Appendix 1: The Cayley transform
Throughout this paper, we work in the frequency domain \(\mathbb {C}\) common in the engineering literature, where the imaginary axis takes values iω. In some of the literature, a domain related by a bijective conformal transformation is used instead. The imaginary axis is mapped to the unit circle, and the right halfplane \(\mathbb{C}^{+}\) is mapped to the unit disk \(\mathbb{D}\). This mapping is known as the Cayley transform, and the two domains are often called the sdomain and zdomain, respectively.
The mapping from \(\mathbb{C}^{+}\) to \(\mathbb{D}\) is given by
$$f(z) = \frac{z1}{z+1}. $$
The mapping from \(\mathbb{D}\) to \(\mathbb{C}^{+}\) is given by
$$f^{1} (z)=\frac{1+z}{1z}. $$
Sometimes in the literature the upper halfplane is used instead of the right halfplane, which slightly changes the transformation.
In our notation, a zero \(z_{k}\) of the Blaschke factor in the disc \(\frac{z_{k}z}{1\overline{z_{k}} z} \frac{\vert z_{k}\vert }{z_{k}} \) is transformed to a zero in the plane \(\lambda_{k} = \frac{1+ {z_{k}} }{1 {z_{k}} } \) of the factor \(e^{i \phi_{k}} \frac{z{\lambda_{k}}}{z + \overline{ \lambda_{k}}}\), where \(e^{i \phi_{k}} = \frac{\vert 1\lambda_{k}^{2}\vert }{1\lambda_{k}^{2}}\) contributes only a phase.
Appendix 2: Unitarity and boundedness implies a function is inner
(Based on [23], Lemma 3 on page 223.)
This section will prove the assertion that
$$\begin{aligned} T(z) T^{\dagger}(z)= I \quad \text{for } z \in i \mathbb{R} \quad \implies \quad T(z) T^{\dagger}(z) \le I \quad \text{for } z \in\mathbb{C}^{+}, \end{aligned}$$
(42)
assuming \(T(z)\) is an \(N \times N\) matrixvalued analytic (except possibly at infinity) function bounded in \(\mathbb{C}^{+}\).
For any appropriately sized unit vectors u, v, we have from the CauchySchwartz inequality and the maximum modulus principle that
$$\begin{aligned} \bigl\vert \bigl\langle u,T(z) v \bigr\rangle \bigr\vert ^{2} \le \Vert u\Vert \bigl\Vert T(z) v \bigr\Vert \le1. \end{aligned}$$
(43)
Since the u, v were arbitrary, the assertion follows.
Appendix 3: Potapov factorization theorem and nonpassive linear systems
In this section we cite the Potapov factorization theorem for Jcontractive matrices. This factorization, when applicable, consists of several terms which each have a different property. For this paper we will only need a special case for the theorem, but we cite the full version because it may be conductive to extensions of the work in this paper. In particular, a related frequencydomain condition of physical realizability is discussed in [13] and [24].
3.1 3.1 Definitions
First, we introduce some terminology common in the literature. Let \(D = \mathbb{D}\) or \(\mathbb{C}_{+}\) (the unit disc or right halfplane, respectively). Further let J be the signature matrix
$$\begin{aligned} J = \begin{pmatrix} I_{m} & 0 \\ 0 &  I_{r} \end{pmatrix} , \end{aligned}$$
(44)
for some m, r. A matrixvalued function \(M(z)\) is called:

1.
Jcontractive, when \(M(z)JM(z)^{\dagger}\le J'\) for z in D,

2.
Junitary when \(M(z)JM(z)^{\dagger}= J'\) for z on ∂D,

3.
Jinner (or Jlossless) when \(M(z)\) is Junitary and Jcontractive.
When \(J = I\), we drop the J in the definition.
Definition
(Stieltjes multiplicative integral (from [18]))
Let \(K(t)\) (with \(a \le t \le b\)) be a monotonically increasing family of JHermitian matrices with \(t = \operatorname {tr}[K(t) J] \) and \(f(t)\) (\({a \le t \le b}\)) be a continuous scalar function. Then the following limit exists
$$\begin{aligned} \lim_{\max\Delta t_{j} \to0 } e^{f(\theta_{0}) \Delta K(t_{0})} e^{f(\theta_{1}) \Delta K(t_{1})}\cdots e^{f(\theta_{n1}) \Delta K(t_{n1})}, \end{aligned}$$
(45)
where we take \(a = t_{0} \le\theta_{0} \le t_{1} \le\cdots \le t_{n} = b\). The limit is denoted
$$\overset{\curvearrowright}{ \int_{a}}^{b} e^{f(t)\,dK(t)} $$
and is called the multiplicative integral.
3.2 3.2 Potapov factorization
In much of the mathematical literature the domain of the transfer function is transformed via the Cayley transform (see Appendix A). This changes how some of the terms in the factorization are written, but not the fundamental features of the factorization.
Next we will cite some of the theorems by Potapov. In the case \(J=I\) the function T satisfies the unitarity condition \(T(z)T^{\dagger}(z) = I\) on the boundary of the disc or halfplane (depending on the domain taken).
The fundamental factorization theorem by Potapov characterizes the class of J contractive matrices.
Theorem
(Adapted from [18])
Let
\(T(z)\)
be a [meromorphic] Jcontractive matrix function in the unit circle
\(\vert z\vert < 1\), and suppose that
\(\det( T(z))\)
does not vanish identically; then we can write
$$\begin{aligned} T(z) = B_{\infty}(z) B_{0}(z) \overset{ \curvearrowright}{ \int_{0}^{\ell}} \exp \biggl( \frac{z+e^{i\theta(t)}}{ze^{i\theta(t)}} \,dK(t) \biggr). \end{aligned}$$
(46)
Here
$$\begin{aligned} B_{\infty}(z) = \prod_{k} U_{k} \begin{pmatrix} I & 0 \\ 0 & \frac{1  \overline{\mu}_{k} z}{\mu_{k}  z}\frac{\mu_{k}}{\vert \mu_{k}\vert }I_{q_{k}'} \end{pmatrix} U_{k}^{1} \end{aligned}$$
(47)
is a product of elementary factors associated with the poles
\(\mu_{k}\)
of the matrix function
\(T(z)\)
inside the unit circle, \(q_{k}' \le q\), and
\(U_{k}\)
is a Junitary matrix;
$$\begin{aligned} B_{0} (z) = \prod_{k} V_{k} \begin{pmatrix} \frac{\lambda_{k}  z}{1  \overline{\lambda}_{k} z} \frac{\vert \lambda _{k}\vert }{\lambda_{k}} I_{p_{k}' } & 0 \\ 0 & I \end{pmatrix} V_{k}^{1} \end{aligned}$$
(48)
is a product of elementary factors associated with the zeros in
\(\vert z\vert <1\)
of the determinant of the matrix function
\(T_{\infty}(z) = B_{\infty}^{1} (z) T(z)\), which is holomorphic in
\(\vert z\vert < 1\), \(p_{j}' \le p\), \(V_{j}\)
is a
Junitary matrix; the last term is the Stieltjes integral, where
\(K(t) J\)
is a monotone increasing family of Hermitian matrices such that
\(t = \operatorname {tr}[K(t) J] = t\). Here
\(\theta\in[0,2 \pi]\)
is a monotonically increasing function
The integral in the above expression is known as the RieszHerglotz integral. It captures the effects of the zeros and poles that occur on the boundary as well as effects not due to zeros or poles. The next theorem will be useful for allowing us to express the multiplicative integral term above in a special way.
Theorem
(Adapted from [18])
An entire matrix function
\(T(z)\), Jcontractive in the right halfplane and
Junitary on the real axis, can be represented in the form
$$\begin{aligned} T(z) = T(0) \overset{\curvearrowright}{ \int_{0}^{\ell}} e^{z K(t)\,dt}, \end{aligned}$$
(49)
where
\(K(t)\)
is a summable nonnegative definite
JHermitian matrix, satisfying the condition
\(\operatorname {tr}[K(t) J ] = 1\).
Appendix 4: Using the Padé approximation for a delay
For a system involving only feedforward (i.e. no signal ever feeds back), no poles or zeros will be found in the transfer function. For this reason, the zeropole interpolation cannot naturally reproduce a transfer function to approximate the system. Instead, a different approach is needed to obtain an approximation for the statespace representation for systems of this kind. Still, any finitedimensional statespace representation will have poles and zeros in its transfer function. If we use such a system as an approximation of a delay, these zeros and poles will be spurious but unavoidable.
The Padé approximation is often used to approximate delays in classical control theory [25]. When using the \([n,n]\) diagonal version of the Padé approximation to obtain a rational function approximating an exponential, we obtain
$$\begin{aligned} e^{Tz} \approx\exp_{[n,n]}(Tz) = \frac{Q_{n}(zT)}{Q_{n}(zT)}. \end{aligned}$$
(50)
The \(Q_{n}(z)\) is a polynomial of degree n with real coefficients. Because of this, its roots come in conjugate pairs. As a result, we can write the Padé approximation as a product of Blachke factors:
$$\begin{aligned} \exp_{[n,n]}(Tz) = \prod_{n}  \frac{z + \overline{{p}_{n} }}{z+ p_{n}}. \end{aligned}$$
(51)
In particular, note that the approximation preserves the unitarity condition, and is therefore physically realizable. For this reason, it is possible to approximate time delays with this approximation.
Although this approach may be useful for the case of feedforwardonly delays, in the case of delays with feedback this may produce undesirable results. To illustrate this, we introduce an example where the Padé approximation is used in order to produce an approximated transfer function for the network discussed in Section 7.1. For this approximation, the order used for the approximation of each delay is chosen to be roughly proportional to the duration of the delay. Figure 14 illustrates the approximated transfer function. Please compare this result to Figure 8, where we have used the zeropole interpolation.
We see in Figure 14 that the peaks in the approximating functions often do not occur in same locations as the peaks of the original transfer function. This may be problematic when attempting to simulate many physical systems for which the locations of the peaks correspond to particular resonant frequencies that have physical relevance. For instance, if one chooses to introduce other components to the system, such as atoms, the resonant frequencies due to the trapped modes of the network must be described accurately or else the resulting dynamics of the approximating system may not correspond to the true dynamics of the physical system. For this reason, using the Padé approximation may not always be the best choice.
Appendix 5: BlaschkePotapov product in the limit \({\Re(z) \to\infty}\)
In Section 8, we proposed a criterion for checking when the multiplicative integral component in Section 5.1 was not necessary. We assumed that the BlaschkePotapov product converged to a nonzero constant in the limit \({\Re(z) \to\infty}\). In this section, for demonstrative purposes we show this is the case for the example of a single trapped cavity discussed in Section 4.
In order to examine the convergence of the product in Eq. (14) of Section 4, we write it in the following way. We observe by taking the logarithm with an appropriate branch cut that
$$\begin{aligned} \prod_{n} \bigl(1+ a_{n}(z)\bigr) \end{aligned}$$
(52)
converges if and only if the infinite sum
$$\begin{aligned} \sum_{n} a_{n}(z) \end{aligned}$$
(53)
converges, assuming \(\sum_{n} \vert a_{n}(z)\vert ^{2} \) converges. We take
$$\begin{aligned} a_{n}(z) = 1  \frac{z+p_{n} }{z\overline{p_{n}} } = \frac{2 \Re(p_{n})}{z  p_{n}}. \end{aligned}$$
(54)
Using the relation
$$\begin{aligned} \cot(z) = \sum_{n \in\mathbb{Z}} \frac{1}{z  n \pi}, \end{aligned}$$
(55)
we get that
$$\begin{aligned} \lim_{\Re(z) \to\infty} \sum_{n} a_{n}(z) =\ln(r). \end{aligned}$$
(56)
Actually, higherorder terms of the logarithm expansion of Eq. (52) go to zero in this limit, so we get that
$$\begin{aligned} \lim_{\Re(z) \to\infty}\tilde{T}(z) = \lim_{\Re(z) \to\infty}\prod _{n \in\mathbb{Z}} \biggl( \frac {z+\overline{p_{n}} }{zp_{n} } \biggr)= \lim _{\Re(z) \to\infty}  \prod_{n} \bigl(1+ a_{n}(z)\bigr) = 1/r. \end{aligned}$$
(57)
This shows the desired result that the infinite product in this limit goes to a nonzero constant. Also interestingly, we have been able to compute this value and remark that it is indeed equal to \(\lim_{\Re(z) \to\infty} T(z)\).
The above example with a single trapped cavity formed is illustrative of typical behavior for more complicated systems formed by networks of beamsplitters and time delays. The zeropole pairs of the system occur in a region of bounded positive real part, and roughly uniformly along the imaginary axis. This suggests that the BlaschkePotapov product resulting from the zeropole interpolation will converge to a nonzero constant in the \({\Re(z) \to\infty}\) limit under quite general circumstances.