius with a focused CO₂ laser beam. The introduced effective radius variation is measured using the microfiber scanning method [22, 23]. In this method, a biconical optical fiber taper having a micron diameter waist is positioned normal to the test SF, as e.g., WG₁ in Fig. 1. The waist of the taper is translated along the SF and touches it periodically at the contact points where the transmission spectra of the taper are measured. These spectra are used for the determination of the SF radius variation. In our experiments, the resonant transmission amplitude spectra are determined with the Luna Optical Vector Analyzer (wavelength resolution 1.3 pm). The experimental data is taken along the SF at contact points spaced by 2 µm. In previous publications [3,4], the effective radius variation of similar microresonators was approximately determined by enveloping the spectral resonances and rescaling the wavelength variation $Δ λ$ to radius variation $Δ r$ with the relation $Δ λ / λ_{r e s} = Δ r / r_{0}$ . However, the theory enabling the accurate determination of the effective radius variation and the WG/SF coupling parameters from the experimental data has not been developed.

It is shown in this Section that the Schrödinger Eq. (2) and Eq. (4) for the transmission amplitude allow the accurate determination of the characteristics of a SNAP device from the experimental data. To this end, the effective radius variation of the SF is determined by fitting the spectrum found by the numerical solution of the Schrödinger Eq. (2) to the experimental spectrum. Two examples are considered: a single SNAP bottle microresonator and a chain of 10 coupled microresonators.

In the first experiment, a single SNAP bottle resonator is fabricated and investigated. This microresonator was created by a single exposure of the focused CO₂ laser beam. The surface plot of the resonant transmission amplitudes for this microresonator is shown in Fig. 5(a) . It is seen that the spectral resonances are broadened and shifted due to coupling to the microfiber. Therefore, the actual spectrum of the uncoupled bottle resonator is determined in the regions where the coupling is minimized, i.e., near nodes of the field and in evanescent regions. The set of transmission amplitudes, which correspond to the microfiber positions 50 μm, 94 μm, and 120 μm and jointly contain all of such narrow resonances indicated by dashed arrows, are shown in Fig. 6(a) . These resonances are numerically fitted to the eigenvalues of the Schrödinger Eq. (2) where the effective fiber radius variation of the SNAP bottle microresonator is parameterized as a sum of Gaussian and Lorentzian shapes:

Δ r_{e f f} (z) = Δ r_{0} [\exp (- \frac{{(z - z_{0})}^{2}}{ζ^{2}}) + \frac{ε}{1 + [{(z - z_{0})}^{2} / ζ^{2}]}]

The eigenvalues of Eq. (2) are found as the singularities of the bare Green’s function

G (λ, z_{1}, z_{1})

where

z_{1}

is taken close to the center of symmetry of

Δ r_{e f f} (z)

. The result of fitting yields

Δ r_{0} = 14.1 nm, ζ = 10.9 μ m, z_{0} = 120 μ m, ε = 0.22,

and completely determines the effective radius variation, Eq. (25). The theoretical wavelength eigenvalues calculated with these parameters for

G (λ, 130 μ m, 130 μ m)

and experimental eigenvalues found from Fig. 6(a) are compared in Fig. 6(b). Next, parameters

S_{11}^{(0)},

| C_{1} |^{2},

and

D_{1}

of the transmission amplitude

S_{11} (λ)

, Eq. (8), are determined by substitution of the bare Green’s function

G (λ, z_{1}, z_{1})

calculated for the radius variation determined by Eqs. (25) and (26) into Eq. (8) and fitting it to the experimental transmission amplitude at

z = 120 μ m .

The result of fitting shown in Fig. 6(c) performed in the vicinity of the fundamental resonance (darken area in Fig. 6(c)) yields:

S_{11}^{(0)} = 0.879 - 0.084 i, | C_{1} |^{2} = 0.026 μ m^{- 1}, D_{1} = 0.022 + 0.02 i μ m^{- 1} .

As expected, these values satisfy the conditions of Eqs. (10) and (11) following from energy conservation. Note that the absence of several resonant peaks on the theoretical spectral dependence in Fig. 6(c) is explained by the fact that this spectrum corresponds to the microresonator plane of symmetry. At this plane, the antisymmetric axial modes are dark and do not show up on the spectrum. On the other hand, these resonances are clearly seen at the theoretical spectrum in Fig. 6(b). Finally, the calculation of the transmission amplitude, Eq. (8), with parameters given by Eq. (27) and the radius variation given by Eqs. (25) and (26) yields excellent correspondence between the experimental data shown in Fig. 5(a) and the theoretical calculations shown in Fig. 5(b). In addition there is an agreement between the surface plots of the Gaussian mode transmission amplitude shown in Fig. 2(f) with those in Fig. 5 in the vicinity of the fundamental mode wavelength (~1549.5 nm).

Fig. 5 (a) – The surface plot of the experimentally measured resonant transmission amplitude through the microfiber scanned along the SNAP bottle resonator with 2 µm steps. (b) – Surface plot of the theoretically calculated transmission amplitude fitting the experimental data of Fig. 5(a).

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Fig. 6 (a) – The experimental transmission amplitudes of the bottle microresonator from Fig. 5(a) measured at the axial microfiber positions 50 μm, 94 μm, and 120 μm. (b) – The bare Green’s function calculated with the radius dependence defined by Eq. (27) fitting narrow experimental resonances in Fig. 6(a). (c) – Fitting the experimental transmission amplitude at axial position 120 µm.

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In the second experiment, a chain of 10 coupled SNAP microresonators illustrated in the inset of Fig. 7 is fabricated and investigated. This chain was created by ten successive exposures of the focused CO₂ laser beam spaced by 50 μm. The surface plot of the resonant transmission amplitudes for this chain is shown in Fig. 7(a). It is seen that, due to mutual coupling, the fundamental modes of microresonators form a transmission band (outlined by a dashed line) separated from other modes by a bandgap. The magnified surface plot of the fundamental transmission band is shown in Fig. 7(b). The observed spectral plots can be described theoretically by approximating the effective fiber radius variation as:

Δ r_{e f f} (z) = Δ r_{0} {\begin{matrix} \exp (- z^{2} / ζ^{2}), z < 0, \\ \frac{a^{2}}{2 π^{2} ζ^{2}} (\cos (2 π z / a) - 1) + 1, 0 < z < N a, \\ \exp {- {[z - (N - 1) a]}^{2} / ζ^{2}}, z > (N - 1) a, \end{matrix}

where

a = 50 μ m

is the microresonator spacing,

N = 10

is the number of microresonators, and parameters

Δ r_{0}

and

ζ

are the fitting parameters. Equation (28) describes periodic oscillations of the effective radius variation in the exposed region. Parameters

Δ r_{0}

and

ζ

are determined by fitting the experimental distribution of higher-order resonances below the bandgap and also the bandgap position in Fig. 7(a). The fitting, which is performed similar to the previous example (see dashed arrows connecting the experimental and theoretical spectra in Fig. 7(a) and 7(c)), yields:

Δ r_{0} = 6.9 nm, ζ = 19.4 μ m, a = 50 μ m, N = 10.

The resultant effective radius variation is shown in Fig. 7(c). The parameters of the transmission amplitude distribution in Fig. 7(c) are:

S_{11}^{(0)} = 0.83 - 0.15 i, | C_{1} |^{2} = 0.028 μ m^{- 1}, D_{1} = 0.022 + 0.026 i μ m^{- 1} .

The magnified distribution of this plot near the region of fundamental transmission band is shown in Fig. 7(d). Notice a remarkable agreement between the fine features of this plot and of the experimental plot shown in Fig. 7(b). A small deviation of the experimental spectra in Fig. 7(b) from the theoretical distribution in Fig. 7(d) can be approximately taken into account by perturbing

Δ r_{e f f} (z)

with a Gaussian radius variation

δ r_{e f f} (z) = δ r_{0} \exp [- {(z - z_{0})}^{2} / ζ_{0}^{2}]

with

δ r_{0} = 0.1 nm,

z_{0} = 350 μ m,

and

ζ_{0} = 100 μ m

, shown at the bottom of Fig. 7(e). The small angstrom amplitude of

δ r_{e f f} (z)

allows us to suggest an angstrom precision in the fabrication of this microresonator chain.

Fig. 7 (a) – The surface plot of the experimentally measured resonant transmission amplitude through the microfiber scanned along the chain of 10 SNAP microresonators with 2 µm steps. (b) – Surface plot of Fig. 7(a) magnified in the region of the fundamental transmission band. (c) – Surface plot of the theoretically calculated transmission amplitude fitting the experimental data of Fig. 7(a). (d) – Surface plot of Fig. 7(c) magnified in the region of the fundamental transmission band. (e) – Surface plot of Fig. 7(d) perturbed by the effective radius variation shown at the bottom of Fig. 7(e).

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7. Discussion and summary

The theory developed in this paper can be directly applied both to modeling of SNAP devices and to their evaluation from the experimental data. The basic assumptions of this theory following from the experimental conditions are (a) adiabatic separation of variables and reduction of a 3D problem for the bare Green’s function to a 1D Schrödinger equation and (b) the axial dimension of SF/waveguide coupling region being much smaller than the characteristic axial wavelength of this equation. The latter assumption allows us to arrive at the 1D Schrödinger equation for the renormalized Green’s function of the SNAP device, Eq. (4). In this equation, coupling to a waveguide is described by a short-range potential characterized by a complex-valued coupling constant. As a result, the transmission amplitudes of a SNAP device coupled to N waveguides are expressed through the values of the bare Green’s function calculated at the waveguide positions and 3N complex constants $S_{1 n}^{(0)}$ , $C_{n}$ , and $D_{n}$ . General inequality relations between these parameters following from energy conservation are obtained. The developed theory is applied to the investigation of the simplest SNAP devices: a separated resonance state, a uniform fiber, and a bottle microresonator coupled to one or two waveguides. The cases of ideal (lossless) coupling as well as lossy coupling to waveguides are considered. It is found that these simplest devices can exhibit quite complex resonance spectra depending on the values of the coupling parameters. The problem of the theoretical determination of constants $S_{1 n}^{(0)}$ , $C_{n}$ , and $D_{n}$ through the SF and waveguide parameters is not addressed in this paper since it is often more straightforward and accurate to determine these values experimentally. As an example, these constants and the SF radius variation are determined from the experimental data of a SNAP bottle microresonator coupled to a single waveguide. In this case, the excellent agreement between the theory and the experiment is demonstrated. Generally, it is quite interesting to determine the parameters $S_{1 n}^{(0)}$ , $C_{n}$ , and $D_{n}$ theoretically and, in particular, to get a better understanding of how they are related to each other. The developed theory provides a way to investigation of the transmission properties of SNAP devices with a complex variation of the SF radius, e.g., chains of coupled microresonators considered in Section 6, including the optimization of coupling between microresonators and waveguides [16] to achieve the required filtering and group delay characteristics.

Appendix 1 Transmission amplitude of a SNAP device

The WGMs in the SF coupled to waveguides can be found by adiabatic separation of variables, similar to Eq. (1):

U_{m, p, q}^{(r e n)} (r) = Ψ_{m, p, q}^{(r e n)} (z) Ξ_{m, p} (ρ) \exp (i m φ)

where function

Ψ_{m, p, q}^{(r e n)} (z)

satisfied Eq. (4) and

r = (z, ρ, φ)

is expressed in cylindrical coordinates. The Green’s function is constructed from these solutions as

G (λ, r_{1}, r_{2}) = \sum_{m, p, q} \frac{U_{m, p, q}^{(r e n)} (r_{1}) U_{m, p, q}^{(r e n) *} (r_{2})}{E (λ) - E_{m, p, q}}

Using Eq. (A1.1) we can summarize Eq. (A1.2) over q,

G (λ, r_{1}, r_{2}) = \sum_{m, p} G_{m, p}^{(r e n)} (λ, z_{1}, z_{2}) Ξ_{m, p} (ρ_{1}) Ξ_{m, p} (ρ_{2}) \exp [i m (φ_{1} - φ_{2})],

where

G_{m, p}^{(r e n)} (λ, z_{1}, z_{2})

is the renormalized Green’s function of Eq. (4) (see also Appendix A2 in [2]).

We define $χ_{n} (r)$ as the wave propagating along the single mode waveguide WG_n in the absence of SF when the waveguides do not couple to each other. The SF is introduced with a potential V(r) so that the T-matrix, which determines the inter-waveguide coupling due to the presence of the SF, is expressed through the Green’s function of Eq. (A1.2) as $T = V + V G V$ [24]. Then, the transmission amplitudes between the waveguide modes are determined by the elements of the S-matrix [24]:

S = 1 - i T = 1 - i (V + V G V)

From Eq. (A1.4), for the amplitude of the direct transmission through the waveguide WG₁ we have:

S_{11} (λ) = 1 - i \int d r_{1} | χ_{1} (r_{1}) |^{2} V (r_{1}) - i \int d r_{1} d r_{2} χ_{1} (r_{1}) V (r_{1}) G (λ, r_{1}, r_{2}) V (r_{2}) χ_{1}^{*} (r_{2})

In the vicinity of the resonance

E (λ_{m_{0}, p_{0}}) = E_{m_{0}, p_{0}}

, only one term in Eq. (A1.3) with

m = m_{0}

and

p = p_{0}

has resonance behavior, while all others are slow functions of E which can be set to constants. In SNAP, the axial size of the waveguide/SF coupling area is small compared to the axial wavelength (i.e., to the characteristic axial variation length of the Green’s function

G_{m, p}^{(r e n)} (λ, z_{1}, z_{2})

). For this reason, the coordinates

z_{j}

in Eq. (A1.5) can be set equal to the axial coordinate of WG₁,

z_{1}

, and

G_{m, p}^{(r e n)} (λ, z_{1}, z_{1})

can be factored out of the integral. Thus, substitution of Eq. (A1.3) into Eq. (A1.5) yields:

\begin{array}{l} S_{11} (λ) = S_{11}^{(0)} - i | C_{1} |^{2} G_{m_{0}, p_{0}}^{(r e n)} (λ, z_{1}, z_{1}), \\ C_{1} = \int d r χ_{1} (r) V (r) Ξ_{m_{0}, p_{0}} (ρ) \exp (i m_{0} φ) . \end{array}

Here

S_{11}^{(0)}

includes contribution of non-resonant terms. If this contribution is negligible, e.g., for very small “ideal” [20] waveguide/SF coupling, we have

S_{11}^{(0)} = 1

, while generally

| S_{11}^{(0)} | < 1

. It is important that, due to its non-resonant nature,

S_{11}^{(0)}

is constant in the considered wavelength interval with good accuracy.

Similarly to Eq. (A1.6), the amplitude of transmission from WG₁ to WG_n is

\begin{array}{l} S_{1 n} (λ) = S_{1 n}^{(0)} - i C_{1} C_{n}^{*} G_{m_{0}, p_{0}}^{(r e n)} (λ, z_{1}, z_{n}), \\ C_{n} = \int d r χ_{n} (r) V (r) Ξ_{m_{0}, p_{0}} (ρ) \exp (i m_{0} φ), \end{array}

where constant

S_{1 m}^{(0)}

is the non-resonant component. In the main text, the indices

m_{0}

and

p_{0}

are omitted and the notation

\bar{G} (λ, z_{1}, z_{n}) = G_{m_{0}, p_{0}}^{(r e n)} (λ, z_{1}, z_{n})

is used.

Appendix 2 Calculation of the Green’s functions

The bare Green’s function of Eq. (2) is defined through the solutions of Eq. (2) $Ψ_{1} (λ, z)$ and $Ψ_{2} (λ, z)$ , which satisfy the boundary conditions in the left-hand side and in the right-hand sides, respectively, and have the unity Wronskian,

Ψ_{1 z} (λ, z) Ψ_{2} (λ, z) - Ψ_{1} (λ, z) Ψ_{2 z} (λ, z) = 1,

as

G (λ, z_{1}, z_{2}) = {\begin{matrix} Ψ_{1} (λ, z_{1}) Ψ_{2} (λ, z_{2}) & z_{1} > z_{2} \\ Ψ_{1} (λ, z_{2}) Ψ_{2} (λ, z_{1}) & z_{1} < z_{2} \end{matrix} .

The renormalized Green’s function of Eq. (4) is expressed through the linear combination of the same functions,

K_{1 n} Ψ_{1} (λ, z) + K_{2 n} Ψ_{2} (λ, z)

, where the coefficients

K_{1 n}

and

K_{2 n}

correspond to the axial interval

z_{n} < z < z_{n + 1}

. These coefficients are determined by 2n matching conditions at the points of coupling

z_{n}

:

\begin{array}{l} \bar{G} (λ, z, z_{n}) |_{z ↘ z_{n}} - \bar{G} (λ, z, z_{n}) |_{z ↘ z_{n}} = 0, n = 1, 2, ..., N \\ {\bar{G}}_{z} (λ, z, z_{1}) |_{z ↘ z_{1}} - {\bar{G}}_{z} (λ, z, z_{1}) |_{z ↘ z_{1}} = 1 + i D_{1} \bar{G} (λ, z, z_{1}) \\ {\bar{G}}_{z} (λ, z, z_{1}) |_{z ↘ z_{n}} - {\bar{G}}_{z} (λ, z, z_{1}) |_{z ↘ z_{n}} = i D_{n} \bar{G} (λ, z_{2}, z_{1}), n = 2, 3, ... N \end{array}

where the tilted arrow in

z ↘ z_{n}

and

z ↗ z_{n}

means the limit to point

z_{n}

for

z > z_{n}

and

z < z_{n}

, respectively. It is assumed in Eqs. (A2.3) that WG₁ is the only input and output waveguide, while all others are output only waveguides.

From Eqs. (A2.3), for a SNAP device with a single waveguide WG₁, $N = 1$ , the renormalized Green’s function is defined by Eq. (7).

For a SNAP device with two waveguides WG₁ and WG₂, $N = 2$ , the continuity condition (the first equation in Eqs. (A2.3)) is incorporated for the renormalized Green’s function in the form:

\bar{G} (λ, z, z_{1}) = {\begin{matrix} [A_{1} Ψ_{1} (λ, z_{1}) + A_{2} Ψ_{2} (λ, z_{1})] Ψ_{1} (λ, z) Ψ_{2} (λ, z_{2}), \\ [A_{1} Ψ_{1} (λ, z) + A_{2} Ψ_{2} (λ, z)] Ψ_{1} (λ, z_{1}) Ψ_{2} (λ, z_{2}), \\ [A_{1} Ψ_{1} (λ, z_{2}) + A_{2} Ψ_{2} (λ, z_{2})] Ψ_{1} (λ, z_{1}) Ψ_{2} (λ, z), \end{matrix} \begin{matrix} z < z_{1}, \\ z_{1} < z < z_{2}, \\ z > z_{2} . \end{matrix}

Then Eqs. (A2.3) are reduced to the system of two linear equations for the coefficients

A_{1}

and

A_{2}

. The solution of this system yields Eqs. (18) and (19) for the renormalized Green’s function.

Appendix 3 Limitations on the transmission amplitude and coupling parameters

For an inactive SF coupled to a single waveguide WG₁, the conservation of energy requires $| S_{11} (λ) | < 1$ . This inequality applied to Eq. (8) yields:

Λ_{0} + 2 Re (Λ_{1} G) + Λ_{0}^{- 1} (| Λ_{1} |^{2} - | C_{1} |^{4}) | G |^{2} > 0

where, for brevity,

G = G (λ, z_{1}, z_{1})

and

\begin{array}{l} Λ_{0} = 1 - | S_{11}^{(0)} |^{2}, \\ Λ_{1} = i | C_{1} |^{2} (^{S_{11}^{(0)}) *} - i Λ_{0} D_{1} . \end{array}

The Green’s function of an inactive SF obeys the general condition

Im (G)

<0 [12], while there are no formal restrictions on

Re (G)

. Thus, Eq. (A3.1) should be satisfied for all complex values of G with

Im (G)

<0.

Let us start with the case $Im (G)$ =0. Then Eq. (A2.1) is simplified to

Λ_{0} + 2 Re Λ_{1} Re G + Λ_{0}^{- 1} (| Λ_{1} |^{2} - | C_{1} |^{4}) {(Re G)}^{2} > 0,

which should be valid for all

- \infty < Re (G) < \infty

. This condition is equivalent to

Λ_{0} > 0,

| Im Λ_{1} | > | C_{1} |^{2} .

The minimum value of the left-hand side of Eq. (A3.3) is achieved for

Re G = - Λ_{0} Re (Λ_{1}) {(| Λ_{1} |^{2} - | C_{1} |^{4})}^{- 1}

and is equal to

Ξ_{0} = Λ_{0} (1 - \frac{Re (Λ_{1})}{| Λ_{1} |^{2} - | C_{1} |^{4}}) .

Generally, Eq. (A3.1) is satisfied under the conditions determined by Eq. (A3.4) and Eq. (A3.5) and

Ξ_{0} - 2 Im Λ_{1} Im G + Λ_{0}^{- 1} (| Λ_{1} |^{2} - | C_{1} |^{4}) {(Im G)}^{2} > 0

for all

Im (G) \leq 0

. Eq. (A3.7) is equivalent to the condition

Im Λ_{1} > | C_{1} |^{2},

which is stronger than Eq. (A3.5).

To summarize, the inequality $| S_{11} (λ) | < 1$ following from the conservation of energy imposes a limitation on parameters C₁, D₁, and $S_{11}^{(0)}$ of the transmission amplitude $S_{11} (λ)$ , which are determined by Eqs. (A3.4) and (A3.8) equivalent to Eqs. (10) and (11), respectively.

Acknowledgments

The author is grateful to Y. Dulashko for assisting in the experiments and to D. J. DiGiovanni and J. M. Fini for useful discussions.

References and links

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Theory of SNAP devices: basic equations and comparison with the experiment

Abstract

1. Introduction

2. WGMs in a SNAP fiber in the absence of input/output waveguides

3. Theory of a SNAP device

4. Transmission amplitude of a SNAP device coupled to a single waveguide

4.1. Transmission through localized states of an SF

4.2. Transmission through a uniform SF

5. Transmission amplitudes of a SNAP device coupled to two waveguides

5.1. Transmission through the localized states of an SF

5.2. Transmission through a uniform SF

6. A SNAP bottle microresonator and a chain of 10 coupled microresonators: theory vs. experiment

7. Discussion and summary

Appendix 1

Transmission amplitude of a SNAP device

Appendix 2

Calculation of the Green’s functions

Appendix 3

Limitations on the transmission amplitude and coupling parameters

Acknowledgments

References and links

Cited By

Figures (7)

Equations (49)

Optics Express