The transfer matrix model

Abstract

Chapter 2, and in particular Sect. 2.4, developed the necessary tools to describe the interaction of an atom with the electromagnetic field, as parametrised by the atoms characteristic polarisability. In one dimension, one can succinctly describe the fields interacting with a linear scatterer through what is called the transfer matrix approach [1]. Restricting ourselves to one spatial dimension is not an overly restrictive approximation, despite the quote at the beginning of this chapter; the formalism that is discussed in this chapter allows us to describe a wealth of physical situations. The purpose of this chapter is to extend this model significantly, enabling it to account for moving as well as static scatterers; this is done in Sect. 4.1 and the model that results is solved generally in Sect. 4.2. The extended model discussed here takes into account the first-order Doppler shift but not relativistic effects, and it is therefore correct only up to first order in the velocity of the scatterer.

The reader might wonder why it is of interest, physically, to consider $n$-manifolds for which $n$ is larger than 4, since ordinary spacetime has just four dimensions. In fact many modern theories [...] operate within a ‘spacetime’ whose dimension is much larger than 4.

R. Penrose, The Road to Reality (2004)

Appendix: Cavity Properties from the Transfer Matrix Model

In this section we will derive a consistent set of relations used to describe cavities, based on the transfer matrix formalism. Specifically, we will relate the cavity HWHM linewidth $\kappa$, its finesse $\mathcal F$, and its $Q$-factor to one another and to the reflectivity of the cavity mirrors.

1.1 Cavity Finesse

The finesse of a cavity is defined as

$$\begin{aligned} \mathcal F =\frac{\Delta \lambda }{\delta \lambda }, \end{aligned}$$

(A.1)

where $\delta \lambda$ is the FWHM linewidth of the cavity transmission peak and $\Delta \lambda$ is the free spectral range (FSR) of the cavity. We model the cavity as a Fabry–Pérot resonator having mirrors of reflectivity $r_1$ and $r_2$ and a length $L$. On resonance, we can find some integer $n$ such that

$$\begin{aligned} L=\tfrac{1}{2}n\lambda . \end{aligned}$$

(A.2)

The FSR is defined [35] as the wavelength interval such that

$$\begin{aligned} L=\tfrac{1}{2}(n+1)(\lambda -\Delta \lambda ). \end{aligned}$$

(A.3)

We approximate $\Delta \lambda \ll \lambda$, whereby

$$\begin{aligned} \Delta \lambda =\frac{\lambda ^2}{2L}. \end{aligned}$$

(A.4)

The cavity is described by the transfer matrix equation:

$$\begin{aligned} \left(\begin{array}{c} A\\ B \end{array}\right)= \frac{1}{t_1t_2} \left[\begin{array}{cc} t_1^2-r_1^2&r_1\\ -r_1&1 \end{array}\right] \left[\begin{array}{cc} e^{-\mathrm i kL}&0\\ 0&e^{\mathrm i kL} \end{array}\right] \left[\begin{array}{cc} t_2^2-r_2^2&r_2\\ -r_2&1 \end{array}\right] \left(\begin{array}{c} C\\ D \end{array}\right). \end{aligned}$$

(A.5)

We set $C=0$ and write the transmitted field $D$ in terms of the only input field, $B$:

$$\begin{aligned} |D|^2=\frac{|t_1t_2|^2}{|1-|r_1r_2|\exp (2\mathrm i kL)|^2}|B|^2, \end{aligned}$$

(A.6)

where the phase shifts induced by $r_1$ and $r_2$ have been absorbed in $L$. For a reasonably good cavity ($|t_{1,2}|\ll 1$, $\delta \lambda \ll \lambda$), the transmission is therefore Lorentzian, with a FWHM linewidth

$$\begin{aligned} \delta \lambda =\frac{\lambda }{kL}\frac{1-|r_1r_2|}{\sqrt{|r_1r_2|}}. \end{aligned}$$

(A.7)

Finally, we substitute Eqs. (A.4) and (A.7) into Eq. (A.1) to obtain

$$\begin{aligned} \mathcal F =\frac{\pi \sqrt{|r_1r_2|}}{1-|r_1r_2|}. \end{aligned}$$

(A.8)

If the approximations $|t_{1,2}|\ll 1$ and $\delta \lambda \ll \lambda$ no longer hold, it can be similarly shown that Eq. (A.6) implies

$$\begin{aligned} \mathcal F =\frac{\pi /2}{\sin ^{-1}\biggl (\frac{1-|r_1r_2|}{2\sqrt{|r_1r_2|}}\biggr )}. \end{aligned}$$

(A.9)

1.2 Physical Meaning of the Cavity Finesse

The factor $\rho =|r_1r_2|$ present in the above relations is related to the power lost by the cavity after one round-trip, $1-\rho ^2$. Let us set $N=\sqrt{\rho }/(1-\rho )$. The power remaining in the cavity after $N$ round-trips is then

$$\begin{aligned} \rho ^{2N}=\rho ^{\frac{2\sqrt{\rho }}{1-\rho }}\rightarrow \frac{1}{e^2}, \end{aligned}$$

(A.10)

where we have taken the good-cavity ($\rho \rightarrow 1$) limit. In other words, we can write

$$\begin{aligned} \mathcal F =\pi N, \end{aligned}$$

(A.11)

where $N$ is the number of round-trips the light makes inside the cavity before the intensity decays by a factor of $1/e^2$.

1.3 Cavity Linewidth and Quality Factor

The HWHM cavity linewidth in frequency space can be defined in terms of the FWHM linewidth in wavelength space by means of the relation

$$\begin{aligned} 2\kappa =\frac{\omega }{\lambda }\,\delta \lambda , \end{aligned}$$

(A.12)

whereupon

$$\begin{aligned} \delta \lambda =\frac{\kappa \lambda ^2}{\pi c}, \end{aligned}$$

(A.13)

and substituting this expression for the linewidth into Eq. (A.1) gives

$$\begin{aligned} \mathcal F =\frac{\pi c}{2\kappa L},\;\text{ or}\;\mathcal \kappa =\frac{\pi c}{2\mathcal F L}. \end{aligned}$$

(A.14)

The quality factor, or $Q$-factor, is defined as the ratio of the cavity frequency to its FWHM linewidth in frequency space: $Q=\omega /(2\kappa )$, or

$$\begin{aligned} Q=\frac{2\mathcal F L}{\lambda }. \end{aligned}$$

(A.15)