Theory (for students) | Rayko Research Group

Ultrafast Optics and Electromagnetism.

Tue, 20 Jun 2023 00:00:00 +0000

This is the chapter about what is meant by Ultrafast Physics in the context of Terahertz pulses.

Table of Contents

I am going to talk about femtosecond optics and physics

Ultrafast Optics

What is Ultrafast physics goes into here, which is you know yet to be updated…. You know I need to talk about femtosecond pulses and stuff

Electromagnetism

This entire work lays its foundations upon Maxwell’s macroscopic equations:

\begin{equation} \nabla \cdot \mathbf{D} = \rho_f, \tag{ max-1 } \end{equation} \begin{equation} \nabla \cdot \mathbf{B} =0, \tag{ max-2 } \end{equation} \begin{equation} \nabla \times \mathbf{E} =-\frac{\partial \mathbf{B}}{\partial t}, \tag{ max-3 } \end{equation} \begin{equation} \nabla \times \mathbf{H} =\frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}_f, \tag{ max-4 } \end{equation}

where $\rho_f$ and $\mathbf{J}_f$ are respectively the free charge and current densities within some space. The macroscopic fields $\mathbf{D}$ and $\mathbf{H}$ are defined as

\begin{equation} \mathbf{D} \equiv \epsilon_0 \mathbf{E} + \mathbf{P}=\epsilon \mathbf{E}, \tag{ d_def } \end{equation} \begin{equation} \mathbf{H} \equiv \frac{1}{\mu_0}\mathbf{B} - \mathbf{M}=\frac{1}{\mu}\mathbf{B}, \tag{ h_def } \end{equation}

where $\epsilon_0$ and $\mu_0$ are respectively the permittivity and permeability of free space with $\epsilon$ and $\mu$ being the electric permittivity and the magnetic permeability of a material, respectively. The polarization $\mathbf{P}$ and magnetization $\mathbf{M}$ hold the macroscopic information regarding the properties of the medium in mind.

Boundary Conditions at a surface discontinuity

The above equations are stated for regions of space where there is no discontinuity in the material properties of the medium. However, objects exist causing abrupt changes in the material properties needed to describe the scene in mind. These changes impose boundary conditions to the electric and magnetic fields across the surface of such discontinuities. These boundary conditions are only stated here due to their immense importance in all electro-magnetic phenomena and for the sake of completeness;

\begin{align} \mathbf{n}_{12}\cdot (\mathbf{B}^{(2)}-\mathbf{B}^{(1)}) & =0, & \mathbf{n}_{12}\times (\mathbf{H}^{(2)}-\mathbf{H}^{(1)}) & =\mathbf{j}_s, \\ \mathbf{n}_{12}\cdot (\mathbf{D}^{(2)}-\mathbf{D}^{(1)}) & = \rho_s, & \mathbf{n}_{12}\times (\mathbf{E}^{(2)}-\mathbf{E}^{(1)})& =0 \label{eq:b_cond} \tag{b_cond} \end{align} where $\rho_s$ and $\mathbf{j}_s$ are respectively the surface charge and current densities across the discontinuity and $\mathbf{n}_{12}$ is the vector normal to the surface. In words, these boundary conditions can be stated as: The normal component to the magnetic induction and the tangential electric field are both continuous across the discontinuity, and the normal electric displacement and tangential magnetic fields change abruptly with their discontinuities respectively equaling $\rho_s$ and $\mathbf{j}_s \times \mathbf{n}_{12}$. A full derivation of these boundary conditions can be found in chapter 1.1.3 of reference ¹.

Wave equation and Fabry-Perot

From Maxwell’s equations, we next obtain the wave equation for the electric field. This is accomplished by putting equations (max-4), (d_def) and (h_def) into the curl of (max-2) and simplifying with the vector identity $\nabla \times (\nabla \times \mathbf{A})=\nabla(\nabla \cdot \mathbf{A})-\nabla^2 \mathbf{A}$; \begin{equation} \nabla^2 \mathbf{E}-\epsilon \mu \frac{\partial^2 \mathbf{E}}{\partial t^2}=\mu \frac{\partial \mathbf{J}_f}{\partial t}+ \frac{1}{\epsilon}\nabla \rho_f. \tag{ wave_0 } \end{equation} Further simplifications are made with Ohm’s law $\mathbf{J}_f=\sigma\mathbf{E} $ and neglecting charge density fluctuations, ie. $\nabla\rho_f=0$, to then obtain \begin{equation} \nabla^2 \mathbf{E}-\epsilon \mu \frac{\partial^2 \mathbf{E}}{\partial t^2}=\sigma \mu \frac{\partial \mathbf{E}}{\partial t}, \tag{ wave_1 } \end{equation} where $\sigma$ is the electrical conductivity. An identical wave equation is obtained is for $\mathbf{H}$ in an identical manner. In the case of free space propagation, $\sigma=0, \epsilon=\epsilon_0, \mu=\mu_0$, one obtains $c=1/\sqrt{\epsilon_0\mu_0}$ as the speed at which an electromagnetic wave moves through a vacuum. In other words $c$ is the speed of light. However, should $\sigma=0$, $\epsilon \neq \epsilon_0$ and $\mu \neq \mu_0$ then the wave propagates with speed $v=c/n$ where $n$ is the refractive index of the material given by $n^2=\frac{\epsilon \mu}{\epsilon_0 \mu_0}$.

Having an equation only sets up the problem and does not yield insight or information regarding the observable world. For this reason, we look for solutions to the wave equation that are expressed as linearly polarized, monochromatic, plane waves traveling in the z-direction with wave-vector $\mathbf{k}=k_z \mathbf{\hat{z}}$, ie:

\begin{equation} \mathbf{E}(\mathbf{r},t)=\mathbf{E}_0 e^{i(k_z z- \omega t)}. \tag{ plane_wave_0 } \end{equation}

Putting this equation into eq. (wave_1) yields the following dispersion relation \begin{equation} k^2_z=\omega \mu(\epsilon \omega + i \sigma). \tag{ dispersion_0 } \end{equation} This relationship determines how a wave propagates in a medium with specific electromagnetic properties $\epsilon, \mu, \sigma$. In the case of a dielectric or an insulator $\sigma \simeq 0$ hence $k_z$ is purely real. Then the wave propagates as \begin{equation} \mathbf{E}(\mathbf{r},t)=\mathbf{E}_0 e^{i(\omega \sqrt{\mu \epsilon} z- \omega t)} \tag{ propagate_0 } \end{equation} and experiences no decay provided $\mu$ and $\epsilon$ are both positive and real. In a conductor, however, the conductivity is very large such that $\sigma \gg \epsilon \omega$ thus $k^2 \approx i \omega \mu \sigma$. Evidently

\begin{equation} k=k_r+i k_i \approx \sqrt{\frac{\omega \mu \sigma}{2}}(1+i), \tag{ k_conductor_0 } \end{equation}

where $k_r$ and $k_i$ are the real and imaginary parts of the $k$ vector. In this case the EM wave propagates as

\begin{equation} \mathbf{E}(\mathbf{r},t)=\mathbf{E}_0 e^{i(\omega \mu \sigma z/2- \omega t)}e^{-z/d}, \tag{ propagate_1 } \end{equation}

where $d=\sqrt{\frac{2}{\omega \mu \sigma}}$ is known as the attenuation length or skin depth. This value indicates how far the wave will penetrate before being attenuated.

Reflections at Boundaries

We now have a plane wave as a simple solution to our wave equation. If we input this into $\nabla \cdot \mathbf{E}=0$ and $\nabla \cdot \mathbf{H} =0$ we observe the following relation

\begin{equation} \mathbf{k} \cdot \mathbf{E}=\mathbf{k} \cdot \mathbf{H}=0. \tag{ k_vector_0 } \end{equation}

This relationship implies that $\mathbf{E}$ and $\mathbf{H}$ are both perpendicular to the direction of travel, hence EM waves are transverse. A consequence of this is that if one considers transmission through an interface between two media of different refractive indices, then the wave can be polarized perpendicular or parallel in regards to the plane incidence as shown in Figure Fresnel Coefficients. This consequence combined with the continuity boundary conditions at the surface, in section Boundary Conditions, implies that you get different reflection and transmission coefficients depending on how your incident light is polarized.

Figure Fresnel Coefficients: Reflection and transmission of a plane wave at a surface between two mediums with different refractive indices. Shown are the incident, reflected and transmitted $k$ vectors in blue, and shown with the pink and green arrows is wave polarization parallel and perpendicular to the plane of incidence respectively.

This fact when combined with Snell’s law of refraction \begin{equation} n_1 \sin \theta_1=n_2 \sin \theta_2, \tag{ snell_law } \end{equation}

where $n_{1,2}$ are the refractive indices of the two mediums and $\theta_{1,2}$ are the angles of incidence and refraction, yields the famous Fresnel amplitude reflection and transmission coefficients: \begin{equation} r_{\parallel}=\frac{n_2 \cos \theta_1 - n_1 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}, \tag{ r_para } \end{equation}

\begin{equation} t_{\parallel}=\frac{2n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}, \tag{ t_para } \end{equation} \begin{equation} r_{\perp}=\frac{n_1 \cos \theta_1 - n_2 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}, \tag{ r_perp } \end{equation} \begin{equation} t_{\perp}=\frac{2n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}. \tag{ t_perp } \end{equation}

Furthermore one defines the Reflectivity and Transmissivity as $R=|r|^{2}$ and $T=n_2 |t|^2/n_1$.

Fabry-Perot Interference

The above consideration would hold absolutely true if our world was made from solely two materials. Obviously untrue, hence the next step of extending our mathematical model of the world is to consider the scenario when the second material is of a finite thickness $L$. When a plane arrives at the incident interface it will split into a reflected and a transmitted component. Then, the transmitted part will come up against the second exit interface and split again. The consequential reflected component will split again at the other interface. This process will carry on going indefinitely. Further, every time the wave travels through the dielectric it will pick up a phase shift, between each preceding member of the set of reflected or transmitted waves, of \begin{equation} \phi=\frac{2 \pi f}{c}L n_{2}\cos \theta_2, \tag{ fp_phase } \end{equation} where $f$ is the frequency of the wave.

Figure Fabry-Perot: Reflection and transmission of a plane wave undergoing multiple reflections within a dialectic.

A Fabry-Perot resonance is defined as when all the components resulting from each individual splitting of the wave interfere constructively. Now, if one considers the superposition of all these waves then the total transmitted field $E_t$ is given by

\begin{equation} \begin{split} \begin{split} E_t&=E_i t_1 t_2e^{i\phi}(1+r_1 r_2 e^{2i\phi}+ r_1^2 r_2^2 e^{4i\phi}+...\\ &=E_i t_1 t_2 e^{i\phi}\sum_{n=0}^{\infty}(r_1 r_2 e^{2i\phi})^{n}\\ &=E_i \frac{ t_1 t_2e^{i\phi}}{1 - r_1 r_2 e^{2i\phi}},\end{split}\end{split} \tag{ fp_trans } \end{equation}

where $t_1, t_2, r_1, r_2$ are the relevant Fresnel coefficients in Figure Fabry-Perot. A more detailed derivation is given in ch. 7.6.1 of reference ¹ along with the equation for reflection; \begin{equation} E_r=-\frac{r_2(1-(r_2^2+t_1 t_2)e^{2i\phi}}{1-r_2^2 e^{2i\phi}}E_i. \tag{ fp_refl } \end{equation}

These two equations (fp_trans) & (fp_refl) do need to be considered if one wishes to do a reflection or transmission experiment through any material. In THz measurements they are also used to extract permittivity of an unknown material, as outlined in \S\ref{sec:multi-layers}.

Fundamental models of matter

So far, the previous sub-sections have assumed that the materials properties, $\epsilon, \mu$ and $\sigma$, do not change with the frequency of the EM wave. This is false for all materials in an absolute sense, however for certain frequency ranges this can be approximately true and such materials are called dispersion-less. However, most materials do change with frequency since everything contains atoms and electrons which interact with an incident EM wave.

Classical Lorentzian

For an improved mathematical description of the world the classical Lorentzian model was developed. It accounts for the response of charged and bound particles to an incident EM wave. Here, one assumes that a bound charge oscillates about its equilibrium position and thus has a potential energy given by a simple harmonic oscillator of frequency $\omega_0$ and mass $m$: \begin{equation} U(x)=\frac{1}{2}m \omega_0^2 x^2. \tag{ harmonic_potential } \end{equation} Then the charged particle will experience a restoring force $F_r$ from $\mathbf{F}=-\nabla U$. Further more, there will be a damping term $F_d$ and a force from the incident electric field $F_E$. Combining these forces into Newton’s second law gives;

\begin{equation} \begin{split}m\frac{d^2 x}{dt^2}&=F_r+F_d+F_E\\ &=-m\omega_0^2x-m\gamma\frac{dx}{dt}+qE,\end{split} \tag{ newton_law2 } \end{equation}

where $\gamma$ is the phenomelogical damping rate and $q$ is the charge of the charged particle. If we say that we have a scalar monochromatic linearly polarized EM wave, ie. it is of the form of eq. (plane_wave_0), then the solution to eq. (newton_law2) is given by \begin{equation} x(t)=\frac{q E_0 e^{-i\omega t}}{m(\omega_0^2-\omega^2-i\gamma\omega)}. \tag{ harmonic_oscillator } \end{equation}

Now one knows the electric dipole moment per charged harmonic oscillator $p(t)=qx(t)$. Hence, for a medium with $N$ oscillators per unit volume we have an electric polarization of

\begin{equation} \begin{split}P(t)=Nqx(t) &=\frac{Nq^2 E_0 e^{-i\omega t}}{m(\omega_0^2-\omega^2-i\gamma\omega)}\\ &\equiv \epsilon_0 \chi(\omega)E_0 e^{-i\omega t}, \end{split} \tag{ lorentz_oscillator } \end{equation}

where $\chi(\omega)$ is the linear susceptibility of the medium. Now if we consider eqs. (d_def) and (lorentz_oscillator) we can define the relative permittivity of our medium \begin{equation} \epsilon_r (\omega) \equiv \frac{\epsilon(\omega)}{\epsilon_0} =1 + \chi(\omega)=1 + \frac{Nq^2 }{m\epsilon_0(\omega_0^2-\omega^2-i\gamma\omega)} \tag{ eps_rel } \end{equation}

with real and imaginary parts $\epsilon_r=\epsilon_r’+i\epsilon_r’’$, given by

\begin{equation} \epsilon_r'=\frac{Nq^2(\omega_0^2-\omega^2)}{m\epsilon_0((\omega_0^2-\omega^2)+\omega^2\gamma^2)} \tag{ eps_real } \end{equation} \begin{equation} \epsilon_r''=\frac{Nq^2\omega\gamma}{m\epsilon_0((\omega_0^2-\omega^2)+\omega^2\gamma^2)} \tag{ eps_imag } \end{equation}

In terahertz this model is most often used to account for the absorption caused by crystal lattice vibrations.

Drude model

In the section above the charged particle is bound in space, however the scenario of it being free to move about in space is also possible. To account for such a response the Drude model was developed. Its assumptions are that we have a sea of mobile electrons and a set of stationary positively charged ions constituting our medium. The mobile electrons freely move in only straight lines unaffected by any other forces except those in the instantaneous (assumed to be so) collisions with the impenetrable ion cores (electron-electron collisions conserve momentum hence no change to the current). The electrons collide with the ion cores on average after time $\tau$. This is the only mechanism by which they reach thermal equilibrium, hence we assume that each collision randomizes the velocity with a speed appropriate to the local thermal conditions.

With the above assumptions we can find the DC electrical conductivity of a material. This is done by considering the current density created from $N$ electrons per unit volume moving through a surface area perpendicular to velocity of the electrons $\mathbf{v}$. The charge carried by each electrons is $-e$, hence the current density is simply \begin{equation} \mathbf{J}=-Ne\mathbf{v}. \tag{ drude_current } \end{equation}

Now, if we apply an electric field $\mathbf{E}$ then after time $t$ an electron’s velocity will be $\mathbf{v}=\mathbf{v}_0-e\mathbf{E}t/m$, where $\mathbf{v}_0$ is the electron’s velocity after its previous collision. Due to the velocity randomization through each collision $\mathbf{v}_0$ averages out to zero. Since each collision occurs on average after time $\tau$, we have

\begin{equation} \mathbf{v}_{avg}=-\frac{e\mathbf{E}\tau}{m}; \;\;\;\;\;\; \mathbf{J}=\frac{Ne^2\tau}{m}\mathbf{E}, \tag{ drude_avg } \end{equation}

where $\mathbf{v}_{avg}$ is known as the electron drift velocity. From Ohm’s law we can see that the DC conducivity, $\sigma_0$, of a metal is \begin{equation} \mathbf{J}=\sigma_0 \mathbf{E}; \;\;\;\;\;\; \sigma_0=\frac{Ne^2\tau}{m}. \tag{ drude_sigma0 } \end{equation}

In practice we don’t know $\tau$ but by measuring the conductivity we can estimate values for it.

With the above assumptions, the Drude response of a medium to a monochromatic wave can be characterized by that of a Classical Lorentzian oscillator centered at zero frequency and a damping rate equal to the probability of collision per unit time, ie. $\omega_0=0$ and $\gamma=1/\tau$ in eq. (harmonic_oscillator). However, we are interested in the velocity not position so we can use eq. (drude_current) to obtain

\begin{equation} \sigma(\omega)=\frac{Ne^2\tau}{m(1-i\omega\tau)}=\frac{\sigma_0}{1-i\omega\tau} \tag{ drude_ac } \end{equation}

as the AC conductivity of our material. Now, if we turn our attention to the wave equation (wave_1) describing our EM wave and say $\mu=\mu_0$, we can see that it can written in the form

\begin{equation} \nabla^2 \mathbf{E}=-\omega^2 \mu_0\left(\epsilon+ \frac{i \sigma_0}{\omega(1-i\omega \tau)}\right)\mathbf{E}. \tag{ drude_wave } \end{equation}

This yields the following dispersion relation for our monochromatic wave;

\begin{equation} k_z^2=\omega^2\mu_0 \epsilon_0\epsilon(\omega); \;\;\;\;\; \epsilon(\omega)=\epsilon_{\infty}+\frac{i \sigma_0}{\epsilon_0\omega(1-i\omega \tau)}, \tag{ drude_disp } \end{equation}

where $\epsilon_{\infty}$ is the frequency independent dielectric permittivity due to the contribution of bound charges and $\epsilon(\omega)$ is known as the Drude permittivity.

If we now consider the case of $\omega\tau»1$ we can see that our Drude permittivity approximates to \begin{equation} \epsilon(\omega) =\epsilon_{\infty}-\frac{Ne^2}{m\epsilon_0\omega^2}=\epsilon_{\infty}-\frac{\omega_p^2}{\omega^2}, \tag{ drude_plasma } \end{equation}

where $\omega_p=\sqrt{Ne^2/m\epsilon_0}$ is known as the plasma frequency of the material. Since $k_z \propto \sqrt{\epsilon(\omega)}$ then when $\omega^2 \epsilon_\infty>\omega_p^2$ we have a purely real dispersion relation thus the wave propagates inside the material. For $\omega^2 \epsilon_\infty<\omega_p^2$ we have a purely imaginary $k_z$ signifying that the waves decay inside the material at the rate given by $k_i(\omega)=\frac{1}{c}\sqrt{\omega_p^2-\epsilon_\infty\omega^2}$.

Born, M. and Wolf, E. Principles of optics : electromagnetic theory of propagation, interference and diffraction of light. Cambridge University Press (1999). ISBN 9780521642224. ↩︎ ↩︎

Terahertz time-domain spectropscopy.

Tue, 20 Jun 2023 00:00:00 +0000

The basic idea of terahertz time-domain spectroscopy is described here.

Table of Contents

THz Time domain spectrometer

Figure THz-TDS: Schematic of time domain terahertz spectrometer. A beam of ultrashort optical pulses leave an ultrafast laser. The beam is split into three beams: generation, detection and excitation. A chopper is placed in the detection or generation beams, depending on the needs of the experiment. A spatial light modulator, which can be synchronized to the ultrafast-laser, is placed in the excitation beam. Parabolic mirrors are used to collect and collimate the THz radiation.

The fundamental layout of a Terahertz time-domain spectromter (THz-TDS) is shown in Figure THz-TDS. In its essence, a beam of femtosecond optical pulses is split into three beams: generation, detection and excitation. The first is used to generate a picosecond THz pulse, through optical rectification in ZnTe as discussed in sec. \ref{sec:THz_gen}, which then passes through the sample under investigation. Our THz beam is collimated and collected by 90$^\circ$ off-axis parabolic mirrors made from gold. The second beam is used to detect the time profile of the THz waveform. This is achieved by temporally overlapping the much longer THz pulse with the very short detection pulse. The difference in pulse durations allows one to discretely sample the terahertz temporal profile by varying the path lengths with an optical delay line (typical THz transient and detection pulse envelope shown in Figure THz pulse. The electric field amplitude is detected via the methods discussed in sec. \ref{sec:THz_det}. These systems detect the amplitude and relative phase of the THz pulse. This allows for the extraction of a real and imaginary permittivity of a sample as discussed in sec. \ref{sec:extract_perm}. The third beam is used to photoexcite the sample. These three beams are what constitute a typical THZ-TDS system. However, additionally in our system, the excitation beam can be spatially modulated via a digital micromirror device (DMD) and a lens so as to project any binary intensity pattern onto our sample, and our DMD is synchronized to the main laser system. The reason for this is to enable the construction of the single-pixel camera as discussed in sec. \ref{sec:had_mat}.

Single-cycle THz transients

Figure THz transient Red: typical THz pulse detected by our system in normal room conditions (~35% humidity). Blue: THz pulse recorded in a box pressurized with dry air (air passed through desiccant). Green: envelope of detection pulse used to discretely sample the THz waveform. Arrow points to the maximum field strength of our single-cycle THz pulse. Oscillations after the red THz pulse are due to water vapor in the background environment. Inset: Blue(red): Fourier spectrum of the THz pulse without (with) water vapour oscillations.

This section describes the characteristics of our THz pulses. Figure THz transient shows a typical temporal trace of the THz pulses detected by our THz-TDS, ie. a plot of $E(t)$ with $0.04ps$ resolution. The envelope of our probe pulses used to discretely sample the field-strength of our THZ waveform is also shown here in green. The black arrow points to the maximum field value of our THz pulse at $\approx 4.2ps$. In-front and behind this positive value two other peaks with negative values can be seen. This is our main THz waveform which is generated and detected by our system. In the blue line, some other oscillations behind the main THz pulse can be seen. These are dependent of optics and alignment of the system, therefore they they do not change regularly but only when one performs a major change/update to the system¹. In the red curve, one can see many other oscillations after the main THz pulse. The experimental difference between the red and blue curves is that the red pulse was measured in normal room conditions, where as the blue one was measured in a dry air enclosure². Therefore, the numerous oscillations seen in the red pulse are due to the water vapor in the ambient atmosphere ($\sim$35% humidity). These are the rotational modes of the water vapor molecules³, and consequently the red THz pulse has reduced field-strength.

This measurement is performed in the time domain. Therefore, by Fourier transformation one can obtain the spectrum, including the phase information, of the frequencies in our THz pulse, or mathematically \begin{equation} |E(\omega)|e^{i\phi(\omega)} = \int_{\infty}^{\infty} E(t) e^{-2 \pi i t f} \text{d}t, \tag{ fft_eq } \end{equation} where $\phi(\omega)$ is the phase information at each angular frequency $\omega =2 \pi f$. This Fourier transform implies that our spectral resolution is determined by the temporal length of our measurement. %Added this last sentence. The amplitude spectrum, $|E(2 \pi f)|$, of our THz pulse is shown in Figure Fourier Spectrums. The full-width-half-maximum and the central frequency of our pulses are $1.3THz$ and $0.95THz$ ($325\mu m$) respectively. The red line shows the spectrum when the water rotational oscillations are present. They manifest at various frequencies as absorptions lines of different widths. These give artifacts when performing spectroscopic analysis, sec. THz spectroscopic analysis, hence they need to be eradicated in experiment if such analysis is required.

THz spectroscopic analysis

Obtaining the frequency dependent refractive index of a sample is the standard use of a THz-TDS. Consider a pulse of radiation incident upon a slab of material. There will be a transmitted and a reflected pulse (see inset of Figure THz pulses). The amplitudes of the transmitted and reflected pulses will be related to the incident pulse amplitude via the Fresnel reflection and transmission coefficients, sec. Reflections at Boundaries. However, multiple reflections will occur within the material, as outlined in sec. Fabry-Perot Interference, hence the transmission and reflection functions of the material are respectively given by eqs. (fp_trans) and (fp_refl) . Therefore, if one has knowledge the incident wave and measures the transmission through (or reflection of) some material of known thickness, then it is possible to solve eqs. (fp_trans) and (fp_refl) for the refractive index of the material. We next outline how to extract the refractive index of a plastic cover slip with our system.

In our THz-TDS we perform two measurements: one measuring the temporal waveform transmitted through a sample, $E_{\mbox{samp}}(t)$, and the other to obtain a reference waveform without the sample, $E_{\mbox{ref}}(t)$. These two measurements are shown in Figure THz pulses. Here one can see that the sample pulse (in red) is at a later time than the reference pulse (in blue). This is because it has traveled through the sample which has a larger refractive index than air (the reference pulse traveled through air). A small pulse at $\sim 14ps$ can be seen in the reference waveform, and it arises from the the first Fabry-Perot reflection in the material. Note, some fast oscillations can be see behind our main pulses. They were found to change depending on the optical alignment of our system, hence are associated with detector response function of our ZnTe crystal.

Our measurements contain the relative amplitudes and propagation times of our pulses, therefore a Fourier transform, eq. (fft_eq), will yield the frequency spectrum and along with the relative phase of each frequency. The frequency spectrum and the phase of our THz pulses are shown on in Figure Fourier Spectrums. Here the solid blue and red lines show the detected transmission spectrum through free-space and our sample respectively. It can be seen that these two solid curves have the similar shapes, however the red curve has small oscillations in it. These are due to the first Fabry-Perot pulse detected in the time-domain measurement. The blue and red dashed lines show the phase⁴ of the free-space and our sample waveforms respectively. One can see that the red dashed line has larger values of phase. This is because the sample waveform has accumulated a bigger phase delay by propagating through the sample.

Figure THz pulses: THz pulses measured by our system. Blue shows the reference pulse and the red trace shows the THz pulse transmitted through our plastic cover slip (this was a 810$\mu m$ thick plastic petry dish made from non-cytotoxic virgin polystyreneSterilin BS EN ISO 24998:2008). Inset shows a schematic illustration of the measurement.

Figure Fourier Spectrums: Fourier spectrum of the THz pulses on the left. The solid lines are the power spectrum and dashed lines are the unwrapped phase where the blue (red) colour shows the data from the reference (sample) scan.

After the Fourier transformation, we have $E_{\mbox{ref}}(\omega)$ and $E_{\mbox{samp}}(\omega)$ which be substituted in eq. (fp_trans). Note, our reference measurement is related to the incident waveform by $E_{\mbox{ref}}(\omega) e^{-i\omega L/c}=E_i$, where $L$ is the sample thickness, since it has propagated through air with a refractive index of $1$. Therefore we obtain \begin{equation} \frac{E_{\mbox{samp}} (\omega) }{ E_{\mbox{ref}} (\omega) e^{-i\omega L/c}} = \frac{ t_1 t_2e^{i\phi_n} }{ 1 - r_1 r_2 e^{2i\phi_n} }, \tag{ ext_perm } \end{equation} where $\phi_n = \omega n L /c$ is the phase delay the wave accumulates by propagating through our sample, $t_{1,2}, r_{1,2}$ are the relevant Fresnel coefficients from sec. Reflections at Boundaries. The only unknown left is the refractive index of our sample, $n$, which can be solved for numerically at each frequency point. Since our measurement contains the relative amplitudes and the relative phases, our solution can obtain the complex refractive index. However, since the phase term is $2 \pi$ periodic there is an infinite number of solutions that satisfy eq. (ext_perm). Therefore, selecting the correct solution is of vital importance. This accomplished by making an initial guess at the refractive index. This guess can be made by looking at the time difference between the reference and sample waveforms. If we denote the temporal co-ordinates of the peaks of the sample and reference waveforms as $t_s$ and $t_r$ respectively, one obtains \begin{equation} n_r = 1 + \frac{(t_s-t_r)c}{L} \tag{ n_real } \end{equation} as the real part of the average refractive index. For non-absorbing samples this will suffice. However, should the sample be absorbing then one can use the above value and \begin{equation} \frac{|E_{\mbox{samp}}(\omega_c)|}{|E_{\mbox{ref}}(\omega_c)|} = t_1 t_2 e^{-n_{i} \omega_c L/c}, \tag{ n_imag } \end{equation} where $\omega_c$ is the central angular frequency of the pulses and $n_{i}$ is the imaginary part of the refractive index, to obtain an initial guess for the imaginary part⁵. A more accurate estimate is obtained by saying $t_{1,2}$ are also functions of $n_{i}$, however a numerical solver needs to then be used. Although crude, this initial guess will ensure that the numerical solver (e.g. the the vpasolve function built into MATLAB 2016b) of eq. (ext_perm) will find the physically relevant solution. A technical note, this crude initial guess is used only for the first frequency point inputed into the solver. Then the solution from the first frequency point is used as the initial guess for the second frequency point. This process is iterated until one has a solution for all the frequencies in the spectrum. This iterative process ensures that the obtained refractive index is a continuous function.

Multi-layer systems

The previous section is only applicable if there are 3 materials, in other words the sample is free standing in space. However, if our sample is enclosed by two cover slips, ie. it is a multi-layered system, then it is convenient to use the transfer matrix method⁶. As before, our measurement contains the transmission (or reflection) coefficients , however our transmission (or reflection) function of the system is ⁶$^{,}$⁷ \begin{equation} t=\frac{ 2 q_i }{q_i M_{11} + q_f M_{22} + q_i q_f M_{12} + Mj_{21}}, \tag{ multi_t } \end{equation} \begin{equation} r=\frac{ q_i M_{11} - q_f M_{22} + q_i q_f M_{12} - M_{21} }{q_i M_{11} + q_f M_{22} + q_i q_f M_{12} + Mj_{21}}, \tag{ multi_r } \end{equation} where $q_{i,f}= n_{i,f} \cos(\theta_{i,f})$ for S-polarized light and $q_{i,f}= n_{i,f}/ \cos(\theta_{i,f})$ for P-polarized light, $n_{i,f}$ are the refractive index of the initial and final media, respectively, enclosing the multilayer system, $\theta_{i,f}$ are the incident and exit angles, and $M$ is a $2\times2$ matrix associated with the propagation through the entire multilayer system. This matrix is given by the product of the individual layer matrices, $M\equiv M_1M_2M_3…M_N$, describing the propagation through each layer. The characteristic matrix of the $j^{\text{th}}$ layer, $M_j$, with thickness $l_j$ and refractive index $n_j$ is given by \begin{equation} M_j = \left[ \begin{array}{cc} \cos\beta_j & \frac{i}{q_j}\sin\beta_j \\ i q_j \sin\beta_j & \cos\beta_j \end{array}\right], \tag{ m_j } \end{equation} where $\beta_j=\omega l_j n_j \cos(\theta_j)/c$ is the phase delay associated with light propagation inside the $j^{\text{th}}$ layer, and $q_{j}= n_{j} \cos(\theta_{j})$ for S-polarized light and $q_{j}= n_{j}/ \cos(\theta_{j})$ for P-polarized light. By equating the experimental amplitude transmission coefficients $E_{\mbox{samp}}/E_{\mbox{ref}}$ with (multi_layers), we can then numerically solve for the permittivity of the sample as before. Then one can divide the transmission functions of the different systems and equate them to the experimental amplitudes. Note, it is of preference that the materials used in the reference measurement are not absorptive hence improve signal to noise.

This multi-layer analysis allows one model much more complex systems than a simple Fabry-Perot model. We use this multi-layer analysis in…

For example changing a parabolic mirror or replacing the crystal detector. ↩︎
The box was pressurized with air that was passed through desiccant, thus there was no water vapour in the air but in practice the humidity was about 5%. ↩︎
Slocum, D.M., Slingerland, E.J., Giles, R.H., and Goyette, T.M. Atmospheric absorption of terahertz radiation and water vapor continuum effects, Journal of Quantitative Spectroscopy and Radiative Transfer, 127, 49 (2013). ↩︎
The unwrapped angle, shown here, is typically shown since it does not have $2 \pi$ discontinuities hence it has more visual appeal. ↩︎
This equation is only valid if we consider only the first transmitted pulse through our sample, ie. zero out the Fabry-Perot pulses in the temporal measurement ↩︎
Ulbricht, R., Hendry, E., Shan, J., Heinz, T.F., and Bonn, M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy. Reviews of Modern Physics, 83, 543 (2011). ↩︎ ↩︎
Furman, S.A. and Tikhonravov, A.V. Basics of optics of multilayer systems. World Scientific Publishing (1992). ISBN 2863321102. ↩︎

Terahertz Sources and Detectors.

Tue, 20 Jun 2023 00:00:00 +0000

The techniques used for generating and detecting THz radiation in my laboratory is discussed here.

Table of Contents

THz radiation from non-linear optics

The THz generation and detection mechanisms used here are respectively optical rectification and electro-optic sampling in ZnTe. They are both 2$^{\text{nd}}$ order non-linear polarization effects. The classical description of non-linear polarization phenomena is as follows. Electrons are modeled as masses held in place by non-linear springs. In other words, a Lorentzian oscillator \S \ref{sec2:Classical Lorentzian} in a non-linear potential with cubic, and higher, displacement dependence terms. The solution to the equation of motion for an input of two superpositioned $E$-fields with frequencies $\omega_1$ and $\omega_2$ is¹ \begin{equation} P^{(2)}_i(\omega_1 - \omega_2)=\epsilon_0 \sum_{j,k}\chi_{i,j,k}^{(2)}(\omega_1-\omega_2) E_j(\omega_1)E_k^*(\omega_2), \tag{ p2_i } \end{equation} where $\chi_{i,j,k}^{(2)}$ is the susceptibility tensor and the $i,j,k$ subscripts denote the Cartesian components of polarization. Note, this is the difference frequency term. The case of the sum frequency mixing is observed in the mathematical details¹. In the early 1960s, Bass et al² observed difference frequency mixing, named optical rectification, and Franken et al³ observed sum frequencies generation.

THz generation, Optical Rectification

Optical rectification falls out of eq. \eqref{eq:Pnl and chi} when we take the limit of $\omega_2 \to \omega_1$, \begin{equation} P^{(2)}_i(0)=\epsilon_0 \sum_{j,k}\chi_{i,j,k}^{(2)}(0) E_j(\omega_1)E_k^{*}(\omega_1). \tag{ p2_0 } \end{equation} One can now see that a strong electric field at $\omega_1$ gives rise to a DC polarization. In practice, the bandwidth of the input laser determines the distribution of difference frequency components. Lasers emitting ultra-short (100fs) pulses have frequency bandwidths in the THz regime. Such pulses are shone onto a crystal. This causes a polarization and thus the oscillation of the bound charges. Accelerating charges cause the emission of $E$-fields. Since the polarization has low frequency components, the emitted field has THz frequencies.

The above process can only occur in crystals where the 2$^{\text{nd}}$ order susceptibility tensor, $\chi^{(2)}$, is non-zero. This excludes all crystals with inversion symmetry\footnote{All components of $\chi^{(2)}$ equal $0$ in crystals with inversion symmetry}. Listed in the table below are commonly used THz generation crystals and their symmetry point group. The crystal symmetry group determines the crystal’s response to the angular orientation of the incoming radiation. ZnTe, the crystal used in this thesis, emits maximum THz when the optical polarization lies in the {110} plane. For crystals cut in this plane, THz generation is then maximized by rotation of the crystal wafer.

Material	Point Group
LiNbO$_{3}$, LiTaO$_{3}$	$3m$
ZnTe, GaAs, GaP, InP	$\bar{4}3m$
GaSe	$\bar{6}2m$

For further optimization the crystals needs: to be transparent with minimal absorption at all frequencies involved, to have a high damage threshold to withstand the intensities needed and to not have other competing non-linear processes. For final generation optimization, one has to have constructive interference between all the waves generated from all points in the crystal. In other words, the THz phase velocity has to be equal to the group velocity of the laser input packet within the material.

If the phase matching condition is fully satisfied, the THz field gradually gets amplified while propagating through the medium. Now consider an optical wave traveling faster than the THz wave. The effective interaction length is given as the coherence length $l_c$ of when either waves are $\pi/2$ out of phase with each other: \begin{equation} l_c=\frac{c}{2f_{THz}|n_g-n_{THz}|}, \tag{ coh_length } \end{equation} where $n_g$ ($n_{THz}$) is the group (phase) refractive index at the visible (terahertz) frequencies. From eq. \eqref{eq:coherence length} one can see that $l_c \to \infty$ as $n_g \to n_{THz}$. In reality this condition is hard to satisfy. The most commonly used nonlinear crystal for THz generation is ZnTe because it best satisfies this condition at the operational wavelength (800nm) of Ti:Sapphire lasers. In ZnTe $l_c \to \infty$ for visible wavelengths around 800nm and terahertz frequencies around 1.69THz \cite{Nahata1996}, shown in the fig. \ref{fig:Ref_index_ZnTe}. \begin{figure}[h!]\centering \includegraphics[width=0.65\linewidth]{Chapters/Experimental/Ref_index_ZnTe.pdf} \caption{Group refractive index $n_g$ and phase refractive index $n_{THz}$ at optical and terahertz regimes, respectively, of ZnTe. Taken from ⁴} \label{fig:Ref_index_ZnTe} \end{figure}

Finally, one has to consider the crystal’s absorption in the spectral region of interest. The dominant causes of absorption in such crystals are the transverse-optical phonon resonances in the terahertz region. Table below lists the lowest TO-phonon resonances in some EO crystals in commonly used crystals, taken from ⁴.


	ZnTe	GaP	InP	GaAs	LiNbO$_{3}$
$f_{TO}$ (THz)	5.3	11	9.2	8.1	7.7

P. E. Powers. Fundamentals of Nonlinear Optics. CRC Press (2011). ISBN 9781420093513, ch. 3. ↩︎ ↩︎
Bass, M., Franken, P.a., Ward, F., and Weinreich, G. Optical Rectification. Physical Review Letters, 9, 446 (1962). ↩︎
Franken, P.A., Hill, A.E., Peters, C.W., and Weinreich, G. Generation of optical harmonics. Physical Review Letters, 7, 118 (1961). ↩︎
Lee, Y.S. Principles of terahertz science and technology (2009). ISBN 9780387095394. ↩︎ ↩︎