Time-Frequency Analysis of Thz-Time Domain Spectroscopy Data

UPTEC F 18037 Examensarbete 30 hp Juni 2018

Time-frequency analysis of THz-time domain spectroscopy data

Hugo Laurell Abstract Time-frequency analysis of THz-time domain spectroscopy data Hugo Laurell

Teknisk- naturvetenskaplig fakultet UTH-enheten This text investigates THz-TDS signals in the time-frequency domain. Addi- tionally this text discusses the prospects of using time-frequency analysis to Besöksadress: alleviate distortion in THz spectrographic characterizations induced by back- Ångströmlaboratoriet Lägerhyddsvägen 1 reflections in the free space electro optic sampling used in the THz time-domain Hus 4, Plan 0 spectroscopy detection scheme. THz time domain spectroscopy is a technique for characterization of materials in the terahertz regime. The THz regime offers Postadress: interesting properties of materials such as strong phonon-photon interaction and Box 536 751 21 Uppsala resonances for vibration states of molecules. Three time-frequency representations are compared for the analysis of the time-domain signal, the Telefon: short-time Fourier transform, the Wigner-Ville transform and the continuous 018 – 471 30 03 wavelet transform. It is concluded that the Wigner-Ville transform is most suited

Telefax: for analysis of the spectral properties of a single pulse due to the Wigner-Ville 018 – 471 30 00 transforms inherit high spectral resolution. The continuous wavelet transform is most suited for analysis of the time-domain signal since it has no cross-term Hemsida: interference as compared to the Wigner-Ville transform. By masking the http://www.teknat.uu.se/student continuous wavelet transform with a Lorentzian time-frequency mask the back- reflections are dampened and the resolution of the characterization is improved.

Handledare: Ulf Österberg Ämnesgranskare: Bengt Carlsson Examinator: Tomas Nyberg ISSN: 1401-5757, UPTEC F 18037 Tryckt av: Uppsala Contents

1 Introduction3 1.1 Preface3 1.2 Project Description3 1.3 Project Goals3 1.4 Background3

2 Generation of broadband THz single cycle pulses4 2.1 Photoconductive antenna4 2.2 Optical rectiﬁcation5 2.3 Generation in plasma6 2.4 Comparison of the three generation methods7

3 Terahertz time domain spectroscopy7 3.1 The Pockel’s eﬀect8 3.2 Free space electro optic sampling9 3.3 Transmission spectroscopy 10 3.4 Reﬂection spectroscopy 11

4 Mathematical formulation of signals 11 4.1 Complex analytic signals 12 4.2 Deﬁnition and basic properties of analytic signals 12 4.3 The Heisenberg-Gabor inequality 15 4.4 The Shannon-Nyquist sampling theorem 18 4.5 Instantaneous frequency 19

5 Time-frequency representation 19 5.1 The short-time Fourier transform 21 5.2 Cohen’s class of Quadratic time-frequency representations 22 5.3 The Wigner-Ville time-frequency representation 25 5.3.1 Computation of the Wigner quasi-probability distribution 25 5.3.2 The Wigner-Ville distribution for speciﬁc input signals 26 5.4 Wavelet theory 31 5.4.1 The continuous wavelet transform 32 5.4.2 The generalized Morse pseudo-basis functions 33 5.4.3 Masking 33 5.4.4 Gaussian masking 36 5.4.5 Lorentzian masking 36 5.4.6 Lorentzian-Gaussian masking 38 5.5 Comparison of the three representations 39

– i – 6 Results 40 6.1 Transmission spectroscopy 40 6.1.1 Silicon, High resistivity ﬂoat zone (HRFZ), 4.46 mm 40 6.1.1.1 Short-time Fourier transform analysis 44 6.1.1.2 Wigner-Ville transform analysis 44 6.1.1.3 Continuous wavelet transform analysis 45 6.1.2 Gallium phosphide, GaP, 5 mm 49 6.1.2.1 Short-time Fourier transform analysis 50 6.1.2.2 Wigner-Ville transform analysis 51 6.1.2.3 Continuous wavelet transform analysis 52 6.1.3 Potassium titanyl phosphate, KT iOP O4, 500 [µm] 54 6.1.3.1 Short-time Fourier transform analysis 55 6.1.3.2 Wigner-Ville transform analysis 55 6.1.3.3 Continuous wavelet transform analysis 58 6.1.4 Germanium, 1 mm 60 6.1.4.1 Short-time Fourier transform analysis 60 6.1.4.2 Wigner-Ville transform analysis 61 6.1.4.3 Continuous wavelet transform analysis 63 6.2 Reﬂection spectroscopy 64 6.2.1 Potassium titanyl phosphate, KT iOP O4 65 6.2.1.1 Short-time Fourier transform analysis 66 6.2.1.2 Wigner-Ville transform analysis 67 6.2.1.3 Continuous wavelet transform analysis 67 6.3 Noise 69 6.3.1 (αd)max analysis for HRFZ Silicon in transmission 70

7 Discussion 73 7.1 Transmission spectroscopy 74 7.1.1 Silicon, High resistivity ﬂoat zone (HRFZ), 4.46 mm 74 7.1.2 Gallium phosphide, GaP, 5 mm 74 7.1.3 Potassium titanyl phosphate, KT iOP O4, 500 [µm] 74 7.1.4 Germanium, 1 mm 75 7.2 Reﬂection spectroscopy 75 7.2.1 Potassium titanyl phosphate, KT iOP O4 75

8 Conclusion 76 8.1 Further work 77

A Appendix: Properties of the Wigner-Ville distribution 78

B Appendix: Geometrical interpretation of the Wigner-Ville distribution 81

– 1 – Populärvetenskaplig sammanfattning

Terahertz tidsdomän spektroskopi är en spektroskopi teknik där en Terahertz puls väx- elverkar med ett prov och sedan detekteras. Genom att jämföra spektrat från en puls som har växelverkat med ett prov och en referens puls som inte har växelverkat med ett prov kan slutsatser om de optiska egenskaperna hos provet bestämmas. Terahertz pulserna är mycket korta i tid de varar några få picosekunder. Eftersom pulserna är så snabba går det inte att detektera pulserna med vanliga fotodetektorer. Man använder därför en prob puls för att detektera de snabba Terahertzpulserna. Terahertzpulserna och probpulsen låts väx- elverka i en elektrooptisk kristall där Terahertzljuset vrider polarisationen på prob pulsen, sedan kan variationerna i prob pulsens polarisation kan detekteras. När de snabba Tera- hertz pulserna ska detekteras induceras störningar i signalen från detektionsanordningen. Det här bruset påverkar i slutändan den spektrograﬁska karakteriseringen alltså hur brytningsindexet för materialet varierar med våglängden på ljuset. Den här texten undersöker om det är möjligt att minska bruset som induceras av detektionsanordningen med signal behandling av Terahertz tidsdomän signalen. Den typ av signalbehandling som undersöks är så kallad tidsfrekvens representation. Tidsfrekvens representation visar spektral och temporal information av en signal. Genom att dämpa och förstärka de områden som är viktiga för karakterisering i tids-frekvensdomänen kan kvalitén på den spektroskopiska karakteriseringen förbättras.

Genom att representera en signal i tids-frekvens domän kan man se hur centralfrekvensen hos en puls varierar som funktion av tid. Detta kallas för chirp och är intressant då chirpet bär information om de fysikaliska processer som styr egenskaperna hos pulsen. Wigner-Ville transformen är den tids-frekvens representation med störst spektral upplösning av de tre undersökta tids-frekvens representationer och är därför lämpad för tidsfrekvens analys av en Terahertz puls.

Det är viktigt att förstå orsakerna till de spektrala egenskaperna hos THz pulserna eftersom detta kan leda till ny teknik och nya tillämpningar av THz spektroskopi. För att THz spektroskopi ska vara en så eﬀektiv karakteriserings teknik som möjligt är det viktigt att signal behandla Terahertz signalen så att upplösningen är så bra som möjligt. Terahertz spektroskopi är en spektroskopiteknik med syftet att bestämma det komplexa brytningsindexet för material för ljus med Teraherz frekvens.

– 2 – Commonly used abbreviations

• WVD - Wigner-Ville distribution

• THz-TDS - Terahertz time-domain spectroscopy

• CWT - Continuous wavelet transform

• STFT - Short-time Fourier transform

• KTP - Potassium titanyl phosphate

• GaP - Gallium Phosphide

• HRFZ - High resistivity ﬂoat zone

• FSEOS - Free space electro optic sampling

• EO crystal - Electro optic crystal

• NTNU - Norwegian University of Science and Technology

– 3 – 1 Introduction

1.1 Preface This master thesis was written at NTNU in Trondheim under the supervision of Professor Ulf Österberg and with the assistance of Mr. Kjell Mølster. The thesis is a master project in engineering physics at Uppsala University. The subject reader for the project is Professor Bengt Carlsson. Dr. Marcus Björk has assisted with advice on the signal processing contin- uously during the project. This input has been very important for the progress and success of the project. Mr. Kjell Mølster has continously assisted the author with the MATLAB scripts for the spectrographic characterization which are written by Mr. Mølster. This project was made possible by a NORDTEK scholarship.

1.2 Project Description The scope of this project is to improve the analysis method of the THz-TDS characterization. The group in Trondheim has previously computed the spectrographic characterization using zero-padding of the TDS spectums to obtain smooth characterizations. It is investigated if masking of time-frequency representations corresponding to the THz-TDS signal can produce smoother characterizations than zero-padding. Additionally this project aims to investigate the chirp of the THz-TDS main pulse.

1.3 Project Goals This project has two main goals they are as follows.

1. Analyse the THz-TDS signals in time-frequency representation

2. Use the time frequency representation of THz-TDS signals to reduce the eﬀects of distortion in the characterization induced by the detection scheme

1.4 Background Terahertz time-domain spectroscopy (THz-TDS) is a pump probe spectroscopy technique where a single cycle THz pulse, a pulse that alternates once between positive and negative values as a function of time, is measured using an electro optic crystal [17]. When the THz pulse travels through the EO crystal a small portion of the pulse is reflected inside the crystal producing a train of pulses with decreasing amplitudes. When the Fourier transform of the THz-TDS signal is computed the periodic pulses caused by the reflections in the EO crystal will distort the frequency domain. Previously, the deleterious effects from these have been adressed by using windowed Fourier transform around the main peak. By representing the signal in the time-frequency representation both temporal and spectral information of the signal become accessible. This text examines the possibilities of removal of the back-reflections in the TDS signal using time-frequency masking of time-frequency representations corresponding to the TDS signal. THz-TDS spectroscopy is a technique with many applications since THz radiation has resonances in phonon modes and molecular vibrations, intraband transitions in semiconductors and energy gaps in superconductors.

– 4 – THz-TDS can be used to diagnose skin diseases such as skin cancer and burns. Due to the THz resonances in molecular vibrations and especially in water vapour it is excellent for monitoring of the atmosphere. For the applications of THz-TDS to be eﬀective it is important that noise induced in the detection scheme are reduced as much as possible by signal processing. The purpose of this project is to reduce the detection scheme induced noise in the spectroscopic characterization. The data analysed in this text comes from measurements performed by Mølster, [1]. The data consists of two types of time-domains spectroscopy data, transmission and reﬂection spectroscopy. The transmission data investigated comes from transmission time-domain spectroscopy measurements for the following samples.

1. HRFZ silicon, 273 KB

2. Gallium Phosphide, 66 KB

3. Germanium, 86 KB

4. Potassium titanyl phosphate, KTP, 352 KB.

The reﬂection data comes from reﬂection time-domain spectroscopy measurements made on KTP, 15 KB. All measurements are made with a sapling frequency of 30 THz to avoid aliasing since the bandwidth of the signal is at max 15 THz.

2 Generation of broadband THz single cycle pulses

The section discusses how broadband THz single cycle pulses can be generated following the work of [1, 17]. There are three main ways to generate broadband THz single cycle pulses these are through a photoconductive antenna, optical rectiﬁcation and generation of plasma in air. The latter two utilise non-linear optical properties which require high electromagnetic ﬁelds in mediums.

2.1 Photoconductive antenna A photoconductive antenna is a semiconductor with two electrodes imprinted in the semiconductor generating a DC bias. For photon energies high enough an incident photon on the photoconductive antenna can excite an electron hole pair. Since the electron and electron hole is in a potential gradient generated by the DC bias they will be accelerated generating a current that in turn will emit electromagnetic radiation in form of THz radiation. In the far field, λ r, the THz radiation is given by µ0 sin θ E = ∂2 (p(t ))θˆ (2.1) T Hz 4π r tr r where t is retarded time, t = t r/c, p(t ) is the dipole moment of the source. The carrier r r − r density can be related to the far-field THz field by using the continuity equation.

j + ∂ ρ(r, t) = 0 (2.2) ∇ · t

– 5 – Rewriting this equation yields

∂tp(t) = WIPC (t) (2.3) where W is the spotsize and IPC is the current over the photoconductive antenna. It is now possible to insert equation (2.3) in equation (2.1) produces

µ0W sinθ E = ∂ (I (t ))θˆ ∂ (I (t )) (2.4) T Hz 4π r tr PC r ∼ tr PC r Hence the THz radiation is proportional to the time derivative of the current between the anode and the cathode. Figure1 shows the process of generation of broadband THz pulses

Figure 1: Figure showing the generation of THz pulses using a photoconductive antenna. Figure from [8]. with a photoconductive antenna.

2.2 Optical rectification A second method to generate THz pulses is optical rectification. The polarization vector in a dielectric material can be Taylor expanded around the electric susceptibility tensor and the electric field far from a resonance.

P = (χ E + χ2 E E ) + (E3) i 0 ij j ijk j k O For electric ﬁelds in the range from 105 to 108 V/m, the second order term dominates and the polarization can be truncated around the second order term.

P (t) χ2 E E i ≈ 0 ijk j k Assume that the electric field is a two frequency laser field with envelopes A(t) and B(t). The electric field can then be written on the form.

iωt iω0t Ei = Ai(t)e− + Bi(t)e− + c.c.

– 6 – Inserting this in the truncated Taylor expansion of the polarization will give one term that is not a function of either of the incident laser frequencies ω and ω0.

P (t) = A (t)B (t) + ... ⇒ i i i Accordingly for envelopes in the femtosecond range this will result in a THz varying polarization. Since the D-ﬁeld is proportional to the polarization THz radiation will be generated from the dielectric material.

D (t) P (t) i ∼ i Since this process utilises a term to second order in the Taylor expansion it is commonly referred to as a χ2 process.

2.3 Generation in plasma A third way to generate broadband THz single cycle pulses is by focusing a high power laser to such a point that it generates a plasma. The plasma in turn emits the THz radiation. This is the method that the group at Norwegian University of Science and Technology (NTNU) utilises. As the laser is focused down the electric field strength increases. As the electric field increases the Coulomb potential binding the electrons to the Nitrogen nuclei gets distorted. At a some point the influence of the external electromagnetic interaction is large enough for the electrons bound the Nitrogen nuclei to tunnel through the potential barrier generating a plasma.

U(r) U(r)

0 r 0 r

e−

Figure 2: The Coulomb potential with and without an external laser field. The solid curve in the left figure represents the electronic potential for the bound electrons to the Nitrogen nuclei. The dashed curve represents the induced electromagnetic potential of the laser field. The solid curve in the right figure represents the sum of the electronic potential and the external laser field.

Figur2 shows the Coulomb potential with a laser field present and without a laser field. When the laser field is present electrons have a finite probability to tunnel through the

– 7 – potential barrier induced by the external magnetic ﬁeld. As the laser intensity increases the tunnelling probability also increases to a point where dry air is ionized. In ﬁgure

Figure 3: The THz is generated by a plasma caused by by focusing down a high power laser. Figure from [8].

3 the plasma THz generation is shown. A high intensity laser ﬁeld from a mode-locked Ti:Sapphire laser is focused down in a lens. It then passes through a BBO crystal where it undergoes Second harmonic generation. At the focal point the energy is so high that dry air is ionized and a plasma is generated. This process is a χ3 process, i.e. a third order non-linear generation method. The plasma radiates in a spectrum where THz radiation is one component. The bandwidth of the plasma generation method is 15 THz therefore according to the Nyquist-Shannon sampling theorem to avoid aliasing the THz-TDS data must be sampled with a sampling frequency of at least 30 THz. This is the case for all THz-TDS measurements analyzed in this thesis.

2.4 Comparison of the three generation methods

The bandwidth for THz generation through ionization of air is the largest compared to optical rectiﬁcation and generation of THz with a photoconductive antenna. The bandwidth of generation through optical rectiﬁcation and with a photoconductive antenna both have a bandwidth of approximately 2 THz while the plasma generation generates THz up to 15 THz. The large bandwidth for the plasma method is one of its great advantages compared to the other generation techniques.

3 Terahertz time domain spectroscopy

Terahertz time domain spectroscopy is a pump probe spectroscopy technique. After the THz pulse is generated in the χ3 process in the plasma the THz pulse and the pump are separated with a spectral splitter. The THz pulse is then propagated to the sample where it interacts either in transmission or in reﬂection. The THz pulse is extremely localized in time. It has so short pulse duration that it is impossible to detect the pulse with an

– 8 – ordinary photoelectric detection scheme. A clever technique to detect the short THz pulse is needed. The group at NTNU uses free space electro optic sampling for this purpose.

3.1 The Pockel’s effect The following follows from [18] and [1]. The Pockel’s effect is closely related to optical rectification. As optical rectification is a χ2-process for generation of THz radiation the Pockel’s effect is a χ2-process that can be used for for detection of THz. The Pockel’s effect is similar to an inverse of optical rectification. The second order term of the Taylor expansion of the polarization vector in the i-th direction can be written as function of the susceptibility tensor and the electric field as follows.

(2) (2) Pi (ω) = 20χijk(ω, ω, 0)Ej(ω)Ek(0) (3.1) (2) = 0χij (ω)Ej(ω) (3.2)

(2) where, χij (ω), is the field induced susceptibility tensor. From equation (3.2) we can see that a static electric field induces different refractive index for different polarizations, i.e. birefringence. This means that the polarization of the laser pulse will change the polarization state of the probe. Figure4 shows how the THz pulse change the polarization state of

Figure 4: Schematic over the Pockel’s effect. The figure is from [18]. the probe. After the electrooptic crystal the THz have changed the linear polarization of the probe to an elliptical polarization. By decomposing the polarization into two orthogonal directions and measuring the difference of the intensities of the two directions give.

I = I I = I ∆φ (3.3) s y − x 0 where ∆φ is the diﬀerential phase retardation. Consider restricting the analysis to a speciﬁc class of electro optic crystals. The electro optic crystal used in the group at NTNU is

– 9 – Gallium Phosphide, GaP, crystals. GaP is an electro optic crystal in the crystal class 43m. The differential phase retardation ∆φ is then given by. ωL ωL ∆φ = (n n ) = n3 r E y − x c c O 41 T Hz nO is the refractive index of that the probe experiences inside the GaP crystal, r41 is the electro optic coefficient for the GaP. By inserting the differential phase retardation into equation (3.3) gives. ωL I = I n3 r E E s 0 c O 41 T Hz ∼ T Hz Hence by measuring the intensity in the two orthogonal directions of the probe a quantity proportional to the THz-field is detected.

3.2 Free space electro optic sampling Free space electro optic sampling, FSEOS, utilises the Pockel’s eﬀect which through a sequence of measurements are able to measure the amplitude and phase of an optical pulse [1]. FSEOS is a pump probe spectroscopy technique where a sampling probe is used to characterize the THz. In ﬁgure5 the measuring set up for FSEOS is shown. The THz pulse

Figure 5: Schematic over FSEOS. The ﬁgure is from [9]. is propagated and combined with the sampling probe in a beam combiner after which the pulse and the probe is let interact in an electrooptic medium. At NTNU a Gallium Phos- phide crystal is used for the electro optic medium, other alternatives are for example Zink telluride. The interaction between the pulse and the probe is describe by the Pockel’s eﬀect, basically what happens is that the pump through the interaction change the polarization state of the probe. The pulse and the probe is then propagated through a compensator. In the NTNU group a λ/4-plate is used as the compensator. Then the pulse and the probe is propagated through a Polarizing beamsplitter which separate the polarization of the pulse and the probe in two orthogonal polarzation states, the group at NTNU uses a Wollaston

– 10 – prism for this purpose. After the decomposition the intensity in each direction is measured by two balanced photodiodes. Now one point of the THz pulse have been mapped out. To map out the entire pulse the phase of the probe must be changed, this is done with a translation stage. The translation stage varies the optical path length of the probe and thereby the phase of the probe. The probe pulse period is shorter than the THz pulse meaning that the probe experiences a DC electrical field. The translation stage is varied in steps such that the sampling frequency of the THz field is 30 THz. This is to assure that the sampling theorem is fulfilled and no aliasing occurs since the THz has a bandwidth up to 15 THz. An important assumption for the validity of this type of measuring technique is that all THz single cycle pulses are identical, which of course is not the case, but they are similar enough for the technique to produce characterizations.

3.3 Transmission spectroscopy This section on spectrographic characterization for transmission spectroscopy follows the work of [1,4, 17]. The complex refractive index is

N = n + iκ where n is the real part of the refractive index and κ the imaginary part. κ is proportional to material absorption. The purpose of the spectroscopic characterization is to compute n and κ. The transfer function is deﬁned as the quotient of the Fourier transform of the reference and the sample signal. E (ν) H(ν) = s , 2πν = ω Er(ν)

Since the signals measured xsample and xreference are real the Fourier transforms of the reference and sample signals are complex.

Es,Er C ∈ Therefore the transfer function will also be a complex function of frequency. A complex function is always possible to write as an amplitude multiplied with a complex exponential of a phase.

H(ν) = H(ν) eiφ(ν) | | With the amplitude and phase of the transfer function given by.

E (ν) H(ν) = s | | E (ν) r

φ(ν) = Arg (H(ν))

For optically thick samples with low absorption the transfer function can be written as.

iφ(ν) κ(ν) 2πνd i(n(ν) n ) 2πνd H(ν) e = t t e− c e − air c | | 12 23

– 11 – The transmission coefficient into the sample t12 and out of the sample t23 are approximately real valued for low absorptive samples from this form we can determine the refractive index as φ(ν)c n(ν) = + n 2πνd air and solving for κ results in c (n(ν) + n )2 κ(ν) = ln air H(ν) −2πνd 4n(ν) | | where n(ν) is the real part of the complex refractive index, i.e. it determines the group and phase velocity for different frequency components in the material. κ(ν) is the imaginary part of the refractive index and gives a measure on the absorption in the material for different frequencies.

3.4 Reflection spectroscopy Reflection spectroscopy can characterize opaque materials since light is not needed to be transmitted through the sample. The reference signal is obtained by reflecting the radiation on a metal surface. The transfer function is then given by assuming normal incidence. E (ν) nˆ(ν) 1 n(ν) + iκ(ν) 1 H(ν) = sample = r(ν) eiφ(ν) = − = − Ereference(ν) | | nˆ(ν) + 1 n(ν) + iκ(ν) + 1

By separating the real and imaginary part of the transfer function expressions for n(ν) and κ(ν) can be obtained.

1 (1 r 2)2 1 n(ν) = − | | + 2 (1 + r 2 2 r cos(φ(ν)))2 2 s | | − | | 1 2 r sin φ(ν) 1 κ(ν) = 2| | √2 1 + r 2 r cos φ(ν) 1 1 | | − | | − 2n(ν)2 q From these formulas the materials can be characterized in reﬂection.

4 Mathematical formulation of signals

In this section the basic assumptions regarding the signals analysed in this text are presented following the litterature [4]. The relation between spectral and temporal resolution is presented as well as the Shannon-Nyquist sampling theorem which states the lowest possible sampling frequency to use when sampling a signal to avoid aliasing.

– 12 – 4.1 Complex analytic signals A measurement is quantum mechanically deﬁned as the action of a Hermitian operator, i.e. an observable, on a state in a Hilbert space. The eigenvalues of the observable will per deﬁnition be real. This means that all measurements that can be performed will generate a set of real values. Consider the observable A. It then holds that in ket space

A = A† (4.1)

A a0 = a0 a0 (4.2) | i | i and in bra space.

a00 A = a00 A† = a00∗ a00 (4.3) h | h | h | By multiplying equation (4.2) with a it can be shown that the eigenvalues of the Hermi- h 00| tian operator must be real.

a00 A a0 = a00 a0 a0 h | | i h | | i a00∗δ a0δ = 0 ⇒ a00a0 − a00a0 a0∗ = a0 a0 R ⇒ ⇒ ∈ This is a stationary description and time have not been included. To describe an evolving system Heisenbergs equation of motion for the system must be solved which means that the Hamiltonian for the system must be known. If the Hamiltonian of the system is known the observable can be evolved in the Heisenberg picture as follows using the time evolution O operator.

i t dt0H(t0) (t) = e− ~ t0 U R (t) = †(t) (0) (t) O U O U where (0) is the initial condition of the operator . Where the state is held stationary at O O the initial condition, α = α(0) . A sequence of actions with the time evolving observable | i | i (t) in the Heisenberg picture on a state α in the Hilbert space will result in the real O | i H signal x(t), where x(t) are the eigenvalues of the observable (t) at the time instances t. O 4.2 Deﬁnition and basic properties of analytic signals Complex analytic signals are an extension of the real signals to the complex plane. This extension allows for techniques in complex analysis to be applied on the signals. For the real function x(t) as a function of time

x(t) R, t R (4.4) ∈ ∈ The signal is said to be square integrable if the L2- norm of the signal is bounded from above.

x2(t)dt < (4.5) ∞ ZR

– 13 – This is equivalent to the statement that x(t) is in the L2(R) space. 2 x(t) L (R) (4.6) ∈ Suppose that the signal x(t) is square integrable. It is possible to write the signal in Fourier representation as follows.

2πiνt x(t) = dν X(ν)e− (4.7) ZR where X(ν) is the Fourier transform of the signal x(t).

X(ν) = dt x(t)e2πiνt (4.8) ZR Since the signal x(t) is real, the Fourier transform of the signal obeys the reality condition.

X( ν) = X∗(ν) (4.9) − This condition state that the frequency distribution is symmetric around the axis v = 0. Or that no additional information is located in the negative frequencies. It is therefore possible to deﬁne a new signal z(t) that suppresses the negative frequency components, since they do not carry information.

2πiνt z(t) = dν Z(ν)e− (4.10) ZR where the spectrum of the new signal is given by.

X(ν), ν 0 Z(ν) = ≥ (4.11) (0, ν < 0 z(t) is the complex analytical signal corresponding to the real signal x(t). Analyticity means that the function z(t) satisﬁes the Cauchy-Riemann equations. Consider again the assumption that the L2 norm of the signal x(t) is ﬁnite.

dt x∗(t)x(t) < (4.12) ∞ ZR where

2πiνt 2πiνt x(t) = dν X(ν)e− , x∗(t) = dν X∗(ν)e (4.13) ZR ZR Inserting (4.13) in (4.12) gives

2πiνt 2πiν t > dt dνdν0 X(ν)e− X∗(ν0)e 0 (4.14) ∞ 2 ZR ZR = dνdν0 X(ν)X∗(ν0)δ(ν ν0) (4.15) 2 − ZR = dν X(ν)X∗(ν) (4.16) ZR Hence if x(t) is a square integrable function the spectrum of x(t) is also in L2(R) 2 X(ν) L (R) (4.17) ∈

– 14 – Theorem 4.1 (Plancherel). If a signal, x(t), is in L2(R) then the Fourier transform X(ν) is also in L2(R). The Fourier transform is an isometry with respect to the L2-norm. It then trivially follows that the complex analytic signal z(t) of x(t) is also in L2(R). Theorem 4.2. If a signal, x(t), is in L2(R) then the complex analytic signal, z(t), of x(t) is also in L2(R) The signal x(t) can be rewritten using (4.13) as follows 0 2πiνt 2πiνt ∞ 2πiνt x(t) = dν X(ν)e− = dν X(ν)e− + dν X(ν)e− R 0 Z Z−∞ Z ∞ 2πiνt ∞ 2πiνt = dν X∗(ν)e + dν X(ν)e− Z0 Z0 ∞ 2πiνt 2πiνt = dν X(ν)e− + X(ν)e− ∗ 0 Z ∞ 2πiνt = dν 2Re X(ν)e− Z0 = ∞ dν a(ν) cos(φ(ν) 2πνt) − Z0 where

iφ(ν) a(ν)e = 2Z(ν), a(ν) R, φ(ν) R ∈ ∈ From eq. (4.10) it can be deduced that the complex analytic signal z(t) can be written

1 i(φ(ν) 2πνt) z(t) = dν a(ν)e − 2 Z On this form it can be seen that the complex analytic signal is a generalization of x(t). x(t) can be retrieved by taking the sum of the complex analytic signal and the complex conjugate of the complex analytic signal.

x(t) = z(t) + z∗(t) = 2Re(z(t)) Since the measured signal is the real part of the complex analytics signal we can denote the imaginary part of z(t) as y(t) 1 z(t) = (x(t) + iy(t)) 2 The analyticity of z(t) implies that x(t) and y(t) are a Hilbert transform conjugate pair which means that.

1 x(t0) y(t) = PV dt0 π t t ZR 0 − 1 y(t0) x(t) = PV dt0 −π t t ZR 0 − where PV is the principal value at t0 = t. This means that to analytically continue the measured real signal x(t) to the complex analytic signal z(t) all you have to do is to compute the Hilbert transform of the measured signal and take the sum z(t) = x(t) + iH(x(t)).

– 15 – 4.3 The Heisenberg-Gabor inequality The Heisenberg Gabor inequality limit to how accurately it is possible to resolve a set of two canonically conjugate variables. Consider the Cauchy-Schwarz inequality. For the states ψ and φ in the Hilbert spaces and . | i | i H H0

ψ , φ 0 | i ∈ H | i ∈ H The Cauchy-Schwarz inequality is then given by.

ψ φ φ ψ ψ ψ φ φ h | i h | i ≤ h | i h | i Consider a momentum and a position eigenstate.

p0 , x0 0 | i ∈ H | i ∈ H For which it holds

p p0 = p0 p0 | i | i x x0 = x0 x0 | i | i The momentum and position operators are canonical conjugate variables and satisfy the usual commutator relation.

[x, p] = i~

Consider the following shifted momentum and position operators.

P = p p ,X = x x − h i − h i The product of these operators can be decomposed in the real and imaginary part of the operator as follows. 1 1 PQ = P,Q + [P,Q] 2{ } 2 The momentum and position operator are both observables meaning that they are Hermi- tian operators satisfying.

† = O O By taking the Hermitian conjugate of the product of P and Q the real and imaginary part can be identiﬁed. 1 1 (PQ)† = P,Q [P,Q] 2{ } − 2 The Cauchy-Schwarz inequality can be rewritten for operators by acting on the empty state with the operators P and Q.

ψ = Q 0 , φ = P 0 | i | i | i | i

– 16 – The Cauchy-Schwarz inequality then becomes P 2 Q2 PQ 2 |h i||h i| ≥ |h i| The right hand side can be expressed as the sum of the squares of the real and imaginary part. 1 1 = P,Q 2 + [P,Q] 2 4|h{ }i| 4|h i| Since a square is positive definite this is bounded from below by. 1 [P,Q] 2 ≥ 4|h i| 1 = [p, x] + [ p , x ] [ p , x] [ p , x ] 2 4|h h i h i − h i − h i i i| 1 ~2 = [p, x] 2 = 4|h i| 4 P 2 and Q2 are the square of the uncertainty in the momentum and position operators. σ P 2 , σ Q2 p ≡ |h i| x ≡ |h i| In total this yields p p ~ σ σ , Q.E.D. x p ≥ 2 In quantum mechanics all canonically conjugate operators fulfil the Heisenberg Gabor uncertainty relation. The Heisenberg uncertainty relation is a consequence of the wave like nature of matter. The fact that matter is described by wave-functions implicitly lead to the Heisenberg principle from the fact that the momentum and position operators are related via the Fourier transform. From the Pontryagin duality two operators that are canonical conjugates fulfil an uncertainty relation. Consider a signal in time x(t). The Fourier transform is then given by. 1 X(ω) = x(t)eiωtdt 2π ZR In reality it is impossible to store an infinite time domain signal, the signal is always in the context of a finite time interval that can be denoted as I. This restriction in time will, in turn, limit the spectral band. Intuitively this can be explained as follows. Imagine that you listen to a monochromatic audio signal for a time t. If you hear the signal for a very short period of time the signal will sound like a clap, an Dirac excitation in time and a Heaviside in frequency. As you increase the time the signal is sampled the confidence in the frequency composition will increase and converge in frequency as a Dirac delta and in time as a Heaviside function. The uncertainty relation for square integrable functions can be derived as follows [12]. The spread in time and frequency is given by.

1 1 2 2 2 2 2 σt = dt (t t ) x(t) = dt (t t )x(t) R − h i | | R | − h i | Z 1 Z 1 2 2 σ = dν (ν ν )2 X(ν) 2 = dν (ν ν )X(ν) 2 ν − h i | | | − h i | ZR ZR

– 17 – where I for simplicity have assumed the normalization.

dt x(t) 2 = 1 | | Z Assume furthermore that.

ν = t = 0 h i h i Then the time and frequency spread can be written as.

1 2 2 σt = dt tx(t) R | | Z 1 2 σ = dν νX(ν) 2 ν | | ZR Using the duality property of the Fourier transform the spread in frequency can be rewritten to. 1 ∂ x(t) 2 2 σ = dt t ν 2π Z !

This means that

1 σ = tx(t) , σ = (2π)− ∂ x(t) t || ||2 ν || t ||2 Consider again the Cauchy-Schwarz inequality however this time for a time signal and the corresponding spectrum assuming that, x(t) L2(R) and by the Plancherel theorem ∈ X(ν) L2(R). ∈ ψ(t) ψ(t) φ(t) φ(t) ψ(t) φ(t) φ(t) ψ(t) h | i h | i ≥ h | i h | i With the identiﬁcation

1 ψ(t) = tx(t), φ(t) = (2π)− ∂tx(t)

The Cauchy-Schwarz inequality becomes

2 2 2 1 2 1 2 σ σ = tx(t) (2π)− ∂ x(t) tx(t)(2π)− ∂ x∗(t) t ν || ||2|| t ||2 ≥ || t ||2 2 1 = dt tx(t)(2π)− ∂tx∗(t) Z 2 1 Re dt tx(t)(2π)− ∂tx∗(t ) ≥ Z 2 2 t t 2 = dt (x(t)∂ tx∗(t) + x∗(t)∂tx(t)) = dt ∂t( x(t) ) 4π 4π | | Z Z

Since the real part of a complex number is given by. 1 Re(z) = (z + z∗) 2

– 18 – By partial integration this can be further evaluated.

t 2 dt ∂ ( x(t) 2) 4π t | | Z 2 1 ∞ = t x(t) 2 dt x(t) 2 16π2 | | − | | Z −∞

The ﬁrst term is zero from since x( t) L2( R). The second term is unity from the normal- ∈ ization. 1 = 16π2 1 σ σ ⇒ t ν ≥ 4π

Hence the smallest possible time-bandwidth product for square integrable functions is 1/4π which is the case for Gaussian wave forms.

4.4 The Shannon-Nyquist sampling theorem

A reason for distortion in signal processing is aliasing. Aliasing is described through the Nyquist-Shannon sampling theorem. The theorem was discovered by Shannon and presented in the paper [10]. The derivation below follows Shannons original proof.

Theorem 4.3. The Shannon-Nyquist sampling theorem If a function f(t) contains no frequencies higher than B [Hz] it is completely determined by giving its ordinates at a series of points spaced (1/2B)[s] apart.

Suppose that the signal f(t) fulﬁls the Dirichlet conditions and can be expanded in its Fourier integral.

2πB 1 iωt 1 iωt f(t) = dω F (ω)e− = dω F (ω)e− 2π R 2π 2πB Z Z− Consider sampling the signal at the following equally spaced time instants. n t = n 2B

The Fourier series of the spectrum is given by.

iωn iωtn F (ω) = cne = cne 2B n Z n Z X∈ X∈ where the coeﬃcients cn are given by the ordinary Fourier integral.

2πB 1 iωn 1 n cn = dω F (ω)e− 2B = f 4πB 2πB 2B 2B Z−

– 19 – Inserting the Fourier series expansion of the signal in the Fourier integral of the signal gives.

2πB 1 1 n iωn iωt f(t) = dω f e 2B e− 2π 2πB 2B 2B n Z Z− X∈ 2πB 1 n iω( n t) = f dω e− 2B − 4πB 2B 2πB n Z X∈ Z− n sin(π(n 2Bt)) = f − , Q.E.D. 2B π(n 2Bt) n Z X∈ − The sampling theorem is important in time-frequency analysis where aliasing due to un- dersampling must be taken into consideration when analysing distortion in spectrograms. The fact that a signal and the spectrum of a signal are canonically conjugate variables imprints a symmetry on the Shannon-Nyquist theorem. The frequency domain version of the theorem is stated as follows.

Theorem 4.4. The Shannon-Nyquist sampling theorem, (frequency domain) For F (ω) the Fourier transform of the signal f(t) which is only non-zero for t > T the | | spectrum can be completely determined by. nπ sin(T ω nπ) F (ω) = F − (4.18) T T ω nπ n Z X∈ − This is shown analogously to the time domain version of the theorem. This version of the theorem will be shown to be of great importance for this thesis since to increase the spectral resolution of a spectrum i.e. the spacing between the ordinates in the sum (4.18) the time integration domain must be increased.

4.5 Instantaneous frequency The instantaneous frequency of a pulse can be expressed in terms of the phase of the analytic signal corresponding to the real measured signal.

x(t) = A(t)eiφ(t) (4.19)

The instantanous frequency is then given by.

ωi(t) = ωc + ∂tφ(t) (4.20) here ωc is the central frequency. The central frequency is deﬁned as the centroid of the spectrum.

5 Time-frequency representation

Almost all physical processes in nature can be described as the evolution of an energy density with time. Examples of this is music, dissipation and dispersion. In a time-frequency representation both spectral and temporal information is accessible for a signal. This opens

– 20 – up the possibility of analysing how the instantaneous frequency of a signal varies with time. The variation of the instantaneous frequency of a signal as a function of time is called chirp. The chirp may provide information about the physical processes that control the wave packet. There are diﬀerent examples of time-frequency representations. In this text, short-time Fourier transform, the Wigner-Ville transform and the Continuous wavelet transform are examined. In this section the basics of the three time-frequency representations comes from [11].

In quantum mechanics a phase space representation was sought by Wigner and others [13] since a phase space representation can show two canonical variables simultaneously, however this was not so trivial. Since a particle quantum mechanically through the Heisen- berg relation cannot have well defined position and momentum simultaneously it is not possible to define the probability that a particle has q position and p momentum simultaneously which in turn means that it is not possible to define a phase space probability distribution P (p, q). It turns out however that it is possible to define a quasi-probability distribution that can solve this issue. Quantum mechanical effects such as entanglement that has no classical analogy will be represented in the quasi-probabability distribution as negative probabilities. Quasi-probability distribution functions represents a way to describe quantum mechanical averages on a form that is similar to classical averages. Consider the classical function A, the expectation value in a classical phase space representation is then

A = dqdp A(p, q)P (p, q) h iCl Cl Z where ρ(p, q) is a phase-space distribution and A(p, q) is the value of the function at a speciﬁed phase space point. The corresponding expectation value of a quantum mechanical observable is given by.

Aˆ = T r(Aˆρˆ) h iQ where ρˆ is the density matrix of the system and T r is the trace. It turns out that it is then possible to write the expectation value of the quantum mechanical observable as follows.

Aˆ = dqdp A(p, q)P (p, q) h iQ Q Z where PQ(p, q) is a quasi-probability distribution, a distribution that can assume both positive and negative values and normalized to one. The correspondence between the operator Aˆ and the phase-space function A(p, q) is as follows. z z A(q, p) = dz eipz/~ q Aˆ q + h − 2| | 2i Z In time-frequency representation the pair of canonically conjugate operators is interchanged from position and momentum to time and frequency giving the expectation value of a operator as follows.

Gˆ = dt (Gxˆ )(t)x∗(t) = dtdν G(t, ν)ρ (t, ν) h ix x Z Z

– 21 – The function G(t, ν) is the associated function to the operator Gˆ. where the time and frequency observables are deﬁned as follows. i (Tˆ x)(t) tx(t), (Fˆ x)(t) − ∂ x(t) ≡ ≡ 2π t The canonically conjugate operators satisfy the following commutator relation. i [T,ˆ Fˆ] = 2π A time-frequency representation can therefore be seen as method to evaluate expectation values of operators. To transgress from a quantum mechanical Wigner-function to the Wigner-Ville distribution for a classical signal the following identiﬁcation is useful.

x t, p f 7→ 7→ i [x, p] = i~ [t, f] = 7→ 2π 1 ~ = ⇒ 2π Hence to get the Wigner-Ville distribution from the Wigner-function in literature on quantum mechanics simply interchange position and momentum with time and frequency and replace ~ with the 1/(2π). This is exactly the correspondence principle of quantum mechanics. In the limit h 1 the quantum system should become classical. By having a → simple way to switch between the classical and the quantum picture results made for the quantum theory can be used in signal processing.

5.1 The short-time Fourier transform This section discusses the short-time Fourier transform following the book [11]. The short- time Fourier transform is the most straight forward time-frequency representation. It is computed by taking the windowed Fourier transform around several time instances and combining the spectrums to resolve changes in the spectrum time. More precisely it is deﬁned as follows.

h 2πiνu STFT (t, ν) = du x(u)h∗(u t)e− x − ZR where h(t) is the window function in time. In this text the Hamming window is chosen as window function. The spectrogram is then given by.

Sh(t, ν) = STFT h(t, ν) 2 x | x | The total energy of the signal is then obtained as the double integral over the spectrogram.

h Ex = dtdν Sx (t, ν) 2 ZR provided that the window function is normalized.

dt h(t) 2 = 1 | | ZR

– 22 – The short-time Fourier transform is the most widely used time-frequency representation due to its simplicity in interpreting the spectrogram and lacking of cross term. Cross term interference is an inherit problem in quadratic time-frequency representations, since the short-time Fourier transform is a linear time-frequency representation it is free of this problem. A great advantage with the short-time Fourier transform time frequency representation is the relationship between spectral and temporal resolution. This can be seen by writing the signal in Fourier representation and inserting it into the STFT.

h 2πiνu STFT (t, ν) = du x(u)h∗(u t)e− x − ZR 2πi(tk tν) = dk X(k)H∗(k ν)e − − ZR where H (k ν) acts as a bandpass ﬁlter in the frequency domain. The time-bandwidth ∗ − product for the STFT is bound from below according to the Heisenberg-Gabor inequality. 1 σ σ t ν ≥ 4π By increasing the duration of the window function h(t) the spectral resolution can increase. By decreasing the duration of the window function the spectral resolution will decrease and the temporal resolution will increase. This direct relationship between spectral and temporal resolution is useful when analyzing the corresponding spectrograms computed by the STFT. A disadvantage with the STFT is that it is not possible to recover the marginal.

Sh(t, ν)dν = du x(u) 2 h(u t) 2 = x(t) 2 x | | | − | 6 | | ZR ZR Sh(t, ν)dt = dk X(k) 2 H(k ν) 2 = X(ν) 2 x | | | − | 6 | | ZR ZR The quantity obtained by integration over the spectrogram is a smeared version of the marginal.

5.2 Cohen’s class of Quadratic time-frequency representations The following follows from [11]. Cohen’s class, of quadratic time-frequency representations, has an advantage over linear time-frequency representations, such as the short-time Fourier transform and the continuous wavelet transform, in that the marginals can be recovered by integration over the time-frequency representation and that the time-frequency representation is always in R. The general form of a quadratic time-frequency representation is.

TFRx(t, ν) = dt1dt2 K(t1, t2; t, ν)x(t1)x∗(t2) (5.1) 2 ZR τ τ = dtdν (t, ν; ξ, τ) x ξ + x∗ ξ (5.2) R2 K 2 − 2 Z where τ τ (t, ν; ξ, τ) K ξ + , ξ ; t, f (5.3) K ≡ 2 − 2

– 23 – We want the kernel (t, ν; ξ, τ) to satisfy certain convenient properties. These properties K will constrain the kernel and generate the Cohen class. A convenient property of the time-frequency representation would be that if the time domain signal is shifted the time- frequency representation is also time-shifted accordingly, i.e.

y(y) = x(t t ) (5.4) − 0 TFR (t, ν) = TFR (t t , ν) (5.5) y x − 0 By inserting the deﬁnition of the quadratic time-frequency representation into equation (5.5) the ﬁrst constraint of the kernel for the Cohen class is obtained.

t, ν, t, t , ξ, τ, (t, ν; ξ, τ) = (t t , ν; ξ, τ) (5.6) ∀ 0 K K − 0 This constraint can be accounted for by setting ξ = 0 and t0 = ξ. The time-frequency representation the becomes. τ τ TFRx(t, ν) = dtdν k (t ξ, ν, τ) x ξ + x∗ ξ (5.7) R2 − 2 − 2 Z k (t, ν, τ) (t, ν; 0, τ) (5.8) ≡ K Another reasonable constraint would be that if the signal is multiplied with a complex exponential with frequency ν0 the time-frequency representation gets shifted by ν0. This basically imposes a linear frequency axis of the time-frequency representation.

y(t) = x(t)ei2πν0t (5.9) TFR (t, ν) = TFR (t, ν ν ) (5.10) y x − 0 Inserting the constraint time-frequency representation into this expression yields.

t, ν, τ, ν , k(t, ν, τ)ei2πν0τ = k(t, ν ν , τ) (5.11) ∀ 0 − 0 This constraint can be absorbed by letting ν = 0 and ν = ν, the kernel of the time- 0 − frequency representation then becomes.

i2πντ k(t, ν, τ) = k(t, 0, τ)e− (5.12)

It follows that the time-frequency representation is

τ τ i2πντ TFRx(t, ν) = Cx(t, ν) = dτdv φt d(t v, τ)x v + x∗ v e− (5.13) R2 − − 2 − 2 Z where

φt d(t, τ) k(t, 0, τ) = (t, 0; 0, τ) (5.14) − ≡ K From these two constraints we have now arrived at the set called Cohen’s class. The autocorrelation is an interesting function in statistics and signal processing. It gives a measure on the correlation between a signal and the time translated signal. It can for example be used to identify repeating patterns in time domain signals. The autocorrelation of the signal is deﬁned as. τ τ Rx(t, τ) dv φt d(t v, τ)x v + x∗ v (5.15) ≡ R − − 2 − 2 Z

– 24 – If the Fourier transform of the instantaneous autocorrelation function is computed with respect to τ we, again, arrive at Cohen’s class.

i2πντ Cx(t, ν) = dτ rx(t, τ)e− (5.16) ZR τ τ i2πντ = dτdv φt d(t v, τ)x v + x∗ v e− (5.17) R2 − − 2 − 2 Z From the autocorrelation function two other functions can be derived by taking a series of Fourier and inverse Fourier transforms. If the Fourier transform of the auto-correlation function with respect to τ is computed the Wigner-Ville distribution function is obtained. If the Fourier transform of the auto-correlation function with respect to t is computed the ambiguity function is obtained. The functions obtained from these operations will in turn be useful for deﬁning two other representations of Cohen’s class. Figure6 summarizes the

Figure 6: The connection between the Wigner-Ville distribution, the autocorrelation function and the ambiguity function. F indicates the Fourier transform, Ax(τ, ξ) the ambiguity function, Rx(t, τ) the auto-correlation function, and Wx(v, ) the Wigner-Ville distribution function. correspondence between the Wigner-Ville distribution, the autocorrelation function and the ambiguity function. A second way to obtain Cohen’s class is to consider the ambiguity function. The ambiguity function is deﬁned as.

τ τ i2πξt A (τ, ξ) dt x t + x∗ t e− (5.18) x ≡ 2 − 2 Z According to ﬁgure6 by taking the inverse Fourier transform with respect to the variable ξ of equation (5.13) we get back to the auto-correlation function. Inserting this in (5.17) a second representation of Cohen’s class is found.

i2π(tξ ντ) Cx(t, ν) = dξdτ φd D(τ, ξ)Ax(τ, ξ)e − (5.19) − Z where

i2πξt φd D(τ, ξ) dt φt d(t, τ)e− (5.20) − ≡ − Z

– 25 – The third and ﬁnal representation of Cohen’s class is found by taking the inverse Fourier transform of the Wigner-Ville distribution function with respect to λ and inserting the autocorrelation function in (5.17).

Cx(t, ν) = dvdλ φt ν(t v, ν λ)Wx(v, λ) (5.21) − − − Z where the Wigner-Ville distribution is deﬁned as.

τ τ i2πλτ W (v, λ) dτ x v + x∗ v e− (5.22) x ≡ 2 − 2 Z And the kernel

i2πντ φt ν(t, ν) dτ φt d(t, τ)e− (5.23) − ≡ − Z In this thesis the third representation of Cohen’s class, the Wigner-Ville distribution, will be studied due to it’s properties in time-frequency analysis.

5.3 The Wigner-Ville time-frequency representation The Wigner-Ville time-frequency representation is the most famous example of Cohen’s class of quadratic time-frequency representations. It is found by letting the kernel in equation (5.21) become.

φt ν(t v, ν λ) = δ(t v)δ(ν λ) − − − − − It is a central element of the Cohen class, in fact all other elements of the Cohen class can be viewed as the Wigner-Ville distribution smoothed in time and frequency.

5.3.1 Computation of the Wigner quasi-probability distribution The following follows from [2]. The Wigner-Ville transform of a signal x(t) is deﬁned as.

∞ 2πiντ W (t, ν) = x(t + τ/2)x∗(t τ/2)e− dτ x − Z−∞ ∞ 2πiξt W (t, ν) = X(ν + ξ/2)X∗(ν ξ/2)e dξ x − Z−∞ where X(ν) denotes the spectrum of the signal. All signals that can be measured are real.

x(t) R ∈ With the spectrum of the signal deﬁned as. 1 X(ν) = dt x(t)e2πiνt 2π ZR Since the integration is over the real line the spectrum have negative frequency components.

X(ν) C, ν R ∈ ∈

– 26 – From the reality condition

X( ν) = X∗(ν) − The negative frequencies carry no information and spectrum is symmetric around the zero frequency. However the negative frequency components will interfere with the positive frequency components through the cross term of the Wigner-Ville transform. This interaction will distort the spectrogram. To get rid of the negative frequencies the Hilbert transform is applied on the signal to construct the Hilbert conjugate to the real measured signal.

1 x(t0) y(t) = H(x(t)) = PV dt0 π t t ZR 0 − The complex analytic signal to the real measured signal can then be constructed. 1 z(t) = (x(t) + iy(t)) 2 And inserted into the Wigner-Ville distribution

∞ 2πiντ W (t, ν) = z(t + τ/2)z∗(t τ/2)e− dτ z − Z−∞ With this method all Wigner-Ville distributions in this report have been computed to reduce unnecessary mathematical interference.

5.3.2 The Wigner-Ville distribution for speciﬁc input signals In this section the Wigner-Ville time-frequency representation is computed for several different simple input signals to get an understanding on how the Wigner-Ville distribution behaves. The Wigner-Ville distribution’s in this thesis have been computed using the time- frequency toolbox in MATLAB. Consider ﬁrst the input signal as a Dirac delta function in time. Then the Wigner-Ville distribution also becomes a Dirac delta function which can be seen by the following computation. x(t) = δ(t)

i2πντ W (t, ν) = dτ δ(t + τ/2)δ(t τ/2)e− x − ZR i2πντ = dτ δ(2t + τ)δ(2t τ)e− − ZR 4πiνt = δ(4t)e− = δ(t) In the Wigner-Ville distribution this would be represented as a longitudinal line centered around t = 0. Figure7 shows the Wigner-Ville distribution for an Dirac delta function in time. The distortion around the peak intensity of the spectrogram is induced by the time windowing around the main pulse. If the input signal is instead a complex exponential the Wigner-Ville distribution becomes. x(t) = ei2πkt

ik(t+ τ ) ik(t τ ) i2πντ Wx(t, ν) = dτ e 2 e− − 2 e− ZR i2πτ(ν k) = dτ e− − = δ(ν k) − ZR

– 27 – 5 5

4.5 4.5 0.18

4 4 0.16

3.5 3.5 0.14

3 3 0.12

2.5 2.5 0.1

2 2 0.08 Frequency [THz] 1.5 1.5 0.06

1 1 0.04

0.5 0.5 0.02

0 0 0 1 -5 0 5 Spectral Amplitude [a.u.] 1

0.5

0 -5 0 5 Amplitude [V/m] Time [ps]

Figure 7: Wigner-Ville distribution for x(t) = δ(t). In the bottom of the ﬁgure the signal is shown and on the left of the ﬁgure the spectrum of the signal is shown. In the middle the Wigner-Ville distribution is shown. Henceforth this combination of Wigner-Ville distribution plot, spectrum and signal will be referred to as the Wigner-Ville distribution.

In the Wigner-Ville distribution this would be represented as an horizontal line centered around k [Hz]. This can be seen in ﬁgure8. For the analysis of the intrinsic chirp of the

0.7 0.7 15

0.6 0.6

0.5 0.5 10

0.4 0.4

0.3 0.3

Frequency [THz] 5 0.2 0.2

0.1 0.1

0 0 0 100 -400 -300 -200 -100 0 100 200 300 400 Spectral Amplitude [a.u.] 0.5 0 -0.5

-400 -300 -200 -100 0 100 200 300 400 Amplitude [V/m] Time [ps]

Figure 8: Wigner-Ville distribution of sinusoidal signal with frequency 0.17 THz. The distortion is caused by the windowing of the signal.

– 28 – THz main pulses it is interesting to see how the Wigner-Ville distribution represents chirps. An ideal linearly chirped signal can be written as follows. 2 x(t) = ei2πkt With linearly chirped means that in the Wigner-Ville distribution the average frequency as a function of time of the signal is a line. The Wigner-Ville distribution of the linearly chirp input signal can be computed as follows.

i2πk(t+τ/2)2 i2πk(t τ/2)2 i2πντ Wx(t, ν) = dτ e e− − e− ZR i2πτ(2kt ν) = dτ e − = δ(2kt ν) − ZR Hence in the Wigner-Ville distribtution this would look like a line with slope 2k as can be seen in ﬁgure9 If the input signal is a constant, c, the Wigner-Ville distribution would look

5 5 180 4.5 4.5 160 4 4 140 3.5 3.5 120 3 3 100 2.5 2.5 80 2 2 Frequency [THz] 60 1.5 1.5

1 1 40

0.5 0.5 20

0 0 0 100 0 2 4 6 8 10 Spectral Amplitude [a.u.] 0.5 0 -0.5

0 2 4 6 8 10 Amplitude [V/m] Time [ps]

Figure 9: Wigner-Ville distribution of sinusoidal signal with linear chirp. The constant k is chosen to, k = 1/2π THz. as follows. x(t) = c, c C ∈ i2πντ 2 W (t, ν) = dτ cc†e− = c δ(ν) = δ(ν) x | | ZR In the Wigner-Ville distribution this would look like a horizontal line around 0 [Hz]. An- other important speciﬁc function is the Boxcar function. For all practical purposes the TDS signals are limited in the time domain. This means that there will be an induced ringing caused by the limitation in the time domain. The rectangular function is expressed as the diﬀerence of two shifted Heaviside functions. R(t, T ) = H(t + T/2) H(t T/2) − −

– 29 – The Wigner-Ville distribution becomes.

τ τ i2πντ W (t, ν) = dτ R t + ,T R t ,T e− x 2 − 2 ZR 1 = sin(2πν(T 2 t ))(H(t + T/2) H(t T/2)) πν − | | − − Since all signals that are considered in this project are limited in time they can be factored

12 12 30

10 10 25

8 8 20

6 6 15 Frequency [THz] 4 4 10

2 2 5

0 0 0 100 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Spectral Amplitude [a.u.] 1

0.5

0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Amplitude [V/m] Time [ps]

Figure 10: Wigner-Ville distribution of the Boxcar function. The period is chose to T = 2 ps. on the form.

y(t) = x(t)w(t, T ) where w(t, T ) is a window function with window length T . From the multiplication property 9 this means that the Wigner-Ville distribution can be factored as follows.

W (t, ν) = W (t, ρ)W (t, ν ρ)dρ y x w − ZR Inserting the Wigner-Ville distribution of the window function gives.

T 1 2 sin(2π(ν ρ)(T 2 t )) = Wx(t, ρ) − − | | dρ π T (ν ρ) Z− 2 − As the signals examined in this thesis are similar to impulse trains it is interesting to study how the the Wigner-Ville distribution looks for two time shifted delta functions. If the input function is on the form.

x(t) = δ(t t ) + δ(t t ) − 1 − 2

– 30 – 10 7 5 5

4.5 4.5 2.5

4 4

3.5 3.5 2

3 3 1.5 2.5 2.5

2 2

Frequency [THz] 1 1.5 1.5

1 1 0.5 0.5 0.5

0 0 0 104 -10 -5 0 5 10 Spectral Amplitude [a.u.] 10000

5000

0 -10 -5 0 5 10 Amplitude [V/m] Time [ps]

Figure 11: The Wigner-Ville distribution of an impulse train signal. The separation of the impulses is 10 ps.

Figure 11 shows the Wigner-Ville distribution of an impulse train. Between the two impulses there is a clear interference pattern caused by the cross terms of the Wigner-Ville distribution. Explicitly this cross term can be seen by inserting the signal in the deﬁnition of the Wigner-Ville distribution.

∞ 2πiντ W (t, ν) = x(t + τ/2)x∗(t τ/2)e− dτ x − Z−∞ ∞ τ τ τ τ 2πiντ = δ t + t + δ t + t δ t t + δ t t e− dτ 2 − 1 2 − 2 − 2 − 1 − 2 − 2 Z−∞ 2πiν(2t1 2t) 2πiν(2t2 2t) = δ(t t )e− − + δ(t t )e− − − 1 − 2 t1 + t2 2πiν(2t1 2t) 2πiν(2t2 2t) +δ t e− − + e− − − 2 The ﬁrst two terms correspond to the solid geometries in the spectrogram. The last term represent the interference peak in the spectrogram since it is a superposition of two frequencies ν1 and ν2. When t = t1 we are at the ﬁrst pulse. The Wigner-Ville distribution becomes.

W (t1, ν) = δ(0)

This represents the solid geometry at the ﬁrst pulse in the spectrogram. When t = t2

W (t2, ν) = δ(0)

– 31 – This represents the solid geometry at the second pulse in the spectrogram. Between the t1+t2 ﬁrst and the main pulse at t = 2 the spectrogram becomes.

t1 + t2 2πiν(t1 t2) 2πiν(t2 t1) W , ν = δ(0)(e− − + e− − ) 2 2πiν(t1 t2) 2πiν(t1 t2) = δ(0)(e− − + e − ) = 2δ(0) cos (2πν(t t )) 1 − 2 Now it can clearly be seen that the cause of the interference peak is the cross terms in the Wigner-Ville transform. The Dirac delta function is singular so according to this expression there would be no interference pattern. The Dirac function should be replaced with a bounded value since all physical pulses have a ﬁnite duration. Cross term induced interference patterns also appear if the input signal is a pure two frequency signal.

x(t) = sin(2πν1t) + sin(2πν2t)

The interference pattern would then be centered around. ν + ν ν = 1 2 c 2 The resulting spectrogram would look like the spectrogram in ﬁgure 11 rotated 90 degrees. In fact this is exactly the reason for ﬁrst computing the complex analytic signal corresponding to the real measured signal and then computing the Wigner-Ville distribution of the complex analytic signal. Otherwise the interference between the positive and negative frequency band would cause unnecessary distortion to the spectrogram.

5.4 Wavelet theory Wavelet theory was developed during the 1980’s with the motivation to ﬁnd a time-frequency representation free from the constraint imposed by the Heisenberg-Gabor inequality as well as the cross-terms imposed by quadratic time-frequency representations. In this section the reference [11] has been followed. The continuous wavelet transform, similarly to the short-time Fourier transform, is a linear time-frequency representation meaning that the cross-term induced mathematical interference in for example the Wigner-Ville transform is not present in the continuous wavelet transform. The Fourier transform presents a method to break down a signal in its harmonic components.

x(t) = dνX(ν)ei2πνt Z However the Fourier transform of a signal is non-local meaning that it is not possible from the Fourier transform of a signal to get information regarding the spectral evolution. This can as was seen in section 5.1 be adressed by taking the Fourier transform for several time windows of the signal and combining the result. In wavelet theory the scale of the signal is introduced giving the possibility to draw conclusions on local behaviour of the signal. Consider the Fourier expansion of the real signal x(t).

x(t) = dνX(ν)ei2πνt (5.24) ZR

– 32 – By introducing a reciprocal scaling of the integration variable ν the integral is invariant.

ν0 ν0 i2πν0t ν0 ν0 i2πν0t a a = d X e = da 2 X e (5.25) R a a R −a a Z Z ∞ 2ν0 ν0 i2πν0t a = Re da 2 X e (5.26) 0 a a Z ∞ 2ν0 ν0 t = Re da 2 X ψ (5.27) 0 a a a Z where

t i2πν0t ψ e a (5.28) a ≡ This rewriting of the inverse Fourier transform is the stand-point from which the continuous wavelet transform begins. By switching basis function ψ from a harmonic function to a non harmonic function the continuous wavelet transform is obtained.

5.4.1 The continuous wavelet transform Consider the following deﬁnition of the pseudo-basis function ψ. 1 t t ψ = ψ − 0 t0,a √a a To include locality of the time-frequency representation we have also deﬁned the time-shift parameter t0. It is then possible to expand the signal x(t) as follows.

∞ da ψ x(t) = dt0 CWT (t0, a)ψ (t) a2 x t0,a ZR Z0 ψ where CWTx (t0, a) is the continuous wavelet transform distribution. If the following condition, denoted as the admissibility condition 0 ∞ dν dν ψˆ(ν)ψˆ∗(ν) = ψˆ(ν)ψˆ∗(ν) = 1 0 ν ν Z Z−∞ | | is fulﬁlled the continuous wavelet transform can be written as.

ψ 1 t t0 CWT (t0, a) = dt x(t) ψ − = x, ψ x √a a h t0,ai ZR The admissibility condition is a necessary condition since the map ψ x(t) CWT (t0, a) 7→ x is overcomplete. The output space is a function of two continuous variables while the domain is function of one continuous variable. The pseudo-basis functions ψt0,a(t) cannot build up a basis since they are linearly dependent, this is the reason why they in this text are called pseudo-basis functions, however they constitute something similar to a basis referred to as a tight frame. Perhaps the most important property of the continuous wavelet transform is how it deals with phase space uncertainty. For low frequencies the spectral resolution is good while the temporal resolution is poor and for high frequencies the spectral resolution is poor while the temporal resolution is good. The reciprocal spectral resolution of the continuous wavelet transform is illustrated in ﬁgure 4.1 in [11].

– 33 – 5.4.2 The generalized Morse pseudo-basis functions The following follows from [15]. In this thesis the generalized Morse pseudo-basis functions have been used in MATLAB to compute the continuous wavelet transform. This is due to the simple implementation of the generalized Morse wavelet in MATLAB using the wavelet library. The generalized Morse functions are given in the form.

iωt β ωγ Ψβ,γ(ω) = ψβ,γ(t)e− = U(ω)aβ,γω e− ZR where U(ω) is the unit step function, aβ,γ a normalization constant and β, γ are denoted as the wavelet parameters. γ is the symmetry parameter of the wavelet. For γ > 3 the wavelet has positive skewness, for γ < 3 the wavelet has negative skewness and for γ = 3 the wavelet is symmetric. Figure 2 in [15] shows the temporal profile of the generalized Morse wavelet for different values of β and γ. From this figure a wavelet similar to the single cycle THz pulses have been chosen. In signal processing with the continuous wavelet transform in this text the wavelet parameters are set to, γ = 1, β = 1, since then the Morse wavelet is similar to the single cycle THz pulses.

5.4.3 Masking Masking is the process of multiplying a time-frequency representation with a two dimensional window function. Four types of masks are investigated, Lorentzian, Lorentzian- Gaussian, 2D Gaussian and Lorentzian-exponential masks. First on, purely temporal Lorentzian, Voigt and Gaussian masks were investigated. From the first temporal analysis it was observed that the Voigt and Gaussian masks dampened the spectrums to fast and much information of the main pulse was lost. The temporal Lorentzian mask was seen to produce results similar to the ones computed from zero-padded spectrums. Therefore the majority of the analysis of the masks were devoted to Lorentzian masks. Two properties make the continuous wavelet transform especially useful for the application of removal of back-reflections, the first being the simple recovery of the marginals and the second being the lack of mathematical interference. Therefore an algorithm can be constructed by masking in the time-frequency domain to remove the back-reflections as follows.

1. Compute the Continuous wavelet transform of the signal x(t)

2. Apply the time-frequency mask m(t, ν). The mask should capture the information of the main pulse while cancel the back-reﬂections.

3. Take the inverse continuous wavelet transform and recover the processed signal x0(t).

4. Compute n and κ which is the purpose of the spectroscopic characterization. n and κ is the real and imaginary part of the refractive index. More on the details of the spectroscopic characterization can be found in section3.

Such an algorithm would be diﬃcult to implement with the short-time Fourier transform and the Wigner-Ville transform. It would be diﬃcult for the short-time Fourier transform since when taking the inverse short-time Fourier transform the marginal is convoluted with

– 34 – the window function. For the Wigner-Ville transform it would be difficult to construct the mask such that information of the back-reflections is removed while information in the main pulse is preserved due to the interference peaks superimposed on the spectrogram. The continuous wavelet transform has no cross term interference and no smearing of the marginals and is therefore best suited for the masking algorithm. By applying different masks to the computed continuous wavelet transform the back-reflections can be drastically reduced. A general temporal mask is a smooth function that converges to zero at . ±∞ Examples of such functions are the Lorentzian function, the Gaussian function and the Voigt function. Given a general temporal mask m(t), where t is retarded time, it is possible to introduce a tuning parameter as follows.

+ t √ct, c R (5.29) 7→ ∈ The mask m(t) then becomes.

m(t) m(√ct) (5.30) 7→ The Fourier transform of the tuned temporal mask can be computed through the Fourier rule regarding scaling of the temporal parameter.

µ(ν) F (m(t)) (5.31) ≡ 1 ω 1 2πν F (m(t, c)) = µ = µ (5.32) √c √c √c √c Hence by varying the tuning parameter spectral properties of the mask are varied. The optimization of the refractive index with respect to the tuning parameter can lead to an interesting conclusion for the general temporal mask. The refractive index is a function of frequency and the tuning parameter.

φ(ν, c)c0 n(ν, c) = + n (5.33) 2πνd air The optimization of the refractive index is given by the equation.

∂cn(ν, c) = 0 (5.34)

The derivative of the refractive index with respect to the tuning parameter becomes.

∂ n(ν, c) = 0 ∂ φ(ν, c) = 0 (5.35) c ⇒ c where the phase of the transfer function is deﬁned as.

H = H(ν, c) eiφ(ν,c) (5.36) | | By taking the complex logarithm to the left and right hand side gives.

log H = log H + iφ (5.37) | |

– 35 – The phase is then given by.

H φ = i log (5.38) − H | | By taking the derivative of the phase with respect to the tuning parameter and setting to zero gives.

1 2πν Es(ν) √c µ √c H = ∗ (5.42) E (ν) 1 µ 2πν r ∗ √c √c The derivative of the transfer function with respect to the tuning parameter is then given by.

E (ν) 1 µ 2πν s ∗ √c √c 0 = ∂cH = ∂c (5.43) E (ν) 1 µ 2πν  r ∗ √c √c 1 2πν 1 2πν = E (ν) ∂ µ E (ν) µ (5.44) s ∗ c √c √c r ∗ √c √c 2 1 2πν 1 2πν 1 2πν − E (ν) µ E (ν) ∂ µ E (ν) µ − s ∗ √c √c r ∗ c √c √c r ∗ √c √c (5.45)

The first big bracket is antisymmetric under interchange of indices s and r which means that if the individual terms in the first big bracket can be forced to be symmetric under the interchange of s and r the expression will vanish. The individual terms in the first big bracket are symmetric under the transformations s r, r s when. 7→ 7→ 1 2πν 1 2πν ∂ µ = µ (5.46) c √c √c √c √c By letting the derivative act on the left bracket this can be rewritten as.

3/2 c− 2πν 1/2 2πν 1/2 2πν µ + c− ∂ µ = c− µ (5.47) − 2 √c c √c √c 2πν ∂cµ √c 1 = 1 + (5.48) ⇒ 2πν 2c µ √c

– 36 – Equation (5.48) is denoted as the parameter equation and provides a general way to ﬁnd the tuning parameter that optimises a general mask function. The refractive index is optimized when the transfer function is maximal. The transfer function can be seen the amount of information that is collected in the spectrographic characterization. When the transfer function is maximised as much information as possible is taken into account in the characterization given a speciﬁc temporal window function which is sought for.

5.4.4 Gaussian masking The Gaussian temporal mask is given by

ct2 m(t, c) = e−

The Fourier transform of the Gaussian temporal mask, µ, is then.

ω ω2 µ = √πe− 4c √c The derivative of µ with respect to the tuning parameter is then given by.

ω ω2 ω ∂ µ = µ c √c 4c2 √c Inserting this into the parameter equation gives.

ω2 1 = 1 + 4c2 2c Which has the solution.

1 1 ω2 c = + −4 ± r16 4 Only the positive branch of he square root function preserved the square integrability of the mask giving the tuning parameter as.

1 1 ω2 c(ω) = + + −4 r16 4 The optimized Gaussian mask becomes.

2 1 + 1 + ω t2 − − 4 16 4 m(t, ω) = e q

5.4.5 Lorentzian masking The general Lorentzian mask in this text is deﬁned as follows.

1 + + mα(t, c) = , α Z , c R (1 + ct2)α ∈ ∈

– 37 – where c is the tuning parameter that determines how much of the main pulse that is captured, t is retarded time and α is a positive integer. The Fourier transform of the Lorentzian mask is then given by.

ω | | α 1 i πe− √c − ω F (m (t, c)) = a | | α √c i √c Xi=0 where ai are some coeﬃcients. It is quite troublesome to compute the Fourier transform for the Lorentzian mask by hand. The polynomial in the Fourier transform of the Lorentzian mask become more complex for increasing values of α. Therefore the symbolic library in MATLAB was used to compute the Fourier transform of the Lorentzian masks of high order using the command fourier(). From equation (5.46) it can be seen that to optimize the real part of the refractive index the Fourier transform and the derivative with respect to the tuning parameter of the Fourier transform of the Lorentzian mask must be computed. This is best done with the command diff() in MATLAB. The parameter equation is then given by. ∂ F (m (t, c)) c α = 0 F (mα(t, c)) For α > 0 this is a non-linear equation with variables ω and c. One way to solve this numerically is using the fzero command in MATLAB where ω is the ﬁxed to the frequency vector computed from the continuous wavelet transform. Then, as output from the fzero command, i.e. the tuning parameter, as a function of frequency, is produced. This is the tuning parameter that optimizes the refractive index. And the general Lorentzian mask can be written as. 1 m (t, ω) = α (1 + c(ω)t2)α

In the derivation of the parameter equation no assumptions have been made of the signal that the mask is acting on. To improve the masking even further the width of the main pulse must be taken into account such that the mask captures the main pulse and cancel the back-reﬂections. This has been done by introducing a second tuning parameter a. The parameter, a, shall be chosen such that the main pulse is captured and the back-reﬂections are greatly reduced. The general Lorentzian mask then becomes. 1 m (t, ω, a) = (5.49) α (1 + c(ω/a)t2)α

From the definition of the imaginary part of the refractive index it is also clear that when n converges, κ converges. The refractive index becomes stationary for Lorentzian time- frequency filtering when the tuning parameter is chosen as above. The process of finding the best Lorentzian mask is then to vary α and a, such that under inspection, the filtered continuous wavelet distribution of the main pulse is captured and the back-reflections are reduced. In practice the masks on this form was the best at alleviating the distortion from the back-reflections. All filtered signals in the results section have been processed

– 38 – with masks on the form given in equation (5.49). The Lorentzian masks are chosen for the masking of the continuous wavelet transform since the Lorentzian masks capture the main pulse better than the other investigated masks.

5.4.6 Lorentzian-Gaussian masking

Consider the Lorentzian-Gaussian mask.

2 c(ν ν0) e− − mLG(t, ν) = 2 1 + c0t

In the following ﬁgures the primary tuning parameter c was varied and the secondary tuning parameter was set to, c0 = 1.

c = 1 c = 0.5

c = 0.3 c = 0.1

Figure 12: Figure showing the Log intensity plot of Lorentzian-Gaussian masks with the primary tuning parameter varied and the secondary tuning parameter ﬁxed to c0 = 1. Time is on the horizontal axis and frequency on the longitudinal axis

Figure 12 shows the Lorentzian-Gaussian mask for different values of the primary tuning parameter. The central frequency of the mask is set to, ν0 = 2 THz. Figure 13 and figure 14 shows the real and imaginary part computed from the signal filtered with the Lorentzian-Gaussian mask for different values of the primary tuning parameter. As more of the low frequencies are captured by the mask the refractive index converges on reasonable values. The Lorentzian-Gaussian mask can reproduce similar refractive index functions as the pure Lorentzian masks. It does however also introduce a second tuning parameter which makes the implementation of the mask more time consuming. Therefore the pure Lorentzian mask is more suited for the removal of back-reflections in a THz-TDS signal. Additionally a 2D-gaussian and a Lorentzian-exponential mask were investigated.

– 39 – 10-3 5 4 c = 1 c = 1 c = 0.5 c = 0.5 4.5 c = 0.3 3.5 c = 0.3 c = 0.1 c = 0.1 4 3

3.5

2.5 3

2.5 2

2 1.5

1.5 Real part of refractive index, n 1 Imaginary part of refractive index,

0.5 0.5

0 0 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Frequency [THz] Frequency [THz]

Figure 13: Imaginary part of Figure 14: Real part of refrac- refractive index computed from tive index computed from masked masked continuous wavelet distribu- continuous wavelet distribution with tion with diﬀerent tuning parame- diﬀerent tuning parameters. ters.

They were on the form

(ν ν )2 ct2 − 0 mGG(t, ν) = e− e− c ν | | e− √c m (t, ν) = LE 1 + ct2 The motivation for investigating these masks was that they have an conserved intensity in the time-frequency plane. However, none of these masks could for any tuning parameter tried reproduce n and κ. In this text the Lorentzian mask will be used for signal processing with the continuous wavelet transform due to the simplicity of its implementation and the small deviation of the processed result as compared with reciprocal Lorentzian mask and the Lorentz-Gaussian mask.

5.5 Comparison of the three representations The short-time Fourier transform, the Wigner-Ville transform and the continuous wavelet transform all have advantages and disadvantages and the time-frequency representations are suitable for different applications. The short-time Fourier transform have a straight forward relation between spectral and temporal resolution namely the Heisenberg-Gabor inequality. It does not suffer from mathematical interference induced in for example the Cohen class. This means that the short-time Fourier transform is suitable for signals with repeating structures. It is not so good at representing a single pulse since it has lower spectral resolution than both the Wigner-Ville representation and the continuous wavelet transform. The Wigner-Ville transform have the highest spectral resolution of the three representations however it suffers from cross term interference. This means that the Wigner- Ville time-frequency representation is good for representation of a single pulse while for longer repeating pulse trains the mathematical interference can come to dominate. The continuous wavelet transform is a linear time-frequency representation meaning that it has no cross term interference. However the spectral resolution varies with frequency. For

– 40 – low frequencies the spectral resolution is good while the temporal resolution is poor. For high frequencies the spectral resolution is poor while the temporal resolution is good. This means that the continuous wavelet transform is a good tool for analysis of both single pulses and pulse trains. If the pulses are broadband, such as the THz single cycle pulses, a pulse in the spectrogram will reveal both high accuracy spectral information and high accuracy temporal information. For the analysis of single pulses the Wigner-Ville transform is the best tool.

6 Results

The results in this section have been obtained by analysis in MATLAB using the Time- Frequency Toolbox, the Wavelet Toolbox. The data analysed in this section comes from measurements performed by Kjell Mølster [1].

6.1 Transmission spectroscopy In transmission spectroscopy an optical pulse is transmitted through the sample from which optical properties of the sample can be deduced.

6.1.1 Silicon, High resistivity ﬂoat zone (HRFZ), 4.46 mm

10-3

Reference Sample 8

2 Amplitude [V/m]

-2

-4 -40 -20 0 20 40 60 80 Time [ps]

Figure 15: TDS sample signal for High resistivity ﬂoat zone Silicon with a thickness of 4.46mm.

Figure 15 shows the TDS sample signal for High resistivity float zone silicon in transmission centred around the main pulse. The main pulse is located at t = 0. At later times there are some clear smaller pulses. These are back-reflections in the GaP crystal and the silicon spectral splitter. At t = 6 ps, 12 ps and 18 ps the first, second and third back-reflection in the GaP is located. At t = 25 ps the first back-reflection in the spectral splitter is located and at t = 30 ps the first back-reflection in both the spectral splitter and the Gap is located. Before the main pulse at t = 10 ps there is small excitations of the pulse. − These are artefacts and are caused by high harmonic generation of the main pulse and will

– 41 – also be removed in the time-frequency masking. They have traveled a shorter distance than the main pulse and therefore arrive earlier. For each reflection the intensity of the pulse falls according to the reflection and transmission coefficients of the materials. Figure 16 shows

10-3

Reference Sample 8

2 Amplitude [V/m]

-2

-4 0 50 100 150 200 Time [ps]

Figure 16: Full TDS sample signal sweep for High resistivity float zone silicon with a thickness of 4.46mm. the full sweep of the TDS sample signal of silicon. The full sweep is approximately over 250 ps. Note that the back-reflection profile is different for the reference and the sample. At around 100 ps there are new back-reflections for the sample signal but not for the reference. This is the first back-reflections in the sample. If the sample and reference signal had the same back-reflection profile a high resolution characterization would be possible since the oscillations in the spectrums due to the Fourier integration over back-reflection peaks would cancel when computing n and κ. Figure 17 shows the corresponding spectrum to

101 Reference Sample

100

10-1

10-2 Spectral Amplitude [a.u.]

10-3

10-4 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 17: Spectrum corresponding to the signal in ﬁgure 15. The fast oscillations in the spectrums is caused by the Fourier integration over the back-reﬂections giving a high spectral uncertainty.

– 42 – the full sweep of the TDS sample signal. Both the sample and reference spectrums have a high spectral uncertainty. This spectral uncertainty is caused by the Fourier integration over the back-reﬂection peaks. At around 7 THz the spectrum reaches the noise ﬂoor above which no meaningful characterization can be performed. This is caused by strong phonon absorption channels in the GaP crystal at frequencies higher than 7 THz. A high spectral resolution will in turn yield an accurate characterization which is sought for. The group at NTNU have previously used a technique called zero padding to get this increased spectral resolution. By zero padding means to cut the signal around the main pulse and in the Fourier transform to increase the number of points. For all zero-padded signals in this text the number of points chosen is N = 214. Figure 18 shows the TDS sample and

10-3

Reference Sample 8

2 Amplitude [V/m]

-2

-4 -4 -3 -2 -1 0 1 2 3 4 Time [ps]

Figure 18: TDS reference and sample signal for HRFZ silicon.

reference signal for HRFZ silicon in transmission. The signal is cut around the main pulse. A disadvantage with zero-padding is the high sensitivity to the choice of the interval around the main pulse in which the signal is cut. By slightly changing this interval n and κ can vary greatly. Figure 19 shows the corresponding spectrum to the zero-padded spectrum of the cut signal in figure 18. As can be seen the strong oscillatory behaviour of the spectrum have now been dampened as compared with figure 17 and the spectral uncertainty has decreased. However when the signal is cut around the main pulse information contained in the main pulse far from the centre of the pulse is removed. This loss of information will limit the spectrographic characterization. Figure 20 shows the imaginary part of the refractive index for the zero-padded signal. The fluctuations increase for higher frequencies which is expected due to the lower spectral intensity at higher frequencies and therefore larger noise contribution. After 11 THz the imaginary part of the refractive index becomes negative indicating that the fluctuations are so large that no spectroscopic conclusion above this frequency can be drawn. Figure 21 shows the real part of the refractive index for the zero-padded signal.

– 43 – 100 Reference Sample

10-1

10-2

10-3

-4 Spectral Amplitude [a.u.] 10

10-5

10-6 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 19: zero-padded spectrum of the sample and reference signal.

10-3 3.5

2.5

1.5

1 Imaginary part of refractive index,

0.5

0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 20: Imaginary part of refractive index computed from zero-padded signal.

3.422

3.42

3.418

3.416 Real part of refractive index, n 3.414

3.412

0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 21: Real part of refractive index computed from zero-padded signal.

– 44 – Figure 22: Short-time Fourier transform spectrogram of TDS sample signal for HRFZ silicon.

6.1.1.1 Short-time Fourier transform analysis

Figure 22 shows the short-time Fourier transform for HRFZ silicon. As the number of bins increases the spectral resolution decreases and the temporal resolution increases according to the Heisenberg-Gabor inequality. It is diﬃcult to draw conclusions regarding the ﬁne structure of the pulses in the short-time Fourier time frequency representation.

6.1.1.2 Wigner-Ville transform analysis

The Wigner-Ville transforms computed in this project have been computed using the Time- Frequency toolbox in MATLAB. More specifically using the tfrwv() command on the complex analytic signal corresponding to the real measured signal. Figure 23 shows the Wigner-Ville distribution of the TDS sample signal for HRFZ Silicon. It is evident that there is interference due to the cross terms between the main pulse and the back-reflections. This interference makes it difficult to analyse the Wigner-Ville distribution. Figure 24 shows the Wigner-Ville distribution of the main pulse of the TDS sample signal for HRFZ silicon. The chirp of the main pulse is plotted in white. The chirp is a measure on how the instantaneous frequency of the pulse varies with time. The chirp is computed by computing the mean frequency for each time instance and combing the result. Figure 25 shows the Wigner-Ville distribution obtained by sampling the signal in the interval [ 5, 10] ps. This − time sampling captures the main pulse and the first back-reflection in the EO. There is a

– 45 – 10 -5 4 4 9

3.5 3.5 8

3 3 7

2.5 2.5 6

5 2 2 4 1.5 1.5 Frequency [THz] 3 1 1 2

0.5 0.5 1

0 0 0 -5 0 5 10 15 20 25 30 10 10-3 Spectral Amplitude [a.u.] 2 1 0 -1 0 5 10 15 20 25 30 Amplitude [V/m] Time [ps]

Figure 23: Wigner-Ville distribution of the TDS sample signal for HRFZ silicon. The first peak is the main pulse that passes straight through both the EO and the spectral splitter. The second pulse around t = 6 ps has been reflected once in the EO. The third pulse have been reflected twice in the EO and the fourth pulse have been reflected three times in the EO. The fifth pulse have passed straight through the EO but have been reflected once in the spectral splitter. The sixth pulse have been reflected once in the EO and once in the spectral splitter. clear interference peak between the measured peaks caused by the cross term of the Wigner- Ville distribution. Figure 26 shows the Wigner-Ville distribution obtained by sampling the signal the signal in the interval [2, 16] ps. This time sampling captures the first and second back-reflection in the EO. There is again a clear interference peak between the measured peaks caused by the cross term of the Wigner-Ville distribution. A signal with repeating structures, such as all the measured THz-TDS signals, will have cross-term interference in the Wigner-Ville distribution. This makes analysis and signal processing difficult since it is difficult to say what is information carried by physical interactions and what is information induced by the Wigner-Ville distribution.

6.1.1.3 Continuous wavelet transform analysis

Figure 27 shows the continuous wavelet transform for the measured signal of HRFZ silicon in transmission. At around 50 ps the main pulse is located. The main pulse is no longer located at t = 0 due to problems implementing the time shift of the continuous wavelet transform this has however no influence on the analysis of the signal. For times later than 50 ps the signature of the back-reflections in the continuous wavelet domain is shown. The grey region in the spectrogram is the cone of interest. Inside the grey region boundary effects

– 46 – Figure 24: Wigner-Ville distribution of the main pulse of the TDS sample signal for HRFZ silicon. The THz pulse have interacted with the Silicon spectral splitter and the Gallium phosphide crystal.

10 -4 4 4

3.5 3.5 5

3 3

4 2.5 2.5

2 2 3

1.5 1.5 Frequency [THz] 2

1 1

1 0.5 0.5

0 0 0 -5 -5 0 5 10 10 10-3 Spectral Amplitude [a.u.] 6 4 2 0 -2 -5 0 5 10 Amplitude [V/m] Time [ps]

Figure 25: Wigner-Ville distribution of the main pulse and the first back-reflection in the EO are to big to analyse the signal confidently. A big difference compared to the Wigner-

– 47 – 10 -5 4 4

3.5 3.5 2.5

3 3

2 2.5 2.5

2 2 1.5

1.5 1.5 Frequency [THz] 1

1 1

0.5 0.5 0.5

0 0 0 -2 -3 -4 2 4 6 8 10 12 14 16 10 10 10 10-4 Spectral Amplitude [a.u.] 10 5 0 -5 2 4 6 8 10 12 14 16 Amplitude [V/m] Time [ps]

Figure 26: Wigner-Ville distribution of the ﬁrst and second back-reﬂection in the EO

Figure 27: Continuous wavelet transform of TDS sample signal for HRFZ silicon

Ville representation there is no interference in the spectrogram, a fact that makes the analysis much easier. The continuous wavelet transform distributions have in this project been computed with the continuous wavelet toolbox in MATLAB with Morse pseudo-basis functions using the command cwt(). The wavelet parameters for the Morse pseudo-basis functions have been set to γ = 1, β = 1. The wavelet parameters were chosen such that the mother wavelet would be similar to the THz main pulse. Figure 28 shows the continuous wavelet transform of the signal masked with the general Lorentzian mask. As can clearly be seen the back-reﬂections in ﬁgure 27 have been dampened. Figure 29 shows the unprocesed signal and the signal computed from the Lorentzian masked continuous wavelet distribution.

– 48 – Figure 28: Continuous wavelet transform of TDS sample signal masked with the general Lorentzian mask with a = π and α = 1. The wavelet parameters for the generalized Morse wavelets were γ = 1, β = 1.

1 Processed signal Signal

0.5 Amplitude [V/m]

-0.5 -5 0 5 10 15 20 Time [ps]

Figure 29: TDS sample signal for HRFZ silicon processed with the general Lorentzian mask as well as the unprocessed signal.

The main pulse of the processed signal looses little of its initial information while almost all of the back-reflections are removed. Figure 30 shows the spectrum of the processed signal. It is clear that the fluctuations seen in figure 19 have been reduced significantly by the signal processing. However still at around 11 THz the predictive power is lost since the random fluctuations is so large that the sample spectrum is larger than the reference spectrum. Figure 31 shows the real part of the refractive index for the processed signal and the zero-padded signal. It is evident that the fluctuations is much smaller for the process signal. Figure 32 shows the imaginary part of the refractive index for the processed sample signal and the zero-padded sample signal. It is evident that the fluctuations is much smaller for the process signal. At 7.57 THz there is strong phonon absorption in the GaP crystal.

– 49 – 100 Reference Sample

10-1

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-4 Spectral Amplitude [a.u.] 10

10-5

10-6 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 30: Spectrum of ﬁltered TDS sample signal for HRFZ silicon

Figure 31: Real part of refractive index computed from processed sample signal and zero- padded sample signal for HRFZ silicon.

This is shown as the peak of k at around 7.57 THz. However at around 9.86 THz there is an even larger peak. This peak cannot be caused by absorption in the GaP since there is no phonon mode at this frequency but it could be caused by phonon absorption in silicon.

6.1.2 Gallium phosphide, GaP, 5 mm Gallium phosphide (GaP) is a dielectric crystal used in THz-TDS in the free space electrooptic sampling detection scheme. It is of high interest to characterise GaP since the phonon modes of GaP and the self-phase modulation in GaP will be imprinted in all TDS signals. The GaP crystal investigated had the orientation [1, 1, 0] and a thickness of 5 mm. Figure 33 shows the TDS sample and reference signal for a 5 mm GaP crystal. At 6 ps the first back-reflection in the detection scheme GaP crystal is detected. At 12 ps the second back-reflection in the detection scheme GaP crystal is detected. At around 24 ps the first

– 50 – Figure 32: Imaginary part of refractive index computed for processed sample signal and zero-padded sample signal.

10-3

Reference 10 Sample

2 Electric field strength [V/m]

-2

-15 -10 -5 0 5 10 15 20 25 30 35 Time [ps]

Figure 33: TDS sample signal for 5 mm GaP crystal.

back-reflection in the Silicon spectral splitter is detected and at around 30 ps the first back reflection in both the detection scheme GaP crystal and the Silicon spectral splitter is detected. The full sweep is only 53 ps which is significantly shorter in time compared to HRFZ Si.

6.1.2.1 Short-time Fourier transform analysis

Figure 34 shows the short-time Fourier transform of the TDS sample signal for GaP. As the number of bins is increased the temporal resolution increases and the spectral resolution decreases according to the Heisenberg-Gabor inequality.

– 51 – Figure 34: Short-time Fourier transform of TDS sample signal for GaP.

10 -4 4 4

3.5 3.5 6

3 3 5

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0.5 0.5 1

0 0 0 -5 0 5 10 15 20 25 30 10 10-3 Spectral Amplitude [a.u.] 2 0 -2 0 5 10 15 20 25 30 Amplitude [V/m] Time [ps]

Figure 35: Wigner-Ville distribution of the TDS sample signal for GaP.

6.1.2.2 Wigner-Ville transform analysis

Figure 35 shows the Wigner-Ville distribution for the GaP signal. The interference between the pulses makes the pulse train diﬃcult to analyze in the Wigner-Ville representation. An

– 52 – interesting feature of the spectrogram is the clear presence of dispersion. High frequencies get shifted to later time instances. The back-reflections will have traveled an integer multiple of the distance traveled of the main pulse inside the dispersive medias, the silicon spectral splitter and the GaP crystal. Meaning that the back reflections high frequencies get shifted to later time-instances that the main pulse high frequencies. This is difficult to see in figure 35 but can be seen if zoomed in. Figure 36 shows the Wigner-Ville distribution for the GaP

Figure 36: Wigner-Ville distribution of the main pulse of the TDS sample signal for GaP. sample signal main pulse. The chirp of the pulse is clear and there is a c-shaped geometry in the spectrogram centred at around 1 ps and 1.5 THz. This pattern is most likely caused by the cross-term interference of the Wigner-Ville distribution. The chirp of the main pulse is almost linear in the interval from 0.5 ps to 0.25 ps. This is a manifestation of the − optical Kerr eﬀect where the refractive index inside the GaP is increased by the presence of the main pulse.

6.1.2.3 Continuous wavelet transform analysis

Figure 37 shows the continuous wavelet transform of the unfiltered GaP signal. The trace of the main pulse can be seen around 20 ps with the first back-reflection in the EO at around 25 ps. The width of the distributions increase for decreasing frequency. This is due to the reciprocal property of the continuous wavelet transform. It is clear that the pulses are affected by dispersion inside the GaP since the high frequencies of the pulses are shifted to later times. Figure 38 shows the continuous wavelet distribution masked with the general Lorentzian mask. The back-reflections are effectively removed. Figure 39 shows the spectrum of the filtered reference and sample signal. There is what looks like phonon

– 53 – Figure 37: Continuous wavelet transform of the GaP signal with wavelet parameters γ = 1 and β = 1.

Figure 38: Continuous wavelet transform of GaP signal masked with the general Lorentzian mask with α = 2 and a = 250π. The wavelet parameters were set to γ = 1 and β = 1. absorption at 3.88, 6.15 and 9.24 THz. At 12 THz both the reference and the sample is absorbed which could be caused by absorption of the sample and reference signal in water vapour. Figure 40 shows the unprocessed GaP signal and the GaP signal computed from the Lorentzian masked continuous wavelet transform. The main pulse is mainly preserved while the back-reﬂections are highly dampened. Figure 41 shows the real part of the refractive index for GaP for processed and zero-padded signal. The refractive index proﬁle is quite smooth with no Lorentzian transitions. Figure 42 shows the imaginary part of the refractive index for GaP for processed and zero-padded signal. The imaginary part of the refractive index, κ, for the processed signal is smoother than for the zero-padded signal. There are

– 54 – Reference 100 Sample

10-1

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10-3 Spectral Amplitude [a.u.]

10-4

10-5 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 39: Spectrum of ﬁltered reference and sample signal for GaP.

1 Processed signal Signal

0.5 Frequency [THz]

-0.5 -5 0 5 10 15 20 Time [ps]

Figure 40: Unprocessed GaP signal and GaP signal computed from the Lorentzian masked continuous wavelet transform.

strong absorption peaks at 3.88, 6.15 and 9.24 THz.

6.1.3 Potassium titanyl phosphate, KT iOP O4, 500 [µm]

KTP is a birefringent material with applications in non-linear optics. KTP has many phonon modes in the THz regime and it is therefore important to characterize the KTP in this regime. The beam was propagated through the z-direction of the KTP crystal with polarization in the x-direction. Figure 43 shows the full sweep of the KTP sample and reference signal in transmission. There are many back-reﬂections in the signal. For time t > 100 ps the average signal intensity is very small and almost no back-reﬂections can be seen.

– 55 – Figure 41: Real part of the refractive index computed for processed signal and zero-padded signal.

Figure 42: Imaginary part of the refractive index computed for processed signal and zero- padded signal.

6.1.3.1 Short-time Fourier transform analysis

Figure 44 shows the short-time Fourier transform for the TDS signal on KTP. The Heisenberg- Gabor inequality is manifested here again as with HRFZ Si and GaP. In the STFT for 20 bins there is an interesting oscillation of the spectrogram. This is probably caused by Fourier integration over the main pulse and the ﬁrst back-reﬂection in the GaP.

6.1.3.2 Wigner-Ville transform analysis

Figure 45 shows the Wigner-Ville distribution of the KTP main pulse and back-reﬂection. The interference between induced by the cross terms distorts the spectrogram. The pulse is

– 56 – 10-3

Reference Sample 8

2 Electric field strength [V/m] 0

-2

-4 0 50 100 150 200 250 Time [ps]

Figure 43: Full sweep of sample and reference signal for KTP in transmission.

Figure 44: Short-time Fourier transform for the TDS signal on KTP. highly dispersed. Figure 49 shows the Wigner-Ville distribution of the KTP main pulse. It is as always with the Wigner-Ville distribution difficult to know which features of the geometry in the spectrogram is caused by cross-terms and which is caused by actual features of the pulse. It is however clear that for frequencies close to 2 THz the pulse is highly dispersed. The ringing of the KTP pulses is significantly longer than for example HRFZ silicon. It is in fact so long that the main pulse and the first back-reflection overlap. This can pose a

– 57 – 10 -4 3 3

3.5 2.5 2.5

2 2 2.5

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1.5 Frequency [THz] 1 1

0.5 0.5 0.5

0 0 0 -5 0 5 10 15 20 25 30 10 10-3 Spectral Amplitude [a.u.] 2

-2 0 5 10 15 20 25 30 Amplitude [V/m] Time [ps]

Figure 45: Wigner-Ville distribution of TDS signal with KTP in transmission as the sample. At around 6 ps the first back-reflection in the EO arrives to the detector which can be seen in the Wigner-Ville distribution as the interference region. At around 25 ps the first back-reflection in the Si spectral splitter arrives

Figure 46: Wigner-Ville distribution of TDS signal main pulse of KTP. problem for separating information of the main pulse and back-reﬂections.

– 58 – 6.1.3.3 Continuous wavelet transform analysis

Figure 47: Continuous wavelet transform of unprocessed KTP sample signal. The wavelet parameters were set to γ = 1 and β = 1.

Figure 47 shows the continuous wavelet transform for the unprocessed KTP time domain spectroscopy signal. The full time sweep duration is 300 ps substantially longer than for GaP. Figure 48 shows the continuous wavelet transform for the KTP TDS signal masked

Figure 48: Continuous wavelet transform of KTP sample signal masked with the general Lorentzian mask with α = 3 and a = 1500π. The wavelet parameters were set to γ = 1 and β = 1. with the general Lorentzian mask with α = 3 and a = 1500π. The main pulse is basically unaffected by the filtering while the back-reflections are heavily dampened. Some oscillatory low frequency behaviour of the main pulse is removed. Figure 49 shows the processed and unprocessed KTP signal. For time instances at the edge of the main pulse, the main pulse

– 59 – 1 Processed signal Signal

0.5 Amplitude [V/m]

-0.5 -5 0 5 10 15 20 Time [ps]

Figure 49: Unprocessed KTP signal and signal computed from the Lorentizan masked continuous wavelet distribution corresponding to the KTP signal.

of the processed signal are slightly dampened. The back-reﬂections of the processed signal are highly dampened. Figure 50 shows the spectrum of the Lorentzian processed reference

100 Reference time-filtered 100 Sample time-filtered Reference zero-padded Sample zero-padded 10-1 10-1

10-2 10-2

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-4 Spectral Amplitude [a.u.] 10 -4 Spectral Amplitude [a.u.] 10

10-5 10-5

10-6 0 2 4 6 8 10 12 14 16 18 10-6 Frequency [THz] 0 2 4 6 8 10 12 14 16 18 Frequency [THz] Figure 50: Spectrum of Lorentzian pro- Figure 51: Spectrum of zero-padded refer- cessed reference ence and sample signal for KTP. and sample signal for KTP. and sample signal and figure 51 shows the zero-padded reference and sample spectrum for KTP. At around 2 THz the spectral intensity of the sample decreases rapidly due to absorption in KTP. Figure 52 shows the real part of the refractive index for the Lorentzian processed and zero-padded KTP signal. The refractive index profile computed by the two methods differ. Figure 53 shows the imaginary part of the refractive index for the processed and zero-padded KTP signal. In the interval between 2 and 8 THz there is strong phonon absorption in KTP with especially strong peaks at 2.30, 2.64, 3.01 and 3.70 THz.

– 60 – Time-filtered 4.8 Zero-padded

4.6

4.4

4.2

4 Real part of refractive index, n 3.8

3.6

3.4 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 52: Real part of the refractive index computed for the Lorentzian processed signal.

Figure 53: Imaginary part of the refractive index computed for the signal computed from the Lorentzian masked continuous wavelet transform.

6.1.4 Germanium, 1 mm Germanium is an element with many applications in photonics and is important to characterize in the THz regime. Figure 54 shows the full sweep of the Germanium transmission signal. The full sweep is approximately 120 ps with many back-reﬂections in the signal. There are back-reﬂections from the GaP crystal, the Silicon spectral splitter and the Ger- manium sample.

6.1.4.1 Short-time Fourier transform analysis

Figure 55 shows the short-time Fourier transform of Germanium. Again the Heisenberg- Gabor inequality is manifested in the spectrograms. At t 25 ps in the 20 bin spectrogram ≈ there is an oscillatory behaviour of the spectrogram. This is probably caused by Fourier

– 61 – 10-3

Reference Sample 8

2 Electric field strength [V/m] 0

-2

-4 -20 0 20 40 60 80 100 Time [ps]

Figure 54: Full sweep of sample and reference signal for Germanium in transmission.

Figure 55: Short-time Fourier transform for the TDS signal on Germanium. integration over two neighbouring peaks.

6.1.4.2 Wigner-Ville transform analysis

Figure 56 shows the Wigner-Ville distribution for the sample signal of Germanium. The regions in the spectrogram corresponding the pulses are extremely temporally localized. Cross term interference patterns are present in the Wigner-Ville distribution. Figure 57

– 62 – 10 -4 4 4

3.5 3.5 2

3 3

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0 0 0 -5 -20 0 20 40 60 80 100 10 10-3 Spectral Amplitude [a.u.] 4

-20 0 20 40 60 80 100 Amplitude [V/m] Time [ps]

Figure 56: Wigner-Ville distribution of TDS signal of Germanium.

Figure 57: Wigner-Ville distribution of main pulse time domain spectroscopy sample signal of Germanium. shows the Wigner-Ville distribution of the main pulse of the time domain spectroscopy signal of Germanium. Germanium does not look like a very dispersive material since the high frequencies are not very time shifted compared to the low frequencies. The shape of

– 63 – the Wigner-Ville distribution is almost triangular.

6.1.4.3 Continuous wavelet transform analysis

Figure 58: Continuous wavelet transform of TDS sample signal of germanium. The wavelet parameters were set to γ = 1 and β = 1.

Figure 58 shows the continuous wavelet distribution for the Germanium signal. The main pulse and the first back-reflection in the GaP crystal is clearly separated for all frequencies. This makes it easier to chose a tuning parameter c for the Lorentzian time-filtering such that it cancel the back-reflections and preserves the main pulse. Figure 59 shows the continuous

Figure 59: Continuous wavelet transform of the sample signal of germanium masked with the general Lorentzian mask with α = 2 and a = 100π. The wavelet parameters were set to γ = 1 and β = 1. wavelet distribution of the processed signal. No trace of the back-reﬂections can be seen.

– 64 – Figure 60 shows the processed and the unprocessed Germanium signal. The main pulse is

1 Processed signal Signal

0.5 Amplitude [V/m]

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Figure 60: Processed and unprocessed Germanium signal captured accurately for the processed signal with an exception for the excitation around 2 ps. Figure 61 shows the spectrum of the processed reference and sample signal. The

100 Reference Sample

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Figure 61: Spectrum of ﬁltered reference and sample signal for Germanium. sample spectrum has a clear dip at around 7.80 and 11.79 THz. Figure 62 shows the real part of the refractive index for the processed and zero-padded Germanium signal. Figure 63 shows the imaginary part of the refractive index for the processed and zero-padded Germanium signal. The dip in the spectra at around 7.80 and 11.79 THz can be seen in κ as two peaks. There also appears to be a smaller peak between these two peaks at around 9.36 THz.

6.2 Reflection spectroscopy Up to this point all the data has been transmission spectroscopy however reflection spectroscopy is an important technique for characterization of surface properties of materials. Reflection spectroscopy are much more phase sensitive than transmission spectroscopy

– 65 – Figure 62: Real part of the refractive index computed for processed signal and zero-padded signal.

Figure 63: Imaginary part of the refractive index computed for processed signal and zero- padded signal. meaning that small differences in the experimental conditions can give radically different results. The reference that have been used for all reflection measurements is gold. For frequencies below the plasma frequency gold has extremely small absorption in the THz regime and effectively reflects all incident THz photons. Reflection spectroscopy is also an important techniques for optical characterization of surfaces.

6.2.1 Potassium titanyl phosphate, KT iOP O4 Figure 64 shows the full sweep of KTP in reflection. A significant difference to transmission spectroscopy is the short sampling times. The amplitude is as expected higher for the reference signal that for the sample signal. In the sample signal a small ringing behaviour

– 66 – 10-5

Reference 8 Sample

0 Electric field strength [V/m]

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-6 -4 -2 0 2 4 Time [ps]

Figure 64: Full sweep of sample and reference signal for KTP in reflection. is induce for t > 0. There are no back-reflections in the reflection signal. The reference signal is obtained by reflecting the THz pulse at a gold mirror.

6.2.1.1 Short-time Fourier transform analysis

Figure 65: Short-time Fourier transform for the KTP reﬂection sample signal.

Figure 65 shows the short-time Fourier transform of the KTP reﬂection signal. When the

– 67 – number of bins is increased the spectral resolution decreases and the temporal resolution increases according to the Heisenberg-Gabor inequality. In the 20 bin plot it looks like for times t < 0 the carrier frequency is higher than for times t > 0. This observation can be conﬁrmed by examining the Winger-Ville distribution and the continuous wavelet distribution of the signal.

6.2.1.2 Wigner-Ville transform analysis

10 -4 8 8 9

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Figure 66: Wigner-Ville distribution of full THz reﬂection TDS sample signal for KTP.

Figure 66 shows the Wigner-Ville distribution of the KTP reﬂection sample signal. The Wigner-Ville distribution of the sample and reference signal looks extremely similar. Figure 67 show the chirp of the KTP reﬂection signal.

6.2.1.3 Continuous wavelet transform analysis

Figure 68 shows the continuous wavelet transform of the full reflection sweep on KTP. In the interval between 8 and 10 ps and the frequency interval from 2 to 8 THz there is a increased intensity as compared to the background. Figure 69 shows the continuous wavelet transform of the full Lorentzian time-frequency windowed reflection sweep on KTP. The region, t [8, 10] ps, ν [2, 8] THz, is now dampened by the Lorentzian. Otherwise the ∈ ∈ processed and unprocessed continuous wavelet spectrogram look similar. Figure 70 shows the processed and unprocessed reflection signal. After the processing with the Lorentzian time-frequency windowing much of the small oscillations is removed. Figure 71 shows the zero-padded spectrum and figure 72 shows the Lorentzian time-frequency filtered spectrum of the KTP reflection sample and reference signal. For frequencies between [2, 8] THz the two spectrums agree quite well, although there is a oscillatory behaviour in

– 68 – Figure 67: Chirp of KTP reﬂection signal

Figure 68: Continuous wavelet transform of the KTP sample reflection signal with wavelet parameters γ = 1 and β = 1. the zero-padded spectrum. In the low frequency limit the Lorentzian time-frequency fil- 3 tered spectrums converge on a spectral intensity of approximately 10− . This is induced by the filtering method. In the low frequency limit the zero-padded spectrums converge on spectral intensities one order of magnitude smaller than for the Lorentzian time-frequency filtered. In the high frequency regime between [8, 18] THz the zero-padded spectrums oscillate violently. The Lorentzian time-frequency filtered spectrums does not.

– 69 – Figure 69: Continuous wavelet transform of Lorentzian time-frequency ﬁltered KTP re- ﬂection signal with α = 2 and a = 10π.

1 Signal Processed signal 0.8

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Figure 70: Processed and unprocessed normalized reﬂection signal.

Figure 73 shows the real part of the refractive index for computed from the zero-padded and the Lorentzian time-frequency filtered spectrums. The Lorentzian time-frequency filtered n looks like an average to the n computed from the zero-padded spectrums. Figure 74 shows the imaginary part of the refractive index computed from the zero-padded and Lorentzian time-frequency filtered KTP reflection signal. Between 2 and 4 THz there are three small resonances approximately located at 2.43, 3.04 and 3.48 THz. At approximately 4.78 THz there is a bigger broad resonance and at 7.39 THz the imaginary part of the refractive index explodes.

6.3 Noise There will always be noise in experimental data. For the THz-TDS data the noise contribution can be divided into two categories, correlated noise and uncorrelated noise. Un-

– 70 – 10-3 10-2 Reference Reference Sample Sample

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10-6

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10-7 10-8 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Frequency [THz] Frequency [THz]

Figure 71: Signal and reference spectrum Figure 72: Signal and reference spectrum of zero-padded signal. of Lorentzian time-frequency ﬁltered signal

Figure 73: Real part of refractive index computed from Lorentzian time-frequency filtered and zero-padded signal. correlated noise is due to random fluctuations of the laser signal while correlated noise is due to errors in the experimental detection scheme. Uncorrelated noise can be reduced by averaging several datasets. The method of averaging is not possible for correlated noise. According to Mølster [1] the source of the correlated noise is phonon absorption in the 300 [µm] GaP crystal as well as phase mismatching between the IR probe and the THz. Inside the GaP the group velocity of the THz and the IR differs. Since the group velocity of the single cycle THz pulse is a function of frequency the phase mismatch also varies as a function of frequency adding complexity to the analysis.

6.3.1 (αd)max analysis for HRFZ Silicon in transmission In [16, 17] a method to determine at which frequency the characterisation has hit the noise ﬂoor is presented. It is used for the characterizations from section6 as a method to bring

– 71 – Figure 74: Imaginary part of refractive index computed from Lorentzian time-frequency ﬁltered and zero-padded signal. certainty in the observed phonon modes. Consider the deﬁnition of the signal to noise ratio.

supν R Ereference(ν) SNR = ∈ σ( Ereference (ν > νnoise ) ) | | And the dynamic range. E (ν) DR = | reference | σ( E (ν > ν ) ) | reference noise | where νnoise is the frequency where the noise floor begins. The maximum absorbance is then given by the logarithm of the dynamic range. 4n(ν) (αd) (ν) = 2 ln DR max (n(ν) + 1)2 where the absorption coefficient is defined as usual. 4πνκ(ν) α(ν) = c And the absorbance given by. 4πdνκ(ν) dα(ν) = c And d is the thickness of the characterized sample. n(ω) is the real part of the refractive index and κ(ν) is the imaginary part of the refractive index of the characterized sample. Regions where α > αmax is denoted as saturated absorption. No spectrographic characterization can be made for regions where αmax < 0. Figure 75 shows the maximum absorption, the absorption and the real part of the refractive index for Silicon. At 7.57 THz there is a non saturated absorption peak. At 9.86 THz there is a saturated absorption peak

– 72 – ( d) ( d) 10 max

5 Absorbance 0

0 2 4 6 8 10 12 14 16 18 Frequency [THz]

3.422 n( ) 3.421

3.42

3.419

3.418

Real part of refractive index, n 3.417 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 75: Absorbance and real part of refractive index for Silicon. The signal have been time-frequency windowed around the main pulse with the Lorentzian mask 3. probably due to absorption in the silicon however the spectral intensity is several orders of magnitude below the peak intensity so it is difficult to draw conclusions. The Lorentzian time-frequency window induces an exponential decay of the dynamic range which manifests as a linear decay of the maximum absorption. This linear decay has been cancelled in figure 75 to be able to compare (αd)max with (αd). Around the 9.86 THz absorption peak the real part of the refractive index undergoes a Lorentzian transition. This is similar to the example in [17] on page 134 figure 7 where at the saturated absorption peak the real part of the refractive index also undergoes a Lorentzian transition. Figure 76 shows the sample

100 Sample Reference Noise floor asymptote

10-1

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10-3 Spectral amplitude [a.u.]

10-4

10-5 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 76: Spectrum of reference and sample signal for Lorentzian time-frequency windowed signal. and reference spectrum of the Lorentzian time-frequency windowed signal. The noise ﬂoor asymptote is showed as a dashed line. When computing n and κ this asymptote cancels since the transfer function is deﬁned as the quotient of the spectrums. However it does not for (αd)max, therefore the noise asymptote must be subtracted to be able to compare (αd)max and (αd) for the Lorentzian time-frequency windowed signal. Figure 77 shows

– 73 – 15 ( d) ( d) max 10

5 Absorbance 0

0 2 4 6 8 10 12 14 16 18 Frequency [THz]

3.422 n( )

3.42

3.418

3.416

3.414

3.412 Real part of refractive index, n 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 77: Absorbance and real part of refractive index for Silicon. The signal have been cut around the main pulse and zero-padded. the maximum absorption, the absorption and the real part of the refractive index for the zero-padded silicon signal. Due to the zero padding the distortion to (αd) and (αd)max is much higher than in ﬁgure 75.

100 Sample Reference Noise floor

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10-5 0 2 4 6 8 10 12 14 16 18 Frequency [THz]

Figure 78: Spectrum of reference and sample signal for the zero-padded signal.

Figure 78 shows the reference and sample spectrum for the zero-padded signal. The spectrums oscillate more than the spectrums in ﬁgure 76 due to the zero-padding.

7 Discussion

In this section the results are discussed and analyzed. A key feature for all results obtained from the signal processed with the Lorentzian time-frequency ﬁltering is that it improves the analysis of the data. The continuous wavelet transform was especially suited for representing the entire pulse train while both the Wigner-Ville and the continuous wavelet transform as complements were good for analysis of the main pulse. The short-time Fourier transform analysis could have been improved by increasing the overlap of the windowing.

– 74 – 7.1 Transmission spectroscopy For the experiments analyzed in this thesis the transmission signals all have longer sampling times than the reflection spectroscopy measurements. The total sampling time for the transmission spectroscopy signals is on the order of 100 ps while for the reflection spectroscopy signal on the order of 10 ps. This means that the difference in spectral resolution and the resolution of the characterization for the two techniques is approximately one order of magnitude.

7.1.1 Silicon, High resistivity float zone (HRFZ), 4.46 mm The Lorentzian processed silicon data show a resonance at 9.86 THz that have not been seen as clearly with the zero-padded signal. However since the spectral intensity for this frequency is several orders of magnitude below the maximum spectral intensity it is not possible to verify this resonance. Furthermore the efficiency of the electro optic crystal is also dramatically reduced in this range [19]. The Lorentzian processed n and κ looks like a smoothened version of the zero-padded characterization. This Lorentzian time-frequency windowing simplifies the analysis of the characterization and identification of resonances. However it makes the analysis of the spectrums more complex since the asymptote of the Lorentzian Fourier transform is convoluted with the unprocessed spectrum. In the computation of n and κ this problem is removed since the transfer function is defined as the quotient of the reference and sample spectrum and therefore the Lorentzian asymptote is cancelled. From the (αd)max analysis the peak at 9.86 THz looks like a saturated absorption peak. This peak is probably caused by phonon absorption of the THz inside the silicon sample however hard to verify due to the low spectral intensity. By solving the optimization problem of the refractive index with respect to the tuning parameter the tuning parameter was optimized and generated the general Lorentzian mask. The general Lorentzian mask both theoretically and experimentally produced the smoothest n and κ.

7.1.2 Gallium phosphide, GaP, 5 mm The Wigner-Ville representation of the main GaP pulse shows clear signs of dispersion inside the GaP crystal. Higher frequencies are shifted to later times. The ring like behaviour in Wigner-Ville spectrogram at around 1 ps is however probably caused by cross term interference since it is not present in the continuous wavelet transform spectrogram. The GaP sample show strong phonon absorption at 3.88, 6.15 and 9.24 THz. At 7 THz absorption in the detection scheme GaP crystal is shown as κ becomes negative. Between the resonances at 3.88 and 6.15 THz there are smaller resonances. For the zero-padded κ the ﬂuctuations are larger than for the Lorentzian time-frequency ﬁltered n and κ.

7.1.3 Potassium titanyl phosphate, KT iOP O4, 500 [µm] The Wigner-Ville spectrogram of the main pulse shows that KTP is highly dispersive in the THz regime. Frequencies close to 2 THz are shifted almost 2 ps. There is some ringing after the main pulse which is most likely caused by cross term interference since it is not present in the continuous wavelet spectrogram. For the GaP sample the dispersion

– 75 – appeared almost linear for the upper branch however for KTP the dispersion seems log- arithmic. The spectrum of KTP is significantly narrower than all the other investigated samples. At around 2 THz there is a three order of magnitude decrease of spectral intensity. This is caused by phonon absorption in KTP. The zero-padded and time-frequency filtered spectrums for KTP do not differ as much as for other samples. The ringing induced from the zero-padding dominates at higher frequencies. The real part of the refractive index differ for the different methods for KTP up to around 2.5 THz which is where the spectral intensity for the sample signal drops several orders of magnitude. After this frequency the real part of the refractive index is more similar for the two methods. The imaginary part of the refractive index is similar for the two methods since the absorption becomes dominant for higher frequencies where Lorentzian time-frequency filtering is advantageous. There are peaks in the imaginary part of the refractive index at 2.30, 2.64, 3.01 and 3.70 THz. These peaks are probably caused by phonon absorption.

7.1.4 Germanium, 1 mm The Germanium pulses have not undergone much dispersion as can be seen in figure 57. The higher frequencies have not been shifted to later times. There is some ringing at around 1 ps probably caused by cross term interference since it is not present in the continuous wavelet transform spectrogram. The Germanium main pulse is similar in the Wigner-Ville representations with the Wigner-Ville representation of the Silicon main pulse. The Ger- manium sample spectrum shows a clear dip at 11.79 THz which could be caused by phonon absorption however this resonance cannot be verified due to the low spectral intensity partly caused by the low efficiency in the GaP. There is also a smaller dip at 7.80 THz. The real part of the refractive index computed with zero-padding and Lorentzian time-frequency filtering never converges on each other. The imaginary part however agrees well for the two methods. The time-frequency filtered κ is easier to analyse than the zero-padded. From κ it looks like there are three resonances in Germanium, at 7.80, 9.36 and 11.79 THz. These peaks could be caused by phonon absorption in Germanium however they are hard due verify due to the low spectral intensity partially caused by the low GaP efficiency at high frequencies.

7.2 Reflection spectroscopy A difficulty in the analysis of the reflection spectroscopy data are the short time scans. A shorter time sweep give a lower spectral resolution hence a lower resolution of the characterization. Therefore techniques that artificially enhance the spectral resolution such as zero-padding is especially important for reflection spectroscopy. The phase sensitivity of THz reflection spectroscopy is a big issue that must be resolved in order to confidently characterize materials in reflection.

7.2.1 Potassium titanyl phosphate, KT iOP O4 The Wigner-Ville distribution of the KTP reflection signal show that the main pulse have not undergone significant dispersion which is to be expected from a reflection measurement. This can be seen as the higher frequencies are not shifted to later times. There are some

– 76 – ringing behaviour in the Wigner-Ville distribution probably caused by cross term interference since the ringing is not present in the continuous wavelet transform spectrogram of the reflection signal. In the continuous wavelet transform spectrogram of the unprocessed reflection signal there is an odd intensity profile in the time frequency region t [8, 10] ∈ ps, ν [2, 8] THz. It is not known what this intensity profile comes from. However after ∈ the Lorentzian time-frequency windowing is applied this region is dampened. There are no back-reflections in the KTP signal which would imply that Lorentzian time-frequency filtering is ill-suited for the reflection signals. Zero-padding however should be well suited for the processing of the reflection signals since the signal is cut around the main pulse and zero padding inserts artificial points when computing the Fourier transform. However, the spectrums of the Lorentzian time-frequency filtered sample and reference signal show, surprisingly, that when compared to the zero-padded spectrums the oscillatory behaviour is significantly reduced. The graphs of the real and imaginary part of the refractive index show that there is a significant improvement to the chraracterization for the Lorentzian time-frequency filtered signal as compared to the zero-padded n and κ. The oscillations in n and κ computed from the zero-padded spectrums are very large and make it difficult to draw any conclusions from the characteriaztion. Unexpectedly the Lorentzian time- frequency filtering improves the characterization even though there are no back-reflections in the signal. In the figure for the imaginary part of the refractive index some clear resonance peaks can be identified below 8 THz. These peaks are located at 2.43, 3.04, 3.48 and 4.78 THz. The cause of these resonances is likely phonon absorption in KTP.

8 Conclusion

The Lorentzian time-frequency filtering significantly simplifies the analysis of the characterization of THz-TDS data. The Lorentzian time-frequency filtered n and κ behaves like an average to the n and κ computed from the zero-padded spectrums, being more stable and easier to identify possible absorption peaks in κ and Lorentzian transitions in n. From the analysis of HRFZ silicon a phonon mode can be reported at 9.86 THz however due to the low spectral intensity partly caused by the low efficiency of the GaP crystal this phonon mode cannot be verified. The imaginary part of the refractive index for GaP show a resonance at around 9.24 THz however due to the low spectral intensity this resonance cannot be verified. The analysis of KTP in transmission shows resonances at 2.30, 2.64, 3.01 and 3.70 THz. From the analysis of Germanium there seem to be phonon absorption at 7.80, 9.36 and 11.79 THz since the spectral intensity is several orders of magnitude below the peak intensity these peaks cannot be verified. From the analysis of KTP in reflection five phonon absorption peaks were identified at 2.43, 3.04, 3.48, 4.78 and at 7.39 THz. The Wigner-Ville distribution is good for representing the main pulse of the THz-TDS signal since it offers both high temporal and spectral resolution. It is however poor at representing the entire pulse train due to cross term induced interference that a quadratic time-frequency representation always suffer from. The continuous wavelet transform is good for both representation of the main pulse and the entire pulse train. It is good at representing the entire THz-TDS signal since it is a linear time-frequency representation lacking of cross-term interference.

– 77 – The short-time Fourier transform has a straight forward relationship between temporal and spectral resolution. The comparison between a pure temporal Lorentzian masking and a time-frequency dependent mask showed that for the specific choices of masks there were not much difference in n and κ for the two masks. However, by solving the optimisation problem of the refractive index the tuning parameter could be fixed such that it produces a converging real part of the refractive index. Therefore it is recommended to use general Lorentzian mask for time-frequency removal of back-reflections in a THz-TDS signal.

8.1 Further work

The masking performed of the continuous wavelet transform in this thesis could possibly be improved by investigating more time-frequency masks. For example, temporal Airy masks could be investigated. Especially masks on the form m(ν, t) = Airy(cνt2) would be interesting. It would also be interesting to investigate masks that follow the chirp of the main pulse such that for highly dispersive samples more of the main pulse is captured. The continuous wavelet parameters has in this text been chosen to γ = 1 and β = 1. These have been chosen since the mother wavelet then is similar to the THz single cycle pulse. An interesting point is that the characterization vary slightly when the wavelet parameters are varied for the time-frequency mask processing, the characterization do not however vary for temporal windowing. It would be interesting to find an explanation to this phenomena. A deeper analysis of the intrinsic chirp of the main THz pulse could reveal a solution to the difficulty of the phase sensitivity of the reflection spectroscopy as well as explain the temporal Gouy phase shift.

Acknowledgments

Thanks to Professor Ulf Österberg for his enthusiasm and competence in the topic of this thesis. It is diﬃcult to imagine a better supervisor for a master thesis. Thanks to Mr. Kjell Mølster for his helpfulness in learning the laser system at NTNU, the MATLAB scripts that Mr. Mølster has written, to Mr. Mølster for his excellent thesis [1] of which this thesis is based and for interesting discussions on quantum optics. Thanks to Dr. Marcus Björk for his help in the signal processing part of this project. His input on the cross-term interference of the Wigner-Ville distribution was an important contribution to this thesis. Thanks to Professor Bengt Carlsson for good feedback on the ﬁrst draft of the thesis. Thanks to NORDTEK for sponsoring a part of my stay in Trondheim. Finally a special thanks to my wife Gjertrud Louise Langaas.

References

[1] Mølster, Kjell, 2017, “THz time domain spectroscopy of materials in reﬂection and transmission”, Norwegian University of Science and Technology, Department of Electronic Systems [2] Cohen, Leon, 1995, “Time-frequency analysis”, Prentice Hall PTR

– 78 – [3] Qiang Lin, Jian Zheng, Jianming Dai, I-Chen Ho, and X.-C. Zhang, 2010, “Intrinsic chirp of single-cycle pulses”, Physical review A, Vol. 81 Iss. 4 [4] Mandel, Leonard and Wolf, Emil, 1995, “Optical Coherence and Quantum Optics”, Cambridge University Press [5] Diels, Jean-Claude, 2006, “Ultrashort Laser Pulse Phenomena”, 2nd edition, Academic press [6] Feng, S. and Winful, H. G, 2001, “Physical origin of the Gouy phase shift”, Optics Letters Vol. 26, Issue 8, pp. 485-487 [7] Schleich, Wolfgang P., 2001, “Quantum optics in phase space”, Wiley, VCH Verlag Berlin GmbH, Berlin [8] T. I. Oh, Y. S. You, K. Y. Kim, 2012, “Two-dimensional plasma current and optimized terahertz generation in two-color photoionization”, Opt. Express 20, 19778-19786 [9] http://photonicswiki.org/index.php?title=File:FSEOS_of_THz.png [10] A.J. Jerri, Nov. 1977, “The Shannon sampling theorem-Its various extensions and applications: A tutorial review”, Proceedings of the IEEE, Volume: 65, Issue: 11 [11] Hlawatsch, Franz and Auger, François, Jan. 2010, “Time-Frequency Analysis: Concepts and Methods”, John Wiley and Sons, Inc. [12] Vineet Kumar, Kashinath Murmu, “Chapter 3: The Uncertainty Principle and Time-Bandwidth Product”, https://www.ee.iitb.ac.in/uma/~pawar/Wavelet%20Applications/Chapters_review/ [13] Hillery, M., O’Connell, R. F., Scully, M. O., and Wigner, E. P., 1984 “Distribution functions in physics: Fundamentals”, Physics Reports, Volume 106, Issue 3, Pages 121-167 [14] Royer, A., 1977, “Wigner function as the expectation value of a parity operator”, Physical Review A, Volume 15, Number 2 [15] Lilly, J. M., & Olhede, S. C., 2012, “Generalized Morse wavelets as a superfamily of analytic wavelets”, IEEE Transactions on Signal Processing, 60 (11), 6036-6041. [16] Uhd Jepsen, Peter and Fischer, Bernd M., 2005, “Dynamic range in terahertz time-domain transmission and reﬂection spectroscopy”. Opt. Lett., 30(1):29-31 [17] Jepsen, Peter Uhd, et. al., 20122 “Terahertz spectroscopy and imaging - Modern techniques and applications”, Laser Photonics Rev. 5, No. 1, 124-166 [18] Lee, Yun-Shik, 2009, “Principles of Terahertz Science and Technology”, Springer Science+Business Media, LLC [19] B. Wu, L. Cao, Q. Fu, P. Tan and Y. Xiong, 2014, “Comparison of the Detection Performance of Three Nonlinear Crystals for the Electro-optic Sampling of a FEL-THz Source,”, Proceedings, 5th International Particle Accelerator Conference (IPAC 2014) : Dresden, Germany, June 15-20

A Appendix: Properties of the Wigner-Ville distribution

The Wigner-Ville distribution fulﬁls a list of properties which are of interest in time frequency analysis [2].

– 79 – Property 1. Conservation of instantaneous power and spectral energy density The time and frequency marginal is obtained by integration over the Wigner-Ville distribution with respect to frequency and time respectively.

x(t) 2 = W (t, ν)dν | | x ZR X(ν) 2 = W (t, ν)dt | | x ZR Property 2. Conservation of energy The total energy of the signal is obtained by integration of the Wigner-Ville distribution with respect to both time and frequency.

2 2 = dνdt Wx(t, ν) = x(t) dt = X(ν) dν E 2 | | | | ZR ZR ZR Property 3. Invertibility The amplitude and the phase of the signal or spectrum can be obtained from the following weighted integral over frequency or time.

t i2πνt W , ν e dν = x(t)x∗(0) x 2 ZR ν i2πνt Wx t, e dt = X(ν)X∗(0) R 2 Z Property 4. Conservation of instantaneous frequency 1 1 dν νW (t, ν) = ∂ (arg(x(t))) x(t) 2 x 2π t | | ZR Property 5. Conservation of group delay 1 1 dt tW (t, ν) = ∂ (arg(X(ν))) X(ν) 2 x −2π t | | ZR Property 6. Moment

n n 2 dtdν t Wx(t, ν) = t x(t) dt 2 | | ZR ZR n n 2 dtdν ν Wx(t, ν) = ν x(t) dt 2 | | ZR ZR Property 7. Reality The Wigner-Ville distribution is a real distribution.

Wx∗(t, ν) = Wx(t, ν)

Property 8. Conservation of signal support If the signal is zero in a domain the correspoding region in the spectrogram will be zero.

x(t) = 0, t W (t, ν) = 0, t ∈ D ⇒ x ∈ D

– 80 – Property 9. Multiplication If the signal can be factored on on the form

y(t) = x(t)h(t) then the corresponding Wigner-Ville distribution is given by the following convolution in frequency.

W (t, ν) = W (t, ρ)W (t, ν ρ)dρ y x h − ZR Property 10. Convolution If the signal is a convolution of two signals

y(t) = x(t τ)h(τ)dτ − ZR then the corresponding Wigner-Ville distribution is given by the following convolution in time.

W (t, ν) = W (ρ, ν)W (t ρ, ν)dρ y x h − ZR Property 11. Correlation If the signal is given by

y(t) = x(t + τ)h∗(τ)dτ ZR then

W (t, ω) = W (ρ, ω)W ( t + ρ, ω)dρ y x h − ZR Property 12. Covariance under time translations If the signal is time shifted by t0

y(t) = x(t t ) − 0 then the corresponding Wigner-Ville distribution is also timeshifted by t0.

W (t, ν) = W (t t , ν) y x − 0 Property 13. Covariance under frequency translations If the signal is modulated by ν0

y(t) = ei2πν0tx(t) then the corresponding Wigner-Ville distribution is frequency shifted by ν0

W (t, ν) = W (t, ν ν ) y x − 0

– 81 – Property 14. Covariance under scale transformations If the signal is scaled by

y(t) = √ax(at), a > 0 then the Wigner-Ville distribution is scaled accordingly ν W (t, ν) = W at, y x a Property 15. Covariance under time inversion If the signal is time inverted

y(t) = x( t) − then the Wigner-Ville distribution transforms as.

W (t, ν) = W ( t, ν) y x − − Property 16. Covariance under complex conjugation If the signal is the complex conjugate of another signal.

y(t) = x∗(t) then the Wigner-Ville distribution transforms as.

W (t, ν) = W (t, ν) y x − Property 17. Conservation of scalar product

2 dtdν Wx(t, ν)Wy∗(t, ν) = dt x(t)y∗(t) 2 ZR ZR

B Appendix: Geometrical interpretation of the Wigner-Ville distribution

This subsection follows from [14]. Consider the Wigner function in quantum mechanics in position representation.

2 2ips f(r, p) = ds e− ~ ψ∗(r s)ψ(r + s) (B.1) h − Z Or in momentum representation

2 2ikr = dk e− ~ φ∗(p + k)φ(p k) (B.2) h − Z Moyal was the ﬁrst to realized that this could be expanded into a more meaningful form.

2 i(kr+sp)/~ i(krˆ+spˆ)/~ f(r, p) = h− dkds e− ψ e− ψ (B.3) h | | i Z

– 82 – In this representation the Wigner-function can be viewed as the two-dimensional Fourier i(krˆ+spˆ)/~ transform of the expectation value of the operator e− . However the interpretation of the Wigner-function can be extended even more. In fact it turns out that the Wigner- function is the expectation value of the parity operator around a phase space point. Consider deﬁning the operator Πrp as follows

2ips/~ Π ds e− r s r + s (B.4) rp ≡ | − i h | Z 2ikr/~ = dk e− p + k p k (B.5) | i h − | Z 1 i (k(ˆr r)+s(ˆp p)) t = dkds e ~ − − (B.6) 2h Z It is then possible to write the Wigner-function as. 2 f(r, p) = ψ Πrp ψ (B.7) ~ h | | i In this representation clearly the Wigner function is the expectation value of the operator Πrp. We can now ask ourselves the question the nature of the operator Πrp. At the phase space origin r = p = 0 the Πrp becomes.

Π = ds s s (B.8) 00 |− i h | Z = dk k k (B.9) | i h− | Z 1 i (krˆ+spˆ) = dkds e ~ (B.10) 2h Z From equation (B.8) and (B.9) it is clear that the operator Π00 is the Parity operator. The parity operator is deﬁned as. P x = x (B.11) | i |− i P p = p (B.12) | i |− i For a space or position operator the parity operator inverts space. From the operator Π00 we can construct the operator Πrp as follows.

Π = ( (r) (p))†Π (r) (p) (B.13) rp T T 00T T where (r) and (p) is the unitary position and momentum translation operators deﬁned T T as follows. irpˆ (r) = e− ~ (B.14) T iprˆ (p) = e ~ (B.15) T This means that to construct the operator Πrp we make a phase space translation of the parity operator Π00 = P to the phase coordinates (r, p). Now under the correspondence principle we can go to the limit of classic electromagnetic signals by the identities speciﬁed in section5, the Wigner-function becomes the Wigner-Ville distribution. W (t, ν) = 2 x Π x (B.16) x h | tν | i

– 83 – where the time and frequency translated Parity operator is deﬁned as

Π = ( (t) (ν))†P (t) (ν) (B.17) tν T T T T And the time and frequency translation operators acts as follows on a time and frequency eigenket.

(t) t0 = t0 + t (B.18) T | i | i (ν) ν0 = ν0 + ν (B.19) T | i | i And with the action of the parity operator deﬁned as.

P t0 = t0 (B.20) | i |− i P ν0 = ν0 (B.21) | i |− i

– 84 –