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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 2 — Feb. 1, 2014
  • pp: 411–420
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Mathematical modeling of Fabry–Perot resonators: II. Uniformly converging multimode equivalent-circuit models

G. Hugh Song  »View Author Affiliations


JOSA A, Vol. 31, Issue 2, pp. 411-420 (2014)
http://dx.doi.org/10.1364/JOSAA.31.000411


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Abstract

Based on complex-variable analysis of a Fabry–Perot resonator as a multimode nonsymmetric two-port waveguide device, two versions of equivalent-circuit configurations are presented: Starting from a renewed study on single-mode two-pole circuits, we develop two respective multimode equivalent circuits of an almost identical configuration: one for the reflection coefficient and the other for the pass-through transmission coefficient. In the mathematics language of complex-variable analysis, the two models successfully “approximate” the two scattering coefficients through two “uniformly converging” partial-fraction series expansions.

© 2014 Optical Society of America

1. INTRODUCTION

Almost all resonator-based waveguide devices in the frequency range from microwave to optical frequencies are basically designed based on the Fabry–Perot resonator (FPR) structure [1

1. C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).

] or on its four-port variation [2

2. H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992). [CrossRef]

]. To name a few waveguide resonators with two-holed diaphragms [3

3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.

], standard bulk-optic FPRs, planar-optic ring resonators [4

4. E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969). [CrossRef]

,5

5. I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).

], photonic-crystal resonators [6

6. K. Hwang and G. H. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005). [CrossRef]

], Bragg resonators [7

7. A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of Bragg gratings and its applications to Fabry–Pérot cavities,” J. Opt. Soc. Am. A 20, 273–281 (2003). [CrossRef]

], etc., belong to the broader class of FPRs in metal-tubic, microwave-monolithic, bulk-optic, and integrated-optic implementations throughout the radio to optical frequency ranges.

A good equivalent-circuit model for such a linear time-invariant electromagnetic device is always an efficient aid in understanding key features of the physical device under consideration. Exemplary use of such equivalent circuits for waveguide devices in microwave and optical science and engineering can be found, for example, recently in Ref. [7

7. A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of Bragg gratings and its applications to Fabry–Pérot cavities,” J. Opt. Soc. Am. A 20, 273–281 (2003). [CrossRef]

] as well as historically in Ref. [8

8. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds., Principles of Microwave Circuits (McGraw-Hill, 1948).

] during the early years of microwave engineering.

Some elementary equivalent circuits for FPRs were employed in the analysis for a pair of holed diaphragms in microwaves [9

9. W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961). [CrossRef]

,10

10. R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).

], a mirror cavity in laser optics [11

11. A. E. Siegman, Lasers (University Science, 1986), pp. 934–939.

], and quarter-wave-shifted distributed-Bragg reflectors in integrated optics [12

12. H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992). [CrossRef]

]. In ultrafast optics, use of such an equivalent-circuit model turned out to be especially effective in understanding the working principle of mode-locked lasers [13

13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

].

So far in literature, equivalent-circuit single-mode [13

13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

] and multimode [3

3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.

] FPR models are analyzed mostly from phenomenological viewpoints. A certain multimode model, as the one in Ref. [3

3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.

], may only be called an “input-equivalent multimode circuit,” because the “output circuit” for pass-through is not realized explicitly. Losing too much rigor, such models do not even pass the basic impedance/admittance equivalence test.

In this respect, we find that two ideal multimode electric-circuit models realizing the uniformly converging complex rational functions in an exact manner for a given FPR are desired:
  • 1. one for the “reflection-equivalent model” for the reflection coefficient using only lumped circuit-elements and
  • 2. the other for the “pass-through-equivalent model” that simulates the pass-through coefficient over the entire range of a free-spectral range.

A prerequisite complex-variable analysis for the spectral data is essential and the relevant work has been documented by the present author in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

]. Some key results from the latter work are presented below along with an introduction of important definitions and useful formulas for the work of the present part.

A. Proposition of Definitions and Needed Formulas for Modeling Fabry–Perot Resonators

Inside a cavity of a FPR, filled with a homogeneous medium of length d between two mirrors A and B, let β(ω) be the complex propagation constant for the lightwave at optical frequency ω. Without loss of generality, we suppose that any phase change inside the mirrors is absent, as was imposed in Eq. (8) in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

]. Then the jth resonance frequency ωj is determined by
d·Reβ(ωj)jπ,j=0,1,2,.
(1)

Henceforth in this paper, we presume that the FPR medium is dispersion-free for every resonance frequency for simplicity. Then,
ωjjω1=2πj/τ,j=0,1,2,
(5)
for round-trip time τ2d[Redβ/dω] turning constant, whereas those reciprocated decay time constants
γAlnrA/τ,γBlnrB/τ,
(6)
γin2dImβ(ω)/τ
(7)
become mode-independent.

From our study in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

], for a FPR with two mirrors and the inner medium made of such a dispersion-less, reciprocal media, application of the logarithmic differentiation to Eq. (2) has yielded a partial-fraction series
1R⃗(ω)ηA+[1rArAhA2]j=ω1/2ijω1+γ⃗iω,
(8)
displaying the zeros of the complex-valued function explicitly, where
ηA12[1rA+rAhA2]
(9)
is the properly determined leading constant term worked out at Eq. (27) in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

]. Here,
γ⃗γinγA+γBτ1lnhA2
(10)
is nonzero in general from Eqs. (6) and (7) and is common for all resonance modes in such a FPR.

In the following presentation, we use Laplace-transform variable s for iω rather than kReβ(ω) in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

], giving an explicit expression for frequency dependence. We then introduce a finite series
Σ⃗(N)(s)=[1/rArA/hA2]τ1×{1γ⃗+s+j=1N2s+2γ⃗[s+γ⃗]2+j2ω12}
(11)
to pick up the summation term from Eq. (8) as
1/R⃗(ω)=ηA+Σ⃗()(iω).
(12)

2. TWO-POLE SINGLE-MODE LUMPED-ELEMENT EQUIVALENT CIRCUITS

Only for the present section, we presume that the mirrors are lossless, so that the hypotenuses in Eq. (3)
hA=hB1.
(13)
Actually, a lossy mirror plate may be treated as another lossy FPR, viz., a Fabry–Perot étalon, with two symmetric lossless reflecting surfaces. Therefore, in the fundamental level, the above condition is not so limiting.

Then, the two scattering coefficients in Eqs. (4) and (2) are expressed simply and conveniently as
T(ω)sinh[γAτ]sinh[γBτ]sinh{[γxiω]τ/2},
(14)
R⃗(ω)sinh{[γ⃗iω]τ/2}sinh{[γxiω]τ/2}=γ⃗γxj=1iω/[γ⃗+ijω1]1iω/[γx+ijω1],
(15)
respectively, for γ⃗γin+τ1ln[rA/rB] from Eq. (10) with hA1, whereas
γxγin+γA+γB
(16)
regardless of Eq. (13). The numerator in the right-hand side of Eq. (14) is to be implemented by
1/rArA/hA21/rArA2sinh[γAτ]
(17)
as hA1 according to Eq. (6).

Around a single selected resonance frequency ωx of a highly reflective FPR, sinh[γAτ]γAτ in Eq. (17), the reflection coefficient for an incident wave from side A becomes
Γ⃗(s)2γAs+γx+iωx1
(18)
approximately after retaining a single fraction. Evidently, this is equivalent to Collin [10

10. R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).

] 2nd ed., Eq. (7.82) and to Haus [13

13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

] Eq. (7.37). They both presented a simple equivalent LGC-circuit approximating the above expression. All subsequent papers [2

2. H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992). [CrossRef]

,6

6. K. Hwang and G. H. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005). [CrossRef]

,16

16. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14). [CrossRef]

] of Haus and his co-workers accepted this proposal in their works.

Actually, the single-fraction formula in Eq. (18) is not physically realizable with any circuit elements, because a scattering data from any physical device, regardless of a real FPR or an equivalent circuit, must be symmetric with respect to the real axis in the complex-s plane. Therefore, the proper single-mode expression must carry a pair of symmetric poles at γx±iωx for both scattering coefficients for the simplest possible circuit in Fig. 1.

Fig. 1. Simple lumped-element equivalent circuit approximately representing a Fabry–Perot resonator (FPR) near one resonance frequency ωx1/LxC among infinitely many such resonance frequencies. To match with the external wave admittance Yc, two ideal admittance transformers with winding ratios θA and θB are attached.

A. Single-Mode Equivalent Admittance Circuit

First for Fig. 1, the circuit pass-through coefficient for voltage signals for the device inside the dashed-rectangular box is found as
ϒ⃗V(s)=2YA[YA+YB+G]+sC+1/[Rx+sLx].
(19)
We take
GG+YB+YA
(20)
in the denominator, so that
ϒ⃗V(s)=2YAC·Rx/Lx+s[ssx][ssx*]
(21)
in terms of two poles, one at
sx=12[GC+RxLx]+i1CLx14[GCRxLx]2
(22)
and its complex-conjugate sx*.

The corresponding reflection coefficient is found as
Γ⃗V(s)ϒ⃗V(s)1=ss⃗ssx·ss⃗*ssx*
(23)
in terms of sx above and
s⃗12[G⃗C+RxLx]+i1CLx14[RxLxG⃗C]2,
(24)
G⃗G+YBYA=G2YA.
(25)

The ultimate pass-through coefficient including the effect of the two impedance transformers is found by
ϒP(s)=ϒ⃗V(s)YB/YA,
(26)
distinguished from ϒ⃗V(s) at Eq. (21) based on the voltage signal. The modification makes the coefficient based on square-root-power amplitudes between the input and the output.

Based on some obvious connections to the parameters of a FPR, let us set
LxC1/ωx2,
(27)
{YA/CYB/C}2τ{sinh[γAτ]sinh[γBτ]}{2γA2γB}.
(28)

Next, we have two basic lossy circuit elements in the box at Fig. 1: resistance Rx in series to an inductance and conductance G in parallel to a capacitance. They together represent attenuation of the wave amplitude in the internal medium of the FPR, showing up as Rx/2Lx and G/2C in Eqs. (22) and (24). Hence, we set
Rx/2Lx+G/2Cγin
(29)
from Eq. (7) as a reasonable condition, with which the scattering amplitudes of the spectral response near the resonance must fit with those of the corresponding true spectral response based on Eqs. (14) and (15).

B. Poles Set at the Resonance Frequency

It appears that we still have a freedom in choosing Rx/Lx. One can choose it so that either Imsx or Ims⃗ may be set exactly at ωx1/LxC, although, in a highly effective FPR with highly reflective mirrors, the difference becomes negligible. For the first circuit model however, we now choose
Imsx=ωx,
(30)
which will let the peak of resonance pass-through show up exactly at ωx at Eq. (24). Indeed, the latter choice gives a simpler analysis than the other choice.

Let us thus choose
Rx/Lx=G/Cγxγin+γB+γA,
(31)
G/CγinγBγA,
(32)
satisfying Eqs. (16) and (29). As a model, having a negative parameter in either G yielding Res⃗>0 does not matter. Overall passiveness is determined by Resx<0. Hence, the imaginary part of the pole set at
sxγx+iωxG/C+i/CLx
(33)
at Eq. (22) coincides with ωx1/LxC exactly, whereas
s⃗=γ⃗+iωx24γA2
(34)
follows from Eq. (24) with G⃗(s) at Eq. (25).

With the choice of Eqs. (31) and (32), the partial-fraction expansion of Eq. (21) gives
ϒ⃗V(s)2sinh[γAτ]τ·2s+2γx[s+γx]2+ωx24γAsxsx*[sxssxsx*ssx*],
(35)
Γ⃗V(s)=ϒ⃗V(s)1.
(36)
Therefore, under γx2ω1, which is absolutely reasonable in a functioning resonator, we have
Γ⃗V(iωx)ϒ⃗V(iωx)12γAγx1=γ⃗γx,
(37)
which is consistent with R⃗(ωx)=γ⃗/γx at Eq. (15).

Finally, the ideal admittance/impedance transformers transform the admittance and impedance. For instance,
YAYcθA2,YBYcθB2,
(38)
where θA:1 and 1:θB are the winding ratios indicated for mirrors A and B, respectively, in Fig. 1(b). Neither the overall power pass-through transmittivity nor the reflectivity is affected by presence of these transformers.

In terms of square-root power amplitudes, we multiply γB/γA in the right-hand side according to ϒP(s)ϒ⃗V(s)YB/YA at Eq. (26):
ϒP(s)2γAγB[1+iγx/ωxs+γxiωx+1iγx/ωxs+γx+iωx].
(39)

C. Zeros Set at the Resonance Frequency in the Dual Circuit

One can make up a circuit dual to Fig. 1 almost trivially, as shown in Fig. 2. This could have been made with ϒ⃗V(s) and Γ⃗V(s) being replaced trivially by ϒ⃗I(s) and Γ⃗I(s) after standard substitution of Rx/Lx, G/C by G1/C1, [R+RΓ]/L in Fig. 2.

Fig. 2. Circuit configuration that is dual to that of Fig. 1. To simulate the two paired zeros of Γ⃗I(s), circuit parameters are not chosen exactly dual to those of Fig. 1

We now ask whether we can obtain an alternative two-pole model with the focus on the equivalent reflection coefficient rather than the equivalent pass-through coefficient. It is the second choice,
Ims⃗=ω1,
(40)
which would result in a dip of the reflectivity appearing exactly at ω1 at Eq. (24). We thus propose the dual circuit with a new set of parameters to represent the reflection coefficient more closely than the pass-through coefficient as follows:

First, we partition R into two parts: R and RΓ in expressing the circuit pass-through coefficient as
ϒ⃗I(s)=2ZAZA+ZB+RΓ+R+sL+1G1+sC1
(41)
by considering R as a part of an elementary resonator circuit with the four parameters satisfying R/L=G1/C1. This resonator circuit may pass the signal at resonance without minimal reflection when the so-called reflectionless matching condition
ZA+ZB+RΓ=2ZA
(42)
is satisfied at resonance. This condition lets Γ⃗I(s)ϒ⃗I(s)1 be expressed simply by its fraction-reciprocal
1Γ⃗I(s)=1+2ZAR+sL+1/[G1+sC1].
(43)
A few basic circuit parameters are set as usual:
LC11/ω12,
(44)
{ZA/LZB/L}2τ{sinh[γAτ]sinh[γBτ]}{2γA2γB},
(45)
all in parallel to Eqs. (27) and (28). The basic pair of zeros of Γ⃗I(s) for the single-mode resonator is set by
R/L=G1/C1γ⃗γinγA+γB
(46)
in contrast to G/C=R1/L1 at Eq. (31). Then,
1Γ⃗I(s)1+2γA[1s+γ⃗+iω1+1s+γ⃗iω1],
(47)
contrasting the prior choice of s⃗ and s⃗* for the zeros of Γ⃗V(s) at Eq. (34). Even more directly than Eq. (37), under γx2ω1 of a good resonator, one can confirm that the model Eq. (47) gives
1Γ⃗I(iω1)1+2γAγ⃗=1+γxγ⃗γ⃗=γxγ⃗,
(48)
which is approximately consistent with Eq. (15) and with Eq. (37). These points are of importance when we develop any multimode equivalent circuits for a FPR shortly.

Finally, the same ideal transformers transform the termination impedance:
ZAZc/θA2,ZBZc/θB2
(49)
in a fashion that is fraction-reciprocal to Eq. (38). From now on, we will consider, not the original characteristic admittance and impedance but the mirror admittance and impedance.

So far, the presented two-pole RLGC resonator-circuit models in Figs. 1 and 2 may be considered as trivial improvements over the prior approximate ones in textbooks utilizing either an RLC configuration in Ref. [17

17. A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968). [CrossRef]

] or an LGC configuration in literatures, e.g., in Ref. [13

13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

] Fig. 7.2 and in Ref. [10

10. R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).

] 2nd ed., Fig. 7.28.

3. MULTIMODE REFLECTION-EQUIVALENT CIRCUIT

In such single-pole or single-mode approximation, it did not make much sense to make any effort to obtain an approximation better than what we say something decent. Through the present study on the multimode equivalent-circuit models for general FPRs, some fundamental issues in guided-wave optics are to be addressed and be answered:
  • 1. whether it is truly possible to devise a lumped-circuit multimode-equivalent model for a FPR, which represents both the reflection and transmission coefficient,
  • 2. whether the LGC model is sufficient or whether the RLGC model is necessary in principle,
  • 3. how the delay-line circuit element represents the physical size of the resonator,
  • 4. how and whether we can accommodate the offset constant of the partial-fraction expansion in Ref. [15

    15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

    ] in multimode equivalent circuits for a FPR, etc.
  • 5. whether we can make a full generalization for any type of cavity resonators, beside a regular-looking FPR, with somewhat nonuniform resonance frequencies.

Any actual FPR is inherently a multimode device. Such a multimode FPR model requires a keen eye into some of the fundamental principles of electromagnetics. Each single-mode circuit of Fig. 2 impedes the passage of the signal unless the frequency of the signal matches the resonance frequency. Therefore, for the number of resonance modes being simulated, we may augment as many such series-RLGC resonator circuits in parallel as would “admit” a signal around any of those resonance frequencies, as illustrated in Fig. 3, in which we have proposed an auxiliary conductance on top of the resonator block.

Fig. 3. Multimode reflection-equivalent lumped-circuit model for a FPR supporting a large number of modes by implementing as many unit resonators, in the shape of Fig. 2.

A. Reflection Coefficient for an Enlarged Resonator-Circuit Array

Let us suppose that we have a finite number of N resonators in the array. Then, the total driving-point impedance of the circuit with a finite number of RLGC-resonators in the array must be Zdp(s)RΓ+1/Y⃗Γ(N)(s)+ZB, where
Y⃗Γ(N)(s)G¯Γ+1/Z0(s)++1/ZN(s)
(50)
is the impedance for an array of N resonators indicated in the diagram in Fig. 2 with
1Z0(s)12R+s2L=1/2LR/L+s,
(51)
1Zj(s)=1/LR/L+s+[1/LCj]/[Gj/Cj+s]
(52)
for j=1,2,,N.

Incidentally, a formula similar to Eq. (50), without the crucial first term G¯Γ, appeared in Ref. [18

18. R. Beringer, “Resonant cavities as microwave circuit elements,” in Principles of Microwave Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds. (McGraw-Hill, 1948), Chap. 7, Sect. 7.3.

] for the simulation of a short-circuited lossless transmission line, which could be interpreted as a resonator by itself. The formula did not correctly recognize the role of G¯Γ for the resulting transmission-line resonator.

Such an enforced symmetric impedance termination in turn lets the functional shape of 1/Γ⃗I(N)(s) agree with that of Eq. (8):
1/Γ⃗I(N)(s)=1+2ZAY⃗Γ(N)(s)ηA+Σ⃗(N)(s)
(56)
with
2ZAG¯ΓηA1,
(57)
2ZAj=0N1Zj(s)Σ⃗(N)(s),
(58)
which is consistent with Eq. (11). The sum of the latter two expressions is to be projected at
2ZAY⃗Γ(N)(s)Σ⃗(N)(s)+ηA1
(59)
of Eq. (11) by noticing that the functional shape of each term of Σ⃗()(s) in Eq. (11) is exactly identical to that of Eqs. (51) and (52). Hence, in a straightforward manner, the similarity suggests one should implement as many RLGC-resonator circuits as possible to simulate the true reflection coefficient of the FPR.

That is, impedance ZA and ZB expressing out-coupling through the mirrors are determined by
{ZA/LZB/L}1τ{1/rArA/hA21/rBrB/hB2}
(60)
slightly modified from Eq. (45) for nonunity hA2 in the multimode equivalent circuit. For Eqs. (51) and (52), the resonance frequencies of a FPR must give the capacitance values as
1/LCjωj2j2ω12
(61)
extended from Eq. (44). Further, we need again
Gj/Cj=R/Lγ⃗γin+γBγAτ1lnhA2
(62)
also modified from Eq. (46) for nonunity hA in all places in the circuit array, except two arms: the one with
2LG¯Γτ[ηA1]1/rArA/hA2=τ21/rA+rA/hA221/rArA/hA2
(63)
from Eq. (57) combined with Eq. (60) and the one denoted by 2R and 2L in Fig. 3. To match the ratio between the first fraction and the rest in Eq. (11), we should have 2R and 2L in the first arm, eventually promoting L as one of the “base” parameters of the model.

Because the resulting circuit in Fig. 3 is an exact representation of Eq. (58), we may expect that we will get the predicted spectral responses as already analyzed in thick solid curve in Fig. 7 in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

] and in the thick dashed curve in Fig. 6 in [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

]. Verification of our conclusion on the two reflection coefficients is made with N=6 this time for the plot of Fig. 4(a). For this plot and all others that follow, we keep the parameter values of the FPR that we used for Fig. 3 in [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

].

Fig. 4. Spectral responses of the reflection-equivalent circuit for a FPR in thick curves from the circuit model of Fig. 3 in comparison with the actual response in thin dotted curves. (a) Real and imaginary parts of the reflection coefficient Γ⃗I(6)(iω) of Eq. (56). (b) Those of the pass-through coefficient for the circuit-based interpretation ϒ⃗P(6)(iω) at Eq. (64). (c) Reflectivity |Γ⃗I(6)(iω)|2 and the pass-through transmittivity |Γ⃗I(6)(iω)+1|2ZB/ZA. For all plots in this study, we use hA=hB=1, rA=rB=0.7304, giving γA/ω1=γB/ω1=0.05 and γin/ω1=0.01. In comparison, the two thin curves are made for the real and imaginary parts of R⃗(ω) in Eq. (15) for the corresponding FPR.

B. Pass-Through Coefficient with the Multimode Reflection-Equivalent Circuit

If we wish to maintain the same circuit of Fig. 3 to simulate the square-root-power-based pass-through coefficient of the same FPR as well as its reflection coefficient, we would just use the well-established standard relationship Eq. (54) with the necessary revision of Eq. (26) from the already proposed reflection-equivalent circuit:
ϒ⃗P(N)(s)[Γ⃗I(N)(s)+1]ZB/ZA.
(64)

The plot for ϒ⃗P(6)(iω) is made and compared with T(ω) from the original FPR at Eq. (4) in Fig. 4(b). We find that the degree of agreement with T(ω) represented by thin curves is sufficiently impressive in the pass-bands around successive resonance frequencies. However, appearance of the unavoidable π phase hops happening at those successive resonances has made a serious disagreement in the imaginary parts and ultimately in the power spectral data, shown in Fig. 4(c), at mid-frequencies between adjacent resonance points.

In order to remove the base disagreement in the plot for the phase of the pass-through coefficient in Eq. (64), one can trivially add a negative delay line of length d in addition to the transformer illustrated in Fig. 3 with a negative winding ratio. The resulting plot for ϒ⃗P(N)(iω)eiωτ/2 is given in Fig. 5. Unfortunately, although the phase-hop is removed, overall agreement with T(ω) is still not satisfactory.

Fig. 5. Real and imaginary parts of [Γ⃗I(6)(iω)+1] eiωτ/2ZB/ZA. The thin curves represent T(ω) from the original FPR.

Considering the implication from the plots of Figs. 5 and 4(b), we must conclude that the proposed reflection-equivalent circuit model of Fig. 3 (without any revision of the circuit configuration) comes up short at simulating the pass-through multimode transmission spectrum of a FPR especially at the mid-frequencies, unless the FPR has an extremely high Q-number.

C. Discussion on the Multimode Reflection-Equivalent Circuit

Note first that, beyond the purpose of numerical verification, from the projection of the circuit model in terms of partial fractions, there is no need for any delay or decay element before the array of resonators in the equivalent circuit, because the array itself represents the action of such delay and decay elements.

Note also that, as before with Eq. (46), the sign of Gj/Cj=R/L follows that of γ⃗ according to Eq. (62). This sign will turn out to be negative if eγinτrA/hA2rB<1. Although it may not look pretty, as a mathematical and computational model, negative values for those negative circuit parameters do not impose any limitation of an equivalent circuit.

Also, as mentioned in Eq. (49), to recover the original characteristic wave impedances Zc of the external waveguides rather than ZA and ZB, we need to use two transformers in Fig. 3 with nonunity winding ratios for θA and θB following Eq. (49). Neither the negative winding ratio for the impedance-matching transformer nor the negative delay line of length d denoted near the end of the equivalent circuit in Fig. 3 affects the reflection coefficient being simulated.

The effect of the auxiliary conductance G¯Γ is well demonstrated in Fig. 4(c). At resonance, power will be zero-impeded toward the side of “mirror B” through the resonator array rather than the side door G¯Γ. Hence, the power pass-through would hardly be affected at resonance. On the other hand, in mid-frequencies between adjacent resonance points, effective pass-through is even further reduced than |T(ω)|2, because G¯Γ is draining the incident wave power without transferring the power to the “B side.” Such a draw of power is required for a faithful representation of the reflection coefficient, but not for a good representation of the pass-through coefficient, which is well demonstrated in Fig. 4(c).

In Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

], we mentioned that the change of preceding constant at Eq. (2) from 1/rA to ηA[1/rA+rA/hA2]/2, resulting in proper parameterization of G¯Γ, used to be called “renormalization” in the infinite-order perturbation. The structure of infinite feedback that is so evident in a multimode FPR is one such example that anyone can easily depict. Yet, we have seen that, in single-mode representations at Eq. (37) and at Eq. (48), if implemented, such an offset would only have caused trouble rather than any improvement in terms of accuracy.

As we worked with dual circuits in Figs. 1 and 2, one can compose a second circuit configuration dual to Fig. 3. That is, the dual circuit employs an array of individual admittance resonators all connected in series as depicted in Fig. 6.

Fig. 6. Circuit dual to the one in Fig. 3. Rj,Lj,G,C,GΓ,R¯Γ here replace Gj,CjR,L,RΓ,G¯Γ, respectively, in Fig. 3.

4. MULTIMODE PASS-THROUGH-EQUIVALENT CIRCUIT

If we return to Fig. 4(b), we find that, beside the π-phase hop at successive pass-bands, the spectral match over the pass-bands is so impressive that we may ask whether we should bother with any improvement in the mid-frequencies. More often, however, FPRs are used as optical filters, in which the pass-through coefficient is more important than the reflection coefficient. Therefore, it is worth finding another ideal equivalent circuit for the true pass-through coefficient T(ω) in Eq. (14):

Up to the task of finding a better revision, we investigate the exact relationship in Eq. (4) between the true scattering coefficients of a FPR. Inserting R⃗(ω)Γ⃗I()(iω) of Eq. (56) into it, we have
T(ω)tBeiβ(ω)dtArB[rA+1ηA+Σ⃗()(iω)]
(65)
with Σ⃗(N)(s) from Eq. (11).

In order to make the above expression well-projected to the pass-through coefficient of a certain circuit to be determined, we rewrite Eq. (65) as
T(ω)=eγinτ/2iωτ/2tBrA/tArB×{1+11[1ηArA]+rAΣ⃗()(iω)},
(66)
where
1ηArA=1/2rA2/2hA2tA2/2hA2
(67)
from Eq. (60) will be used shortly. Here, the shape of the last expression is guided by that of the pass-through coefficient across the resonator array as
ϒ⃗I(N)(s)2ZAZA+ZB+Rϒ+1/Y⃗ϒ(N)(s)=2ZAZA+ZB+Rϒ·Y⃗ϒ(N)(s)Y⃗ϒ(N)(s)+1ZA+ZB+Rϒ
(68)
in terms of
Y⃗ϒ(N)(s)G¯ϒ+1/Z0(s)+1/Z1(s)+
(69)
similar to Eq. (50) beside the leading term G¯Υ given in the pass-through-equivalent circuit of Fig. 7 replacing G¯Γ in Fig. 3.

To compare the latter expression with Eq. (66), we rewrite the latter in terms of the square-root-power-based pass-through coefficient in a shape of
ϒP(N)(s)=2ZAZBZA+ZB+Rϒ×{1+11+[ZA+ZB+Rϒ]Y⃗ϒ(N)(s)}
(70)
with Y⃗ϒ(N)(s) from Eq. (69). In determining the circuit parameters of the new pass-through equivalent circuit that is in many respects similar to the prior one, we must decide what to keep and what to change among the recipe parameters for the reflection-equivalent circuit. Some of the basic parameters such as ZA/2L, ZB/2L, LCj, and Gj/Cj=R/L in Eqs. (60)–(62) are all carried intact.

With this much carry-over, comparison between Eqs. (66) and (70) still keeping Eq. (58) in
[ZA+ZB+Rϒ][G¯ϒ+1/Z0(s)+1/Z1(s)+]rAΣ⃗()(s)tA2/2hA2
(71)
lets us choose
[ZA+ZB+Rϒ]/2ZArA,
(72)
rather than unity at Eq. (42), i.e.,
RϒZA=2rA11/rBrB/hB21/rArA/hA22rA1γBγA,
(73)
with ZA as given in Eq. (60) and ZB/ZA˜γB/γA at Eq. (17). This choice then breaks the condition of ZA=ZB+RΥ in Eq. (55) for impedance matching for minimal reflection at resonance. Evidently, in the pass-through-equivalent circuit, unlike the case of the reflection-equivalent circuit in Eq. (55), such property is not maintained.

Next, we need to make up for the difference in the overall factor by a power attenuator/amplifier with the gain A⃗ in Fig. 7, so that the prefactors agree between Eq. (66) and Eq. (70),
2ZAZBZA+ZB+RϒA⃗tBrAtArBeγinτ/2,
(74)
i.e.,
A⃗rArA/rB·eγinτ/2hB/hA,
(75)
and install it.

Fig. 7. Pass-through-equivalent circuit for a true FPR supporting infinitely many modes.

Finally, the auxiliary conductance G¯ϒ is determined by utilizing Eq. (72) and Eq. (60) onto
[ZA+ZB+Rϒ]G¯ϒ=2ZA+ZB+Rϒ2ZAZA2L2LG¯ϒ,
(76)
which must agree with tA2/2hA2 at Eqs. (66) and (67). With respect to the “base” parameter 2L, G¯Υ is determined by
2LG¯ϒτ/2.
(77)
The negative conductance in the equivalent circuit would lower down the reflectivity at mid-frequencies, as indicated in Fig. 8(c). Evidently, if the loss coefficient of the internal medium of a FPR is large, A⃗ can become greater than unity.

Fig. 8. Spectral responses of the pass-through-equivalent circuit for a FPR in thick curves from the circuit model with six RLGC resonators and an amplified/attenuated phase-delay line in comparison with the actual response in thin dotted curves. (a) Real and imaginary parts of the pass-through coefficient ϒP(6)(iω) from Eq. (66), (b) the resulting reflection coefficient Γ⃗I(6)(iω), and (c) their power spectra in thick curves from the pass-through-equivalent circuit model.

With everything implemented properly, we should be obtaining the pass-through transmission coefficient as plotted in Fig. 8(a). The plot for Γ⃗(6)(iω) and the power spectra are given in Figs. 8(b) and 8(c), respectively, for this pass-through-equivalent circuit. To match the proper complex ratio between the reflection coefficient and the pass-through coefficient, the negative delay line, representing the phase advance of kd, after the equivalent circuit is necessary. The best physical interpretation on this ambiguity would be that the lumped circuit is actually simulating a “FPR squeezed into a single point in space.”

Another minor point that deserves a mention is the fact that the zeroth resonance frequency of ω01/LC0=0 is implemented by the “infinite capacitance,” i.e., absent capacitor, at the top of the array of Fig. 3. It can be explained by the fact that any static electric field may exist between two parallel mirrors as an exhibition of the zeroth resonance mode.

5. REMARKS ON THE MULTIMODE EQUIVALENT CIRCUITS

We have presented two versions of multimode equivalent circuits simulating a single FPR: a reflection-equivalent version and a pass-through-equivalent version that employs active and delay elements. Interestingly, in terms of required circuit elements, the two circuits show almost the same configuration. Only a few circuit parameters are different. Depending on the application, we are to choose one between the two choices.

Therefore, we provide answers to the questions that were given when we started Section 3:
  • 1. Unlike the case of a single-mode equivalent circuit, for multimode equivalence, one circuit configuration cannot serve both scattering coefficients.
  • 2. Considering Eqs. (32) and (46), a circuit with all four RLGC elements is far superior to either an RLC or an LGC circuit.
  • 3. A negative-delay line is required for exact agreement of the formulas for the pass-through coefficient. However, it is basically a dummy device.
  • 4. Implementation of an offset conductance element, G¯Γ or G¯Υ, is crucial for multimode-equivalent circuits, whereas any such augmentation would produce an inferior result in a single-mode equivalent circuit.
  • 5. The basic configuration of a FPR can be generalized to a more complicated one with detailed information on, say, γ⃗j and γjx, denoted by mode index j, both depending on each of {ω0,ω1,ω2,} that may not be equidistant in general. The offset 1/2 at Eq. (C8) in Ref. [15

    15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

    ] is known to be valid for the case with equidistant zeros of the reflection coefficient along a straight line in the complex-k plane as Fig. 2 in Ref. [15

    15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

    ]. Therefore, one must be careful in setting the right offset number for such a general case of a resonator.

Quite naturally, one may wish to use a common circuit model for both the reflection and pass-through coefficients, although the circuit becomes only nonuniformly approximating to either coefficient. Obviously, one would eliminate the auxiliary conductance altogether, i.e., G¯Γ=0 in Fig. 3, because such a value is more or less the median between true G¯Γ and G¯ϒ.

Figure 9 shows the resulting plots, in which agreement in the mid-frequencies is not satisfactory in all three plots. However, the disagreement would become negligible for FPRs with high Q number, e.g., γAω1=γBω1=γinω1=0.01, as demonstrated in Fig. 10, although the circuit cannot be said to be uniformly approximating to the actual FPR response.

Fig. 9. Real and imaginary parts of (a) Γ⃗(6)(iω), (b) [Γ⃗(6)(iω)+1]ZB/ZA, and (c) the reflectivity |Γ⃗(6)(iω)|2 and the pass-through transmittivity |Γ⃗(6)(iω)+1|2ZB/ZA all with Γ⃗(6)(iω) in solid thick curves from the circuit of Fig. 3 without the auxiliary conductance, viz., G¯Γ=0. The thin dotted curves represent the original FPR.
Fig. 10. Spectral response of a common equivalent circuit with G¯Γ=0 in Fig. 3, for simulating both the reflection coefficient and the pass-through coefficient in thick solid curves over a range 2<ω/ω1<3 for a FPR with more reflective mirrors with rA=rB=0.9391 than those in the FPR with rA=rB=0.7304 in all other plots in the paper. The curves almost coincide with the thin dotted curved from the analytic responses, which are actually hidden behind the thick curves: (a) Γ⃗(6)(iω). (b) ϒ⃗(6)(iω)eiωτ/2.

Yet, such a single representative but “nonuniformly” approximate model evidently allows cascading circuit models in a network with no extra effort. Practically, FPRs are designed to be of even higher Q than the one demonstrated in Fig. 10.

6. CONCLUSION

Based on complex-variable analysis on the formula for scattering coefficients of a FPR in our prior study, we have successfully developed two multimode-equivalent circuit models, with which one can emulate the given FPR with uniform convergence: One model is made for emulating the reflection-coefficient formula and the other one is made separately for emulating the pass-through-coefficient formula. With all the circuit parameters determined by the two proposed recipes with as many resonator modules as possible, the spectral response of the presented circuit models would become exact, viz., are uniformly converging to the true spectral responses of the given FPR.

Hence, depending on applications, one may use the newly found appropriate multimode models in systemic general-purpose computerized circuit-based simulation tasks. Practically, one may appreciate a common approximate model for a given FPR, which has analyzed in addition to the exact models. Indeed, we have shown that a single representative FPR model would be sufficient for many well-designed high-Q resonators for both the passband and the reflection band that appears in mid-range frequencies between every two adjacent resonance frequencies.

The present study in two parts, which includes the study in Ref. [15

15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

], has sprung up from a pedantic curiosity on expressing some fundamental properties of a basic optical device in a rigorous mathematical manner, which may be appreciated in education. Therefore, beyond applications, the developed models serve as ideal aids in illustrating the basic functions of any passive electromagnetic resonators that include FPRs.

ACKNOWLEDGMENTS

I would like to thank Prof. Sang-Yung Shin of the Korea Advanced Institute of Science and Technology for his kind and special comments on the manuscript. Discussions with colleague Prof. Jae-Hyung Jang of GIST are greatly appreciated. This work was supported in part by the Top-Brand Research Program in the Gwangju Institute of Science and Technology (GIST).

REFERENCES

1.

C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).

2.

H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992). [CrossRef]

3.

R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.

4.

E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969). [CrossRef]

5.

I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).

6.

K. Hwang and G. H. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005). [CrossRef]

7.

A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of Bragg gratings and its applications to Fabry–Pérot cavities,” J. Opt. Soc. Am. A 20, 273–281 (2003). [CrossRef]

8.

C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds., Principles of Microwave Circuits (McGraw-Hill, 1948).

9.

W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961). [CrossRef]

10.

R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).

11.

A. E. Siegman, Lasers (University Science, 1986), pp. 934–939.

12.

H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992). [CrossRef]

13.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.

14.

T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/.

15.

G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)

16.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14). [CrossRef]

17.

A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968). [CrossRef]

18.

R. Beringer, “Resonant cavities as microwave circuit elements,” in Principles of Microwave Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds. (McGraw-Hill, 1948), Chap. 7, Sect. 7.3.

OCIS Codes
(050.2230) Diffraction and gratings : Fabry-Perot
(250.5300) Optoelectronics : Photonic integrated circuits
(070.5753) Fourier optics and signal processing : Resonators

ToC Category:
Diffraction and Gratings

History
Original Manuscript: September 17, 2013
Manuscript Accepted: December 12, 2013
Published: January 30, 2014

Citation
G. Hugh Song, "Mathematical modeling of Fabry–Perot resonators: II. Uniformly converging multimode equivalent-circuit models," J. Opt. Soc. Am. A 31, 411-420 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-2-411


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References

  1. C. Fabry and A. Pérot, “Théorie et applications d’une nouvelle méthodes de spectroscopie interférentielle,” Annal. Chim. Phys. 16, 115–144 (1899).
  2. H. A. Haus and Y. Lai, “Narrow-band optical channel-dropping filter,” J. Lightwave Technol. 10, 57–62 (1992). [CrossRef]
  3. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (IEEE, 1992), Fig. 7.30.
  4. E. A. J. Marcatilli, “Bends in optical dielectric waveguides,” Bell Syst. Tech. J. 48, 2103–2132 (1969). [CrossRef]
  5. I. Ohtomo and S. Shimada, “A channel channel-dropping filter using ring resonators for the millimeter wave communication system,” Trans. Inst. Electron. Commun. Eng. Jpn. 52-B, 265–272 (1969).
  6. K. Hwang and G. H. Song, “Design of a high-Q channel add-drop multiplexer based on the two-dimensional photonic-crystal membrane structure,” Opt. Express 13, 1948–1957 (2005). [CrossRef]
  7. A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of Bragg gratings and its applications to Fabry–Pérot cavities,” J. Opt. Soc. Am. A 20, 273–281 (2003). [CrossRef]
  8. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds., Principles of Microwave Circuits (McGraw-Hill, 1948).
  9. W. Culshaw, “Resonators for millimeter and submillimeter wavelengths,” IRE Trans. Microwave Theory Technol. MTT-9, 135–144 (1961). [CrossRef]
  10. R. E. Collin, Foundations for Microwave Engineering, 1st ed. (McGraw-Hill, 1966).
  11. A. E. Siegman, Lasers (University Science, 1986), pp. 934–939.
  12. H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28, 205–213 (1992). [CrossRef]
  13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984), Fig. 9.9.
  14. T. Quarles, D. Pederson, R. Newton, A. Sangiovanni-Vincentelli, and C. Wayne, SPICE3 Version 3f3 User’s Manual (University of California, 1993), The SPICE Page http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/ .
  15. G. H. Song, “Mathematical modeling of Fabry–Perot resonators: I. Complex-variable analysis by uniformly convergent partial-fraction expansion,” J. Opt. Soc. Am. A31, 404–410 (2014)
  16. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999), Eq. (14). [CrossRef]
  17. A. Karp, H. J. Shaw, and D. K. Winslow, “Circuit properties of microwave dielectric resonators,” IEEE Trans. Microwave Theor. Technol. MTT-16, 818–828 (1968). [CrossRef]
  18. R. Beringer, “Resonant cavities as microwave circuit elements,” in Principles of Microwave Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, eds. (McGraw-Hill, 1948), Chap. 7, Sect. 7.3.

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