# Time-Reversal Symmetry Bounds in Temporal Coupled-Mode Theory

Ken X. Wang To whom correspondence should be addressed. Electronic mail:
<EMAIL_ADDRESS>of Physics, Huazhong University of Science and
Technology, Wuhan, Hubei, China

###### Abstract

We provide a general treatment for the temporal coupled-mode theory with
arbitrary numbers of modes and ports, and derive tight bounds for the coupling
phases in addition to coupling strengths under time-reversal symmetry. We
report trade-offs between the cardinalities of the reciprocal regions of the
resonant coupling strengths versus phases. We discover that time-reversal
symmetry enforces projected generalized reflections in the background to
cancel out completely. In double-port systems, the reciprocal regions of the
coupling phases span a quarter of the nonreciprocal regions for any non-hidden
mode.

The temporal coupled-mode theory is a phenomenological theory for Fano-
Feshbach resonances and is widely used to describe and predict physical
devices [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 15,
19, 15, 20, 21, 22, 5, 23, 24, 25, 26, 27, 12, 11, 28, 29, 15, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39]. The input and output wave amplitudes can be observed
at ports [7], characterizing a scattering matrix of the physical system. Waves
propagate through the system via two pathways: a background pathway without
interacting with localized modes, and a resonant pathway with such
perturbative interaction through the coupling between the ports and the leaky
modes. Waves from these two pathways coherently form the overall scattering
amplitudes. These two pathways, however, are not independent but constrained
by symmetries, and the coupling parameters between modes and ports cannot be
arbitrary but are related to the scattering background. In particular, it was
previously shown that there exists a bound on the coupling strengths in the
single-mode double-port case [14]. In this Letter, we provide a general
treatment for any numbers of modes and ports, and show that the symmetry
constraints not only bound coupling strengths but also the phases.

A temporal coupled-mode system is shown in Figure 1. We assume that there are
$m$ modes and $n$ ports. In electromagnetics, for instance, a mode is a
localized solution of the Maxwell’s equations, and a port corresponds to a
propagating solution. It is perhaps worth noting that our abstraction of the
modes and ports might not intuitively match physical or geometrical shapes of
the actual structures.

Figure 1: Structure abstraction in temporal coupled-mode theory. $\mathbf{a}$
is the amplitude vector of the $m$ localized modes supported by a physical
structure(s), which has many possibilities. $\mathbf{s_{+}}$ and
$\mathbf{s_{-}}$ are the incoming and outgoing amplitude vectors of
propagating waves through the $n$ ports, which could also correspond to a
variety of different physical structures.

The coupled-mode equations model the system dynamics as follows:

$\dot{\mathbf{a}}=(j\Omega-\Gamma)\mathbf{a}+K^{T}\mathbf{s_{+}},$ (1)
$\mathbf{s_{-}}=C\mathbf{s_{+}}+D\mathbf{a},$ (2)

where $\mathbf{a}$ is the mode amplitude vector, $\mathbf{s_{+}}$ and
$\mathbf{s_{-}}$ are the input and output wave amplitude vectors at the ports,
$\Omega$ and $\Gamma$ are the frequency and decay matrices of the modes, $K$
and $D$ are the in-coupling and out-coupling matrices between the modes and
the ports, and $C$ is the background or the direct scattering matrix between
the ports without involving the modes. Equation 1 describes the evolution of
the modes with incoming waves from the ports, and Equation 2 coherently
combines scattering along the background and the resonant-assisted pathways.

The system parameters above are not arbitrary but obey physical laws.

$\Omega^{\dagger}=\Omega$ (3)

$C^{\dagger}C=I$ (4)

$D^{\dagger}D=2\Gamma$ (5)

$CK^{*}=-D$ (6)

$||K||=||D||$ (7)

Table 1: Constraints on temporal coupled-mode system parameters under energy
conservation.

If a temporal coupled-mode system obeys energy conservation alone without
time-reversal symmetry, the system parameters are constrained by the equations
in Table 1 [7, 40, 41]. In particular, $\Gamma^{\dagger}=\Gamma$ as implied by
Equation 5.

$\Omega^{\dagger}=\Omega$

$C^{\dagger}C=I$

$C=C^{T}$ (8)

$D^{\dagger}D=2\Gamma$

$CK^{*}=-D$

$K=D$ (9)

Table 2: Constraints on temporal coupled-mode system parameters under energy
conservation and time-reversal symmetry.

With both energy conservation and time-reversal symmetry, which together imply
reciprocity [1, 6, 42, 22, 43, 44, 45], the system parameters are constrained
by the equations in Table 2. In particular, the in-coupling and out-coupling
matrices are equal as shown in Equation 9.

Combining Equations 6 and 9, we have:

$CD^{*}=-D.$ (10)

We observe that the background pathway, described by the background scattering
matrix $C$, and the resonant pathway under reciprocity, described by the
coupling matrix $D$, are not independent of each other, but related by
Equation 10. $C\in\mathrm{U}(n)\subset\mathbb{C}^{n\times n}$ and
$D\in\mathbb{C}^{n\times m}$, in particular, we can write down each element in
$D$:

$D=\left(\begin{array}[]{cccc}d_{11}&d_{12}&\cdots&d_{1m}\\\
d_{21}&d_{22}&\cdots&d_{2m}\\\ \vdots&\vdots&\ddots&\vdots\\\
d_{n1}&d_{n2}&\cdots&d_{nm}\\\ \end{array}\right),$ (11)

where the matrix element $d_{pq}$ is the coupling constant between the $p$-th
port and the $q$-th mode, where $p=1,2,\ldots,n$ and $q=1,2,\ldots,m$. From
Equations 10 and 11, we can derive that:

$C\left(\begin{array}[]{cccc}d_{1q}\\\ d_{2q}\\\ \vdots\\\ d_{nq}\\\
\end{array}\right)^{*}=-\left(\begin{array}[]{cccc}d_{1q}\\\ d_{2q}\\\
\vdots\\\ d_{nq}\\\ \end{array}\right),\;\forall q.$ (12)

Equation 12 holds for each mode, and implies that the constraint is imposed on
each mode independently. On the system level, one could engineer a mode
without necessarily affecting the coupling properties of other modes. We
denote $\theta_{pq}=\angle d_{pq}$ as the phase angle of the matrix element
$d_{pq}$ in $D$, then $d_{pq}=|d_{pq}|e^{j\theta_{pq}}$. We can then expand
Equation 12 into the following:

$C\left(\begin{array}[]{cccc}e^{-j\theta_{1q}}&0&\cdots&0\\\
0&e^{-j\theta_{2q}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{-j\theta_{nq}}\\\
\end{array}\right)\left(\begin{array}[]{cccc}|d_{1q}|\\\ |d_{2q}|\\\ \vdots\\\
|d_{nq}|\\\
\end{array}\right)=-\left(\begin{array}[]{cccc}e^{j\theta_{1q}}&0&\cdots&0\\\
0&e^{j\theta_{2q}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{j\theta_{nq}}\\\
\end{array}\right)\left(\begin{array}[]{cccc}|d_{1q}|\\\ |d_{2q}|\\\ \vdots\\\
|d_{nq}|\\\ \end{array}\right),\;\forall q.$ (13)

Equation 13 can be written as:

$G\left(\begin{array}[]{cccc}|d_{1q}|\\\ |d_{2q}|\\\ \vdots\\\ |d_{nq}|\\\
\end{array}\right)=0,\;\forall q,$ (14)

where

$G=C\left(\begin{array}[]{cccc}e^{-j\theta_{1q}}&0&\cdots&0\\\
0&e^{-j\theta_{2q}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{-j\theta_{nq}}\\\
\end{array}\right)+\left(\begin{array}[]{cccc}e^{j\theta_{1q}}&0&\cdots&0\\\
0&e^{j\theta_{2q}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{j\theta_{nq}}\\\ \end{array}\right).$ (15)

We examine all possible solutions of Equation 14. One solution is that:

$\left(\begin{array}[]{cccc}|d_{1q}|\\\ |d_{2q}|\\\ \vdots\\\ |d_{nq}|\\\
\end{array}\right)=0.$ (16)

Equation 16 describes a $q$-th mode uncoupled from all the ports. Substituting
Equation 16 into Equation 5, we observe that such a mode cannot leak to ports
through other modes, either. Consequently, such modes could not couple to the
ports either directly or indirectly, however, they could still contribute to
the overall scattering process through coupling with other modes if the
overlap integral with some other mode is nonzero [7]. We refer to such mode as
a hidden mode. In fact, if we do not impose any constraints on the coupling
phases, then almost surely $\mathrm{rank}(G)=n$ and the modes are necessarily
hidden. In other words, for non-hidden modes, the coupling phases are
necessarily constrained.

For non-hidden modes, $G$ is singular by Equation 14. Furthermore, if we do
not impose any constraints on the coupling strengths, then we almost surely
have:

$G=0.$ (17)

Equation 17 is only possible if $C$ is diagonal, and one must carefully
engineers the coupling phases. Specifically, the background scattering matrix
has to be given by:

$C=\left(\begin{array}[]{cccc}e^{j\delta_{1}}&0&\cdots&0\\\
0&e^{j\delta_{2}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&e^{j\delta_{n}}\\\ \end{array}\right),$ (18)

where there are no transmissions, and the reflection at the $p$-th port
introduces a phase shift of $\delta_{p}$. With Equations 15, 18 and 17, we
derive:

$\theta_{pq}=\frac{\delta_{p}\pm\pi}{2},$ (19)

for the $q$-th mode and the $p$-th port. It is remarkable that all mode-port
couplings are necessarily in perfect synchronization among different modes.
Equations 18 and 19 establish the condition for Equation 17 to hold.

If Equations 18 and 19 are not satisfied simultaneously, for a non-hidden mode
such that Equation 16 does not hold, we have $0<\mathrm{rank}(G)<n$. There
still exist uncountably infinite many solutions of the coupling strengths
$|d_{pq}|$, but with constraints, although many of which are redundant due to
the rank deficiency and complexness of $G$. The kernel of $G$ has a
dimensionality of $n-\mathrm{rank}(G)$ according to the rank-nullity theorem
[46]. Thus the lower rank $G$ possesses, there are less restrictions on the
coupling strength and more restrictions on the coupling phase. One could
leverage this trade-off to engineer desirable coupling. In general,

$\det(G)=0,$ (20)

and we can compute the determinant $\det(G)$ [47]:

$\det(G)=\sum_{\sigma\subseteq[n]}\left[\mathrm{minor}_{\sigma}\left(C\mathrm{diag}(e^{-j\theta_{1q}},e^{-j\theta_{2q}},\cdots,e^{-j\theta_{nq}})\right)\mathrm{minor}_{[n]\setminus\sigma}\mathrm{diag}(e^{j\theta_{1q}},e^{j\theta_{2q}},\cdots,e^{j\theta_{nq}})\right],$
(21)

where $[n]:=\\{1,2,\cdots,n\\}$, and $M_{\sigma}$ or
$\mathrm{minor}_{\sigma}(M)$ is the principal minor of a square matrix $M$
indexed by $\sigma$, with $C_{\emptyset}=1$. Equations 20 and 21 together can
be simplified as:

$\sum_{\sigma\subseteq[n]}\left(C_{\sigma}e^{-j\sum_{p\in\sigma}\theta_{pq}+j\sum_{p\in[n]\setminus\sigma}\theta_{pq}}\right)=0.$
(22)

Equation 22 has intriguing physical implications. $C_{\sigma}$ can be
interpreted as a generalized complex reflection coefficient upon the
collection of ports $\sigma$. We can always divide the power set of $[n]$ into
two complementary halves $N$ and $2^{[n]}\setminus N$ such that, for any
element $\sigma\in N$, $[n]\setminus\sigma\in 2^{[n]}\setminus N$. Equation 22
can be written as:

$\sum_{\sigma\in
N}\left(C_{\sigma}e^{-j\sum_{p\in\sigma}\theta_{pq}+j\sum_{p\in[n]\setminus\sigma}\theta_{pq}}+C_{[n]\setminus\sigma}e^{-j\sum_{p\in[n]\setminus\sigma}\theta_{pq}+j\sum_{p\in\sigma}\theta_{pq}}\right)=0.$
(23)

Using the mathematical results on complementary principal minors of unitary
matrix in the Appendix, concerning both magnitude and phase, we can further
derive that:

$\sum_{\sigma\in N}|C_{\sigma}|\cos(\eta_{\sigma})=0,$ (24)

where

$\eta_{\sigma}=\frac{\angle(C_{[n]})}{2}-\angle(C_{\sigma})+\sum_{p\in\sigma}\theta_{pq}-\sum_{p\in[n]\setminus\sigma}\theta_{pq},$
(25)

and $\angle\cdot$ represents the phase angle. Equation 24 determines the
constraints on the coupling phases in relation to the background process $C$.
$|C_{\sigma}|\cos(\eta_{\sigma})$ is a projection of the generalized
reflection, and $2^{n-1}$ such terms cancel out in order to satisfy the time-
reversal symmetry.

For single-port systems that $n=1$, the most general background is given by
$C=e^{j\phi}$, and a single-port version of Equation 19 can be derived from
Equation 24. Therefore, in a single-port reciprocal system, there are no
restrictions on the coupling strength of the non-hidden modes, but the
couplings need to be in perfect synchronization.

For double-port systems that $n=2$, the most general $C$ can be written as:

$C=e^{j\phi}\left(\begin{array}[]{cc}r&jt\\\ jt&r\\\ \end{array}\right)$ (26)

where $r^{2}+t^{2}=1$. Substituting Equation 26 into Equation 24,

$\cos(\theta_{1q}+\theta_{2q}-\phi)+r\cos(\theta_{1q}-\theta_{2q})=0.$ (27)

Since $0\leqslant r\leqslant 1$, we have:

$-1\leqslant\frac{\cos(\theta_{1q}+\theta_{2q}-\phi)}{\cos(\theta_{1q}-\theta_{2q})}\leqslant
0.$ (28)

Equation 28 provides the bound on feasible coupling phases for any $q$-th mode
under time-reversal symmetry. If Equation 28 is violated, then no reciprocal
system is possible. Without loss of generality, we can choose appropriate
reference planes such that $\phi=0$, and plot the feasible phases in Figure 2.
Exactly one quarter of the codomain, or the nonreciprocal feasible region,
forms the feasible region with reciprocity.

Figure 2: Bounds on coupling phases under time-reversal symmetry. The colored
regions are the reciprocal feasible regions of the phase angles at $n=2$, and
the uncolored regions are necessarily nonreciprocal. The bounds are identical
for any $q$-th non-hidden mode.

We can also use Equation 14 to derive that:

$\frac{1-r}{1+r}\leqslant\left|\frac{d_{1q}}{d_{2q}}\right|^{2}\leqslant\frac{1+r}{1-r}$
(29)

for each $q$. Equation 29 is a bound on the coupling strengths consistent with
the results previously reported [14, 16], and is more general since it holds
for every single mode and beyond electromagnetics. Equation 29 further implies
that a mode hidden from a port needs to be hidden from both ports.

When $n=3$, we can compute the bounds on coupling strengths and phases
similarly, by utilizing the results on U(3) parametrization [48, 49]. We could
also obtain simpler closed-form bounds for systems with other symmetries,
including parity, geometrical, and permutation symmetries, both for $n=3$ [50,
8, 51, 15, 52], and for systems with $n>3$ [53, 49].

It is also worth noting that, one may construct an overall reciprocal
scattering matrix with nonreciprocal coupling. We first define the overall
scattering matrix $S\in\mathbb{C}^{N\times N}$:

$\mathbf{s_{-}}=S\mathbf{s_{+}}.$ (30)

Applying Equations 1 and 2 into Equation 30, the scattering matrix can be
written as:

$S=C+D[j(\omega-\Omega)+\Gamma]^{-1}K^{T}.$ (31)

From Equation 31, one can see that, even when reciprocity is maximally
violated at the in- and out-coupling, for example, when $K=-D$, $S$ could be
symmetric and reciprocal. In other words, internal symmetry breaking does not
necessarily lead to external symmetry breaking. It is also of interest to
quantitatively study the degree of symmetry breaking and the relaxation of the
constraints in Tables 1 and 2.

As a final remark, although the temporal coupled-mode theory is an approximate
and phenomenological model, its accuracy is sufficient for many applications,
for instance, in electromagnetics [22, 54]. A simpler version of Equation 29
was discovered in the temporal coupled-mode theory [14], and was later
discovered that similar bounds are generally applicable in electromagnetics
from first-principles [16].

In summary, we discover time-reversal symmetry bounds on both coupling
strengths and phases, as well as the trade-off therein. Given the importance
of phase in physics, we expect the results to be exploited in many scenarios.
In addition to the bounds on the system parameters, the distribution of the
parameters might not be uniform within the bounds. Further work could also
include exploring deep and rich results in nonreciprocal systems or with
energy conversation only in Table 1, in topological systems [55, 56, 57], in
generalized, extended, or other versions of temporal coupled-mode theory [58,
59, 60], and beyond the temporal coupled-mode theory [61].

## Appendix: Complementary Minors in Unitary Matrix

Complementary minors of a unitary matrix have equal magnitudes and
complementary phases that add up to the phase of the determinant of the
unitary matrix.

We consider a unitary matrix $\mathrm{U}\in U(n)$:

$U=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right).$ (32)

It suffices to prove for leading principal minors, as the results can be
generalized to minors by permutation [62, 46].

######  Theorem 1.

Complementary minors of a unitary matrix have equal magnitude.

###### Proof.

By unitarity, $A^{\dagger}A=I-B^{\dagger}B$ and $DD^{\dagger}=I-BB^{\dagger}$.
Since the determinant of sum of identity plus a matrix can be expressed by
sums of principal minors of that matrix [47, 63], and all the symmetric
functions of $B^{\dagger}B$ and $BB^{\dagger}$ are equal to sums of respective
principal minors [46], we can conclude that
$\det(A^{\dagger}A)=\det(DD^{\dagger})$ and
$|\det(A)|=|\det(D)|\leqslant|\det(U)|$. ∎

The statement can be generalized to non-square complementary submatrices as
they share an identical nontrivial spectrum. The result can be further
generalized with interlacing eigenvalues.

Although we can define the phase of a matrix [64], we restrict our attention
to the phase of minors.

######  Theorem 2.

Phases of complementary minors of a unitary matrix add up to the phase of the
determinant of the unitary matrix.

###### Proof.

For well-defined phases, $A$ and $D$ need to be invertible. Using Schur’s
complement [46],
$\det(U)=\det(A-BD^{-1}C)\det(D)=\det(A)\det(I-A^{-1}BD^{-1}C)\det(D)$. By
unitarity, and using the determinant of sum of matrices again, we can deduce
that $\det(I-A^{-1}BD^{-1}C)$ is real. Hence
$\angle\det(A)+\angle\det(D)=\angle\det(U)$ when the phase angle is properly
defined. ∎

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