Sensing with Light

Abstract

The interaction of highly coherent light and matter is the key in the science of measurement. In the classical electromagnetic theory, the absorption and dispersion properties of the light propagation in a medium are described by the classical Lorentz oscillator model. The unavoidable phase and intensity fluctuations of a light field, which are commonly characterized by the first- and second-order correlation functions, respectively, broaden the light spectrum and restrict its practical applications in metrology. In order to suppress the spectral linewidth, the light’s frequency is usually stabilized to an optical resonator. Several light-based sensing applications, such as the gravity-wave detection and optical clocks, are briefly introduced.

Problems

1.1

A light field is polarized in the x-direction and propagates along the z-axis, $\mathbf{E} (\mathbf{r} ,t)=\mathbf{e} _{x}E^{(+)}(z,t)+\mathbf{e} _{x}E^{(-)}(z,t)$, in a medium. The electric polarization density of the medium $\mathbf{P}$ may be written in a similar way, $\mathbf{P} (\mathbf{r} ,t)=\mathbf{e} _{x}P^{(+)}(z,t)+\mathbf{e} _{x}P^{(-)}(z,t)$. Inserting the above expressions into (1.40), one obtains

$$\begin{aligned} \left( \frac{\partial }{\partial z}+\frac{1}{c}\frac{\partial }{\partial t}\right) \left( -\frac{\partial }{\partial z}+\frac{1}{c}\frac{\partial }{\partial t}\right) E^{(+)}(z,t)=-\mu _{0}\frac{\partial ^{2}}{\partial t^{2}}P^{(+)}(z,t). \end{aligned}$$

The positive-frequency parts $E^{(+)}(z,t)$ and $P^{(+)}(z,t)$ may be written in the following way:

where and are the slowly varying amplitudes. The wavenumber $k_{0}$ is related to the light’s central frequency $\omega _{0}$ via $k_{0}=\omega _{0}/c$. Derive the following equation:

(1.181)

in the slowly varying envelope approximation (SVEA)

Fig. 1.19

Boundary conditions. Stokes’ theorem is applied on (a) while Gauss’s theorem is used on (b)

1.2

Derive the interface conditions for the electromagnetic fields (1.45) and (1.48) by integrating Faraday’s law (1.1a) and Ampère’s law (1.1d) over an oriented smooth surface S bounded by a closed boundary curve C across the interface [see Fig. 1.19a]. Stokes’ theorem

$$\begin{aligned} \int _{S}(\nabla \times \mathbf{F} )\cdot d\mathbf{S} =\oint _{C}{} \mathbf{F} \cdot d\mathbf{l} \end{aligned}$$

(1.182)

should be applied. The curve C is positively oriented, meaning that the surface S is always on your left side when you are walking along the curve C. The direction of the element vector $d\mathbf{S}$ is parallel to the unit normal vector of the curve C that follows the right-hand rule. The vector field $\mathbf{F}$ may be $\mathbf{E}$ and $\mathbf{H}$.

Derive the boundary conditions (1.46) and (1.47) by integrating (1.1b) and (1.1c) over a volume V surrounded by a surface S across the interface [see Fig. 1.19b]. Gauss’s theorem

$$\begin{aligned} \int _{V}(\nabla \cdot \mathbf{F} )dV=\oint _{S}{} \mathbf{F} \cdot d\mathbf{S} \end{aligned}$$

(1.183)

should be used. Here, the vector field $\mathbf{F}$ may be $\mathbf{D}$ and $\mathbf{B}$. The outward-pointing element vector $d\mathbf{S}$ is normal to the locally planar surface.

1.3

We consider the solution to Poisson’s equation within a volume V. The associated Green’s function $G(\mathbf{r} ,\mathbf{r} _{0})$ is defined as

$$\begin{aligned} \nabla ^{2}G(\mathbf{r} ,\mathbf{r} _{0})=\delta (\mathbf{r} -\mathbf{r} _{0}), \end{aligned}$$

(1.184)

where $G(\mathbf{r} ,\mathbf{r} _{0})$ may be viewed as a potential generated by a point source located at $\mathbf{r} _{0}\in V$. As $|\mathbf{r} -\mathbf{r} _{0}|\rightarrow \infty$, $G(\mathbf{r} ,\mathbf{r} _{0})$ approaches zero. Equation (1.26) is the solution of the above Poisson’s equation. Derive (1.26) by using the fact $\int _{V}\delta (\mathbf{r} )dV=1$ and Gauss’s theorem.

Fig. 1.20

Hansch–Couillaud frequency stabilization. a Schematic diagram with the symbols: LP (linear polarizer), PBS (polarization beam splitter), and PD (photodetector). b Polarization spectroscopy as a function of the detuning $(\omega -\omega _{\text {FP}})$. The amplitude reflection coefficients of mirrors of the FP resonator are both $r_{1,2}=0.9$. The free spectral range of the Fabry–Pérot resonator is $\omega _{\text {FSR}}$

1.4

The frequency stabilization scheme based on the Hansch–Couillaud method [45] is illustrated in Fig. 1.20a. The incident light (frequency $\omega$) output from a laser system is vertically polarized and enters a Fabry–Pérot resonator (central frequency $\omega _{\text {FP}}$) via a reflection mirror. A linear polarizer, whose transmission direction forms a $45^{\circ }$ angle with respect to the incident beam’s polarization, is inserted into the Fabry–Pérot resonator. The reflected wave contains two components with the respective field amplitude $E_{\parallel }=(E_{0}/\sqrt{2})\cdot r_{\text {FP}}(\omega )$ and $E_{\perp }=(E_{0}/\sqrt{2})\cdot r_{1}$, where $E_{0}$ is the amplitude of the incident beam, $r_{\text {FP}}(\omega )$ is the amplitude reflection coefficient of the Fabry–Pérot resonator [see (1.139a)] , and $r_{1}$ is the amplitude reflection coefficient of the resonator’s mirror M1. The reflected light passes through a $\lambda /4$ waveplate and enters a polarization beam splitter (PBS). We assume that the fast axis of the $\lambda /4$ waveplate is parallel to the transmission direction of the intracavity linear polarizer. Using the Jones calculus, the light fields output from the PBS are given by

$$\begin{aligned} \begin{pmatrix} E_{1} \\ E_{2} \end{pmatrix}=\frac{1}{2} \begin{pmatrix} 1&{} -1 \\ -1 &{} 1 \end{pmatrix} \begin{pmatrix} 1&{} 0 \\ 0 &{} i \end{pmatrix} \begin{pmatrix} E_{\parallel } \\ E_{\perp } \end{pmatrix}. \end{aligned}$$

(1.185)

The intensities of the outputs from the PBS $I_{1}\propto |E_{1}|^{2}$ and $I_{2}\propto |E_{2}|^{2}$ are measured by two photodetectors, respectively. The difference $(I_{1}-I_{2})$ gives the error signal [see Fig. 1.20b]. This error signal is finally fed back into the laser, stabilizing the frequency $\omega$. Derive the error signal $(I_{1}-I_{2})$.

1.5

In a $^{87}$Sr optical lattice clock, the $^{87}$Sr atoms are confined in a one-dimensional optical lattice with a potential depth of 20 $\upmu$K. The lattice lasers operate at the red-detuned magic wavelength 813 nm. The clock transition wavelength is 698 nm. Estimate the harmonic-oscillation frequency of the atoms moving inside a lattice site and the Lamb–Dicke parameter.