A WIDE BANDWIDTH SOLIEL-BABINET COMPENSATOR FOR TERAHERTZ SPECTROSCOPY KYRUS KUPLICKI. Bachelor of Science in Engineering Physics. University of Tulsa

A WIDE BANDWIDTH SOLIEL-BABINET COMPENSATOR FOR TERAHERTZ SPECTROSCOPY By KYRUS KUPLICKI Bachelor of Science in Engineering Physics University of Tulsa Tulsa, Oklahoma 2003 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE December, 2008

A WIDE BANDWIDTH SOLIEL-BABINET COMPENSATOR FOR TERAHERTZ SPECTROSCOPY Thesis Approved: Dr. Alan Cheville Thesis Adviser Dr. Charles Bunting Dr. Paul Westhaus Dr. A. Gordon Emslie Dean of the Graduate College ii

ACKNOWLEDGMENTS I would like o first thank Dr. Cheville for being my advisor and for the insight he has given me over the years. I would like to thank both Dr. Reiten and Nick Oswald for the many conversations over my thesis topic. To Dr. Reiten thank you for satisfying my inquisitive nature because know one can gain vast amounts of knowledge without asking many questions. I would like to thank Dr. Westhaus and the Physics Department at Oklahoma State University for the financial support I have been given over the years. Thank you Dr. Westhaus for being on my committee and may you have a happy and long retirement. Lastly I would like to thank my mother for your support over the many years and believing in my ability grow and prosper on my own. iii

TABLE OF CONTENTS Chapter Page I. INTRODUCTION...1 1.1 Evolution of THz-TDS...1 1.2 Phase and Polarization...3 1.2-1 Known use of Polarization and Phase measurements...5 II. THE TRADITIONAL THZ-TDS SYSTEM...7 2.1 The Standard THz-TDS setup...7 2.2 The polarization and phase of a standard THz-TDS system...9 2.3 THz-TDS data collection and analysis...15 III. POLARIZATION AND PHASE MANIPULATION TECHNIQUES...18 3.1 Manipulation of Phase and Polarization...20 3.2 The Wide Bandwidth Soleil-Babinet Compensator...24 3.3 Jones Matrix Theory...26 3.4 Jones Matix for the Wide Bandwidth Soleil-Babinet Compensator...29 3.4-1 Vertical grating state...30 3.4-2 Horizontal grating state...30 3.4-3 Angled grating state...30 iv

Chapter Page IV. EXPERIMENTAL TECHNIQUES...33 4.1 Experimental setup with the Wide Bandwidth Soleil-Babinet Compensator. 33 4.2 Gap Induced reflections between the silicon prism and the Wide Bandwidth Soleil-Babinet Compensator...35 4.3 Alignment Techniques...39 V. EXPERIMENTAL RESULTS...46 VI. CONCLUSION...53 REFERENCES...55 v

LIST OF TABLES Table Page 1-1 Requirements of the three different classifications for an electric field...5 4-1 Special cases of a Lissajou curve...41 vi

LIST OF FIGURES Figure Page 1-1 Three different cases for a standing electric field...4 2-1 The standard THz-TDS system...8 2-2 The DC status of a Thz transmitter...10 2-3 Accelerating electron/hole pair depiction...11 2-4 A side view of a Thz transmitter with a silicon lens...12 2-5 Receiver transmission lines schematic...14 3-1 Depiction of a standing electric field with zero phase difference...19 3-2 Soliel-Babinet compensator design...23 3-3 Depiction of the WB-SBC wafer...24 3-4 An illustration of the WB-SBC attached to the silicon prism...25 3-5 A lab and optic reference frame...28 3-6 An arbitrary grating orientation and the accompanying φ...31 4-1 Experimental setup...34 4-2 A measured experimental pulse...36 4-3 Theoretical intensity transmitted as a function of gap distance...37 4-4 The frequency spectrum with and without numerically attenuating the reflections...38 4-5 Phase shift as a function of frequency with and without numerically attenuating the reflections...39 4-6 A theoretical pulse...42 4-7 A plot of figure 4-6 vs. figure 4-6 with a view along the time axis...43 4-8 Another theoretical pulse...44 4-9 A plot of figure 4-6 vs. figure 4-8 with a view along the time axis...44 5-1 Induced phase difference cause by the WB-SBC...47 vii

Figure Page 5-2 Induced phase shift on the S polarization caused by the WB-SBC...48 5-2 Induced phase shift on the P polarization caused by the WB-SBC...49 5-4 S polarization amplitude caused by the WB-SBC...50 5-5 P polarization amplitude caused by the WB-SBC...51 6-1 P Induced phase difference and electric field ratio...54 viii

CHAPTER I INTRODUCTION 1.1 Background of THz-TDS In 1986 Ketchen et. all published a paper on creating subpicosecond electrical pulses on a coplanar transmission line [1]. The electrical pulses were created by photoconductively shorting a charged coplanar transmission line. The electrical pulses traveled down the coplanar transmission line to an electrical sampling probe. The photoconductive nature of the sampling probe made it possible to see how the electrical pulses broadened as a function of transmission distance. This basic system was the basis for a spectroscopy system. Sprik [2] placed erbium iron garnet (ErIG) over the transmission lines and compared the resulting electrical pulse having traveled down the transmission lines with and without the ErIG on the transmission lines [2]. After the spectroscopy measurements involving ErIG the strong frequency-dependent loss of energy due to Cherenkov radiation was measured [3]. Cherenkov radiation occurs when a subpicosecond electrical pulse travels over the transmission lines faster than the phase velocity of the underlying dielectric. It was around this time that it was suggested that there might be freely propagating radiation due to the rapidly accelerating electrons in the photoconductive switch. An investigation into this led to the generation and detection 1

of freely propagating electrical pulses [4]. A spherical mirror was placed over the excitation spot on the transmission lines so any freely propagating field would return to the transmission lines and it could be measured using methods in [1]. Following the success of observing a freely propagating field in [4] the next step was to have the freely propagating field travel through space and collect the field at a receiver [5]; the setup was observed to be linear. A THz radiation detector was placed in front of the THz radiation source. A truncated sapphire sphere was placed over the THz source to direct the THz radiation into a cone. A truncated sapphire sphere was also placed over the detector to collect the propagated THz radiation and to focus it onto the detector. To increase the measured THz radiation two parabolic mirrors were added to the system in order to increase the confocality of the system [6]. The Hertzian dipole nature of both the THz transmitter and receiver work best when they are in a confocal system. With the basic THz-TDS system seen in [6] most of the work from then on is using the THz as a spectroscopy system. A classic example is the identification of the absorption lines for water vapor [7]. The THz-TDS system correctly identified all of the absorption lines due to water vapor between the frequencies of 0.5 THz and 1.5 THz. With the creation of the standard THz-TDS system in [7] there are two categories of THz-TDS work: improvement of the standard THz-TDS system, or using the standard THz-TDS system for making experimental measurements. Shortly after the investigation into the THz absorption lines of water vapor [7], there was an investigation into pulse reshaping due to total internal reflection (TIR) [8]. A crystalline quartz right-angle prism was used to create the TIR. There was a frequency independent phase shift, due to TIR, along with a frequency dependent phase shift due to 2

the THz pulse propagating through the crystalline quartz. Aside from [8] there has not been much investigation into manipulating THz pulses for spectroscopy measurements until recently. Masson and Gallot use a stack of 6 quartz plates to create an achromatic quarter-wave plate [9]. Their plate creates a π/2 phase shift between the THz frequencies of 0.25 and 1.75 THz. The other recent case of phase manipulation is by Hirota et. all where they use a silicon prism and TIR to achieve a phase shift of π/2 between two orthogonal polarizations of a propagating THz pulse [10], resulting in a circularly propagating THz pulse. 1.2 Phase and Polarization A THz pulse in the far field is well approximated by propagating transverse electromagnetic (TEM) pulse. This means that the propagating electric field is in the plane perpendicular to the direction of the traveling pulse. The electric field can be broken into two orthogonal components as seen in figure 1-1 [11]. 3

Figure 1-1 The three different classifications for an electric field. The three different classifications are: a linear field, a circular field and an elliptical field. The two orthogonal components are often referred to as S polarization and P polarization. Each orthogonal component has two values associated with it, which are phase and field amplitude. Both polarizations have oscillating field amplitude as a function of position and time. For a given position in space the electric field amplitude and field direction is represented in figure 1-1 for three different cases. Each case represents one of the three classifications that can be given to an electric field. The classification of an electric field is based on the values of phase and amplitude for the two polarizations. Table 1-1 shows the requirements for each classification. 4

Linear Phase Difference Amplitude Ratio ES φdiff = mπ E mπ ES Circular φdiff = = 1 2 EP ES ES Elliptical m Integer < < 0 or 0 < < EP EP Table 1-1: A table describing the requirements of the three different classifications for an electric field. P When the phase difference between the two polarizations is an integer multiple of mπ the field is considered a linear field or if one polarization has amplitude equal to zero. The circular classification occurs when the phase difference is an integer multiple of mπ and 2 the two polarizations have the same amplitude. An elliptical classification is everything that is not considered a linear or circular field. The standard THz-TDS system only makes use of linearly propagating THz pulses. In order to do phase dependent absorption analysis a device must be placed into the THz-TDS system. An example of phase dependent absorption analysis is reviewed in the next section. Currently there is not a good way to manipulate the phase values of a THz pulse due to the wide bandwidth nature of a THz pulse. 1.2-1 Known use of Polarization and Phase measurements The uses of polarization and phase dependent spectroscopy measurements have been used in chemistry for chiral materials since the early 1800 s. Chiral molecules have phase dependent absorption. Jean-Baptiste Biot first observed the optical property of chiral materials in 1815 [12]. It was however Louis Pasteur in 1848 that he deduced that 5

the optical property of chiral materials was molecular in nature [13]. J. C. Bose found artificial composite materials in 1898 that were chiral for electric fields in the microwave region [14]. Along with chiral spectroscopy there is vibrational circular dichroism (VCD) spectroscopy. VCD spectroscopy is based on the circular dicroism of biological materials in the vibrational frequency range [15]. These biological materials exhibit phase dependent vibration with them. Being able to manipulate the phase of a THz pulse will allow for THz-TDS to do chiral and VCD based measurements. This thesis reports on a device that will allow a THz-TDS system to make such measurements. Chapter two describes a standard THz-TDS setup, how each piece in a standard THz-TDS system effects the phase and polarization of a THz pulse and how THz-TDS data is analyzed. Chapter three is a mathematical explanation of a propagating THz pulse, ways in which to affect a propagating THz pulse, a description of the wide bandwidth Soliel-Babinet compensator (WB-SBC), and the Jones matrix theory used to describe the WB-SBC. Chapter four is a description of the experimental setup, the ramifications of the experimental setup used, and the alignment procedures used. Chapter five contains the experimental results and a comparison between the experimental results and what is predicted by the Jones matrix theory. 6

CHAPTER II THE TRADITIONAL THZ-TDS SYSTEM This chapter describes the traditional terahertz time domain spectroscopy system. This chapter is divided into three parts. The first part, section 2.1, describes the layout. The second part, section 2.2, describes the polarization and phase of a transmitted terahertz pulse through the traditional terahertz time domain spectroscopy system. The last part, section 2.3, describes the how data is taken and analyzed in the traditional terahertz time domain spectroscopy system. 2.1 The Standard THz-TDS setup This section describes the layout of the traditional Terahertz Time-Domain Spectroscopy (THz-TDS) system. The traditional THz-TDS system can be seen in figure 2-1 [6]. 7

Figure 2-1 An experimental setup of a standard THz-TDS system. The laser is typically a mode locked Ti-sapphire laser that has a repetition rate of about 85MHz. The operating center wavelength of the laser is around 810nm. The emitted pulses from the laser are first incident on the beam splitter. Half of the pulse travels on to the transmitter while the other half of the pulse travels through an optic delay line and onto the receiver. The transmitter is comprised of two 10µm wide aluminum lines spaced 80µm apart on a gallium arsenide (GaAs) substrate. A voltage is applied across the two aluminum transmission lines of about 80V. The applied voltage is slightly less that the ionization break down voltage of air. The incident optical pulse from the femto-second laser creates electron-hole pairs in the GaAs substrate. The free electrons (and to a lesser degree, holes) undergo ballistic acceleration. Terahertz radiation is emitted due to the ballistic acceleration. On the other side of the GaAs substrate is a truncated high resistivity silicon lens. The silicon lens is used to direct the terahertz radiation to the first parabolic mirror in figure 2-1. This parabolic mirror is used to collimate the terahertz pulse. Once collimated, the terahertz pulse interacts with the sample material, which is placed in the center of the terahertz system. After passing 8

through the sample the terahertz pulse is then incident on the second parabolic mirror. This mirror is used to re-image the terahertz pulse onto the receiver. To further focus down the terahertz pulse on the receiver dipole a silicon lens is placed on the receiver substrate. The receiver is comprised of two 10µm wide aluminum transmission lines spaced 50µm apart with two extruding aluminum blocks between the transmission lines. The two transmission lines are on a highly ion implanted silicon on sapphire substrate. In order to achieve proper alignment by the transmitter and receiver they are both housed on a custom fabricated x-y-z translation stage. The optically delayed pulse from the femtosecond laser frees electrons in the silicon on sapphire substrate in between the two transmission lines, while the incident terahertz pulse creates an electric field, which enables the freed electrons to create a current between the transmission lines. A current amplifier then amplifies this current. The resulting amplified current is proportional to the electric field amplitude of the terahertz pulse. 2.2 The polarization and phase of a standard THz-TDS system Most of the useful information that is used in the traditional THz-TDS system is contained in the frequency spectrum. The frequency spectrum data is used to determine the absorption spectrum for materials that absorb electromagnetic radiation within the THz spectrum [7] and to determine index of refraction as a function of frequency [16]. The THz-TDS system can also be used to determine the phase difference between a sample scan and a reference scan as a function of frequency [8]. To use the polarization and phase information more effectively an analysis needs to be done on the THz-TDS system to see how each piece affects the polarization and phase of a terahertz pulse. The 9

THz-TDS system for the most part only cares about the orientation of the transmitter and receiver; this will be explained in detail later. The THz-TDS system is designed to compare the difference between a reference scan and a sample scan. A scan is a measurement of the electric field amplitude over a given optic delay duration. There is a wide range of spectroscopy that involves the phase and polarization measurements. The traditional terahertz spectroscopy setup is ill equipped for measuring polarization unless certain things are done. The following is a description of how each piece in the traditional terahertz spectroscopy system affects the phase and polarization of an emitted terahertz signal. The terahertz transmitter emits a transverse electromagnetic pulse in a dipole radiation pattern. A depiction of this can be seen in figure 2-2 and figure 2-3. Figure 2-2 The DC status of a THz transmitter. A 2D representation of the initial state of the electric field on the transmitter before an incident optical pulse. 10

Figure 2-3 The immediate direction that the electrons and holes inside the incident optical pulse (dashed circle) due to the initial state of the electric field shown in figure 2-2. Figure 2-2 shows an established electric field in between the two transmission lines. An optical pulse from the femto-second laser irradiates a spot between the two transmission lines. The irradiated spot creates electron-hole pairs due to the semiconductor absorbing energy from the irradiating optical pulse. The freed electrons and holes undergo ballistic acceleration toward their respective transmission line. This ballistic acceleration, which is along the x-direction as seen in figure 2-3, creates a terahertz pulse. The pulse travels in the positive and negative z direction with the field polarization in the same direction as the electrons accelerate, the x-direction. The far field radiation of the emitted terahertz pulse is a dipole radiation pattern. To summarize, the emitted terahertz pulse is linearly polarized in the x-direction and travels in the positive and negative z-direction. The first optic that can affect the phase and polarization of an emitted terahertz pulse is the boundary between the silicon lens and air, which is illustrated in figure 2-4. 11

Figure 2-4 Side view of the Transmission wafer and the silicon lens. When the terahertz pulse reaches the silicon/air boundary the boundary conditions for a dielectric/air boundary must be met before an external electric field can be transmitted from the incident electric field. Fulfillment of these boundary conditions can be described by the use of Fresnel transmission coefficients. These transmission coefficients vary as a function of incident angle and polarization. Though the transmission coefficient for a given incident angle will not induce a change of phase on the S or P polarization, the variance in transmission over the surface of the silicon lens will cause pulse reshaping. The pulse reshaping itself is what causes a change in phasethe phase change actually causes the pulse reshaping. An in depth investigation concerning pulse reshaping due to the silicon lens has been investigated by Reiten et. all 12

[17]. It was discovered that the spatial profile of the electric field increased with complexity with increasing frequency [17]. The lower THz frequencies retained a simple Gaussian structure. The phase corresponding to the spatial resolved electric field was also frequency dependant. According to Reiten et. all the higher frequencies exhibited an increasing number of ring structures within the spatial profile [17]. The overall phase as a function of spatial position did not vary more than + 7 for any frequency. In 1975 Gans published a paper regarding the amount of cross polarization due to a gaussian beam reflecting from curved surfaces [18]. Inside the paper Gans used a paraxial ray approximation to derive the maximum amount of cross-polarized signal ratio for a gaussian beam reflecting from a parabolic mirror. It was found to be a function of the incident waist radius (w), the focal length of the parabolic mirror (F), the 10-dB half angle of the incident gaussian beam (θ c ), and the off set angle of the center beam (θ o ). A calculation of the maximum amount of cross-polarized signal ratio (C max ) can be found from equations 2.1 and 2.2 [18]. ( ) ( θ 2) w ln 10 θ c = (2.1) 2 F sec o C max ( 2) ( ) θ c tan θ = 0 (2.2) e ln 10 It is calculated that for a waist radius of 10mm a focal length of 119.4mm and an off set angle of 90 degrees the 10-dB half angle is 0.0635. The corresponding maximum amount of cross-polarized signal ratio is 0.0167. The amount of cross polarization does vary as a function of frequency because the lower THz frequencies have a larger waist radius incident upon the parabolic mirror. A waist radius of 40mm,which corresponds to a lower THz frequency, has a maximum amount of cross-polarized signal ratio of 0.0670. 13

The amount of cross polarization caused by one of the parabolic mirrors varies as a function of frequency but the maximum is less than 10 percent of the incident electric field. The silicon lens on the receiver affects a terahertz pulse, which is described by the Fresnel transmission coefficient. The receiver lens does not induce a change of phase on either polarization and it does not turn any part of one polarization into the other polarization. It however does decrease the amplitude of the transmitted field as a function of incident angle and polarization. The silicon lens on the receiver will decrease the x-polarized field, induce no change in phase and the y-oriented field will still be zero. The receiver will only indicate there is a signal if the incoming terahertz pulse is oriented in the correct direction. Figure 2-5 shows a schematic of the receiver. Figure 2-5 A schematic of the receiver transmission lines. If the incoming field is not oriented in the x' direction some of the incoming signal is lost. A propagating terahertz field polarized in the y' direction will cause freed 14

electrons to accelerate in the y' direction. If the receiver could be described as being a perfect dipole oriented along the x' axis then no signal would be measured; however it was discovered by Rudd that the receiver is actually quadrupolar and not dipolar in nature [19]. Instead of no signal being measured in the case of a dipole, with a y' oriented field, a severely reduced signal will be measured in the case of a quadrupole. A field oriented between x' and y' will create some current. Given a perfectly aligned terahertz system items within the basic THz system will cause a change in phase and or polarization. The change in phase and or polarization caused by a sample can be determined with great accuracy because we are comparing a THz pulse with and without a sample. When there is not good alignment there is a propagation dependant change in phase and polarization factor that cannot be taken out through a comparison analysis. Thus it is imperative to achieve good alignment. Such alignment techniques are described in Chapter 5. 2.3 THz-TDS data collection and analysis This section is divided into three parts: the first part describes how data is collected, the second part explains the data analysis of a traditional terahertz spectroscopy system, and the last part describes the data analysis of the phase and polarization of a terahertz pulse. Traditional data analysis is a measurement of complex refractive index as a function of frequency, which is done as follows: in order to increase the signal to noise ratio a lock-in amplifier measures the resulting electric field amplitude at the receiver as a function of optical delay; a mechanical chopper is placed inside the terahertz system right after the transmitter. The mechanical chopper is used to modulate the 15

terahertz signal going to the receiver. The resulting complex field is plotted as function of time or optic delay. A Fourier transform of the electric field as a function of time gives the complex field as a function of frequency. F ( ν ) f ( t) 2πiνt = e dt (2-1) The measurement of the complex field amplitude as a function of frequency is done for a sample scan and a reference scan. The reference scan is typically the terahertz system without the sample inside the system. The resulting absorption caused by the sample as a function of frequency is found by taking the ratio between the reference scan and the sample scan. reference ( ν ) ( ν ) Fsample F result ( ν ) = (2-2) F The extraction for the polarization and phase information is similar. A measured terahertz pulse has to be done for two orthogonal polarizations for a sample and a reference. The two polarizations are S and P, an explanation of the orientation of these polarizations can be found in chapter 3. The complex electric field amplitude as a function of frequency is obtained by performing a Fourier transform. The resulting phase shift caused by the sample on the S or P polarization is found from equation 2-3: ( ν ) ( ν ) ( ν ) ( ) ν S, P S, P E ( ) Re ference ERe ference φs, P ν = arctan Im Re (2-3) S, P S, P ESample ESample This phase shift is in reference to the S-polarized reference. The phase shift for the P polarization is found the same way except with a P polarized incident pulse. The phase shift for the P polarization will be in reference to the P-polarized reference pulse. The 16

phase shift difference between the S and P polarization is the difference between the two induced phase shifts as seen in equation 2-4. φ ( ν ) = Phaseshift ( ν ) Phaseshift ( ν ) diff. S P (2-4) This phase difference will be in reference to the two reference samples used to determine the phase shift on the S and P fields. The phase shift difference can be found in two different ways. One way is to extract the individual phase shifts with equation 2-3 and then take the difference with equation 2-4. The other way is to extract the phase shift difference immediately from the frequency spectrum sample scans. This involves the use of equation 2-5. φ diff ( ν ) ( ν ) ( ν ) ( ) ν S S E ( ) Sample ESample. ν = arctan Im Re (2-5) P P ESample ESample Without the reference scans the induced phase shift on the S or P polarization cannot be found. The total field amplitude as a function of frequency is the magnitude of the two polarizations. amp 2 2 ( ν ) E ( ν ) E ( ν ) E = + (2-6) s p 17

CHAPTER III POLARIZATION AND PHASE MANIPULATION TECHNIQUES The goal of this study is to create a device that can manipulate the phase and polarization of a terahertz pulse. This chapter is broken down into four parts. The first part is a mathematical description of a terahertz pulse. The second part describes what can affect the phase and/or polarization of a terahertz pulse. The third part is a description of a wide bandwidth Soliel-Babinet Compensator (WB-SBC) created to manipulate the phase and polarization of a terahertz pulse [20]. The fourth and last part of this chapter is a mathematical description of what the WB-SBC does to a terahertz pulse. The electric field of a traveling terahertz pulse can be shown mathematically as equation 3-1: E = P S (3-1) iα iβ i( kz ωt ) ( ν ) x E ( ν ) e + y E ( ν ) e e There are two orthogonal components called the S polarization and the P polarization. These two polarizations account for all of the electric field in a transverse electromagnetic wave. The S polarization is defined as the y component of the electric field, while the P polarization is defined as the x component of the electric field as seen in 18

figure 3-1. Figure 3-1 is a plot of an electric field with zero phase difference between the S and P polarization [11]. Figure 3-1 A plot of an electric field with zero phase difference between the S and P polarizations. The two orthogonal polarizations P and S have a phase value, that phase value is α and β respectively. It is the phase difference between α and β that is important. The phase difference describes the direction the electric field as a function of time. A phase difference of +/- mπ where m = 0,1,2, defines a linearly polarized terahertz pulse. A phase difference of π/2 +/- mπ where m = 0,1,2, is defined as a circularly polarized terahertz pulse [20]. Positive values of mπ reflect an electric field rotating in a clockwise direction while negative values of mπ reflect an electric field rotating in a counter clockwise direction. All other phase differences are indicative of an elliptically polarized terahertz pulse. 19

3.1 Manipulation of Phase and Polarization There are many ways of manipulating the phase and/or polarization of an optical pulse and they include: reflection from a metal surface [20,21], reflection from a dielectric boundary [20], transmission through a dielectric boundary [20,21], passing through a metal grating [20,21], and traveling through a birefringent material [20]. A π phase shift is induced on each polarization when an optical pulse is reflected from a metal surface. The magnitude of the field for each polarization is unaffected by the reflection from a metal surface. When traveling through or reflecting from a boundary between two different materials there are boundary conditions that need to be satisfied. There are four basic boundary conditions that need to be satisfied and they are equations 3-2 through 3-5 [22]: n ( ε E2 ε 1E1 ) = σ f 2 (3-2) ( E ) 0 n (3-3) 2 t E1 t = ( B B ) 0 n µ µ (3-4) 2 2 1 1 = ( 2 B2t µ 2B1t ) K f n µ = (3-5) where σ f is the surface density of free charge and K f is the free surface current density. These four boundary conditions state that the tangential and normal components of the electric and magnetic fields, represented by E and B respectively, across a boundary need to be continuous. Though the boundary conditions are frequency independent they are material and incident angle dependent. The material dependency is found in the two terms permittivity (ε) and permeability (µ). These two terms, ε and µ, can be frequency dependent; µ in most cases is generally regarded as being frequency independent. The 20

boundary conditions can induce a change in phase and or amplitude to an incident electromagnetic field. This change in the field can be described mathematically by the Fresnel transmission and reflection coefficients. The Fresnel coefficients can be derived from the boundary conditions [22]. The Fresnel coefficients allow one to determine the resulting electromagnetic field after an incident electromagnetic field has interacted with the boundary. The incident angle, the two dielectric materials that form the boundary, and the desired polarization are all needed to calculate the values of the Fresnel coefficients. The Fresnel coefficients can be complex and are when dealing with total internal reflection [20]. Total internal reflection is where an incident field angle is greater than the critical angle. This critical angle is described by the following equation [20,21]: 1 n2 θ = c sin (3-6) n1 This critical angle is the angle in which the entire incident field is reflected and none of the field is transmitted. This can only take place when an incident field is traveling from a material with a high index of refraction (n 1 ) into a material with a smaller index of refraction (n 2 ). The index of refraction (n) is a value based upon the relative permittivity (ε r ) and relative permeability (µ r ) values of the material. Now because n is based on the values of ε r and µ r it can be frequency dependent; it will however be assumed that µ r is 1. n = ε µ (3-7) r r Upon reflecting the incident field will incur a polarization dependant phase shift. This induced phase shifts on the S and P polarizations are not equal and lead to a phase difference between the two polarizations. 21

A birefringent material creates an induced phase shift because the two polarizations do not travel through the material with the same phase velocity. In equation 3-1 it is assumed that the phase accumulation for both polarizations of a traveling electromagnetic field is the same for any given distance. Birefringent materials modify equation 3-1 to equation 3-8 [20]. E = p s (3-8) i ( ) ( ) ( α + k ) pz ( ) i( β + x E e y E e k z ) s e i ω ν ν + ν t The amount of phase accumulation for a polarization as a function of distance is described by the ikz e polarizations is the following [20]: k is described by: term. The accumulated phase difference between the two ( β + k z) ( k z) φ (3-9) diff = s α + p k ( ω) π n = 2 (3-10) λ where λ is wavelength in a vacuum and ω is angular frequency. It should be noted that n is frequency dependant in equation 3-10. The polarization with a smaller value of k is considered the fast axis because the phase velocity is greater than the other polarization. On the other hand the polarization with the larger value of k is called the slow axis [20]. A birefringent material can be used to achieve any desired phase difference between the S and P polarization of an electromagnetic field for a given frequency. A Soliel-Babinet compensator is a birefringent material where the distance that the electric field propagates through can be varied. An example of such a design can be seen in figure 3-2. 22

Figure 3-2 Example of a Soliel-Babinet compensator design with the use of a birefringent material. A variable phase difference is the result of being able to vary the distance through the material, which is what a Soliel-Babinet compensator is. Birefringent materials are ill suited in trying to induce a phase difference between the S and P polarizations for a THz pulse for two reasons. A THz pulse has a wide bandwidth so even a small frequency dependency on the index of refraction can cause a phase difference between 0.2 and 2.0THz. To see this equation 3-10 is inserted into 3-9 to get the phase difference shown in equation 3-11. φ diff 2π ( ) z ( β α ) + n ( ω) n ( ω) = s p (3-11) λ If the index of refraction of the S polarization (n s ) and/or the index of refraction of the P polarization (n p ) do not remain constant as a function of frequency there becomes a frequency dependent phase difference. The other reason for why birefringent materials do not make for a good compensator for THz pulses is because of the dispersion term 2π λ located in equation 3-11. Birefringent materials can however be used to create a given phase difference. Masson and Gallot have used a stack of 6 quartz plates (a birefringent material) to create a quarter-wave plate for the terahertz frequencies between 23

0.5 and 1.65 [9]. The use of birefringent materials to create a Soleil-Babinet compensator would be extremely difficult for a wide range of terahertz frequencies. 3.2 The Wide Bandwidth Soleil-Babinet Compensator The WB-SBC explained within this thesis is a combination of a polarizer, reflection from a metal surface and the Fresnel reflection coefficients regarding a total internal reflection. The WB-SBC is a 3.25 diameter silicon wafer with an aluminum grating on it. The grating was placed on a silicon wafer photolithographicly. The grating is 10 µm wide with a period of 20 µm. The grating has a deposition layer of 100 nm of titanium and 500 nm of aluminum. An illustration of the grating structure on the silicon wafer can be seen in figure 3-3. Figure 3-3 A figure of the WB-SBC wafer. 24

The WB-SBC was mechanically attached to the backside of a high resistivity silicon prism using an evaporation technique. A few drops of acetone were applied to the silicon only side of the WB-SBC and then it was placed against the silicon prism until the acetone evaporated. The evaporating acetone caused suction between the silicon only side of the WB-SBC and the silicon prism. If significant pressure was not applied during the evaporative process there became an air gap in between the two surfaces. The cause of the air gap was due to the fact that the silicon only side of the WB-SBC had a polished surface while the silicon prism had an optically polished surface. The ramifications of the air gap are described in section 1 of chapter 4. The silicon prism is a 45 right triangle fabricated of >10KΩ resistivity float zone silicon. This attachment of the WB- SBC to the silicon prism is illustrated in figure 3-4. Figure 3-4 An illustration of the WB-SBC attached to the silicon prism. The vertical state of the grating and the horizontal state of the grating are also shown. As you can see in figure 3-4 an incident electric field propagating along the z-axis will pass through perpendicular to the face of the silicon prism and will then be incident on the WB-SBC. The electric field will then reflect off of the WB-SBC and then pass 25

through perpendicular to the other face of the silicon prism. The two orthogonal components of the incident electric field correspond with the x-axis and the y-axis as seen in figure 3-4. The y-axis corresponds to the S polarization state of the incoming electric field while the x-axis corresponds to the P polarization state of the incoming electric field. They were chosen as such to correspond to the orientation of the grating, which will be described next. When the grating is in a vertical state, as illustrated in figure 3-4, the electric field oriented along the S polarization is oriented in the same direction as the metal grating. The S-polarized field will cause an induced current on the metal grating and will be reflected with a π phase shift. At the same time the electric field oriented along the P polarization will not see the metal grating and will have a phase shift induced by TIR. In a horizontal grating state, as illustrated in figure 3-4, the electric field oriented along the S polarization will not see the metal grating and will have a phase shift induced by TIR. The P polarization is oriented in the same direction as the grating in the horizontal state. Thus the P polarization will see the metal grating and will be reflected with a π phase shift. Now as the WB-SBC rotates from the vertical state to the horizontal state the two polarizations, S and P, of the incident THz pulse will see some combination of the silicon and the metal grating and their induced phase will be some combination of the two. From the grating stand point the incident THz pulse can be broken down into two orthogonal components: one orthogonal component is parallel to the metal grating with the other orthogonal component perpendicular to the metal grating. A Jones matrix approach has been used to describe theoretically what happens in between and at the two states. The Jones matrix theory is described in the next section. 26

3.3 Jones Matrix Theory A Jones matrix approach is used to describe how the polarization and phase of electromagnetic radiation change due to a device inside of an optical system [23]. A 2 1 matrix describes the incident electromagnetic radiation. ES e EPe iβ iα (3-11) Both components of the 2 1 matrix can be complex. The component the phase and magnitude of the incident S polarization while the E e iβ S represents E e iα component P represents the phase and magnitude of the incident P polarization. The S and P polarizations are orthogonal components of the incident optical pulse. The magnitude of a normalized incident optical pulse is 1. The phase difference of the incident optical pulse is: φ diff iα iα E S e ES e = arctan Im Re real (3-12) iβ iβ EPe EPe A component matrix manipulates the incident electric field matrix. The component matrix is a 2 2 matrix. There are 3 basic component matrices: polarizer, retarder, and rotator. Their basic matrix configurations are the following [21]: 1 0 1 0 0 0 0 iφ e (3-13) (3-14) cos sin ( α ) sin( α ) ( α ) cos( α ) (3-15) 27

Expression 3-13 represents the component matrix for a polarizer. The orientation of a polarizer determines the electric field direction that is allowed to pass through it. A polarizer does not affect the phase of the transmitted electric field. Expression 3-14 is the component matrix for a retarder. A retarder affects the phase of the incident electric field. The magnitude of the electric field is unaffected by a retarder. Lastly expression 3-15 represents the component matrix for a rotator. A rotator is a device that rotates the plane of polarization of the incident electric field by an angle alpha. If the orthogonal components of the incident electric field do not match up with the orientation of the optical component there needs to be a coordinate transformation between the incident electric field and the optical component [21]. Figure 3-5 represents the orientation of the two orthogonal field components of an incident electric field Y and X, and the orientation of a birefringent material with fast and slow axes of Y' and X'. Figure 3-5 A lab reference frame represented by the X and Y coordinates and the optic reference frame represented by the X' and Y' coordinates. Because the orthogonal components of the incident electric field do not match up with the orientation of the birefringent material a coordinate transformation is preformed, which is referred to as the lab transfer matrices, to ensure proper overlap [21]. Expression 3-16 is an example of the use of the lab transfer matrices. 28

cos sin ( θ ) sin( θ ) ( θ ) cos( θ ) a c b cos d sin ( θ ) sin( θ ) ( θ ) cos( θ ) (3-16) An inverse coordinate transformation is done to orient the field back to the original coordinate axes. The component matrix of a polarizer, a retarder, or a rotator matrix is placed where the abcd matrix is in expression 3-16. θ represents the angle between the incident electric field orientation and the orientation of the polarizer/retarder/rotator. A multiplication of the two lab transfer matrices will result in the identity matrix. With the incident electric field defined (expression 3-11) and the component matrices defined (expressions 3-13 through 3-16) the resulting electric field can be found from equation 3-17. E E snew pnew e e iγ iξ a = c i b Ese i d E pe α β (3-17) where the abcd matrix represents any number of component matrices. The phase difference between the orthogonal fields of the new electric field is found from the use of equation 3-12. 3.4 Jones Matrix for the Wide Bandwidth Soleil-Babinet Compensator The WB-SBC can be described by using a retarder matrix placed inside the lab transfer matrices. cos sin ( θ ) sin( θ ) ( θ ) cos( θ ) 1 0 0 cos iφ e sin ( θ ) sin( θ ) ( θ ) cos( θ ) (3-17) The lab transfer matrices are used to perform a coordinate transform between the two orthogonal components of the incident THz pulse and the two orthogonal grating components, one parallel to the grating and one perpendicular to the grating. The θ in the lab transfer matrices represents the orientation of the grating. θ is 0 for a vertical grating 29

and increases as the grating rotates clockwise. φ on the other hand represents the phase shift induced by total internal reflection. It takes three different cases to describe how φ changes: the first case is for a vertically oriented grating, the second case is for a horizontally oriented grating, and the third case is for a grating orientation in between a vertical and a horizontal state. 3.4-1 Vertical grating state For a vertical grating state, as seen in figure 3-4, θ is 0 and φ is 169.2575 this corresponds to the induced phase shift due to TIR on the P polarization. Remember as stated in section 3.2 the P polarization sees silicon while the S polarization sees metal. 3.4-2 Horizontal grating state For a horizontal grating state, as seen in figure 3-4, θ is 90 and φ is 84.6287 this corresponds to the induced phase shift due to TIR on the S polarization. Remember as stated in section 3.2 the S polarization sees silicon while the P polarization sees metal. 3.4-3 Angled grating state An illustration of an angled grating state can be seen in figure 3-6. 30

Figure 3-6 An arbitrary grating orientation is shown on the left while on the right the value of φ is determined on the unit circle as an equivalent ratio between θ'/( φ S -φ P ) & θ/90. Figure 3-6 depicts the WB-SBC at some angle θ on the left and a unit circle on the right, which represents all of the possible Fresnel values regarding phase shift. φ S is the Fresnel value regarding the phase shift induced on the THz pulse when the grating is in a horizontal state (-84 ). φ P on the other hand is the Fresnel value regarding the phase shift induced on the THz pulse when the grating is in a vertical state ( 169 ). As the grating rotates from the vertical state to the horizontal state φ rotates from φ S to φ P. The total angle between φ S and φ P is 84, which will be called ψ. The total angle between the grating in a vertical state and the grating in the horizontal state is 90, which will be called χ. The grating angle θ and the θ' on the unit circle are related by the following equation: θ θ = (3-18) χ ψ 31

where χ = 90 and ψ = φ S - φ P. The ratio between θ and χ is the same ratio between θ and ψ. φ is determined from equation 3-19. θ φ = φ p + ψ (3-19) χ When θ is equal to 90 φ has a value of φ S, and a grating that is in a horizontal state. As θ goes beyond 90 φ returns to φ P from φ S therefore equation 3-19 breaks down and equation 3-20 needs to be used to determine φ. o θ 90 φ = φ s ψ (3-20) χ Once θ has reached 180 this again corresponds to a grating that is in the vertical state. Due to the grating symmetry this entire process repeats itself every 180. Chapter 5 contains the experimental results along with a comparison to the theory described above. 32

CHAPTER IV EXPERIMENTAL TECHNIQUES The focus of this chapter is on three things: the insertion of the silicon prism and the WB-SBC into the THZ-TDS system, the ramifications of the air gap between the silicon prism and the WB-SBC, and how proper alignment was achieved. 4.1 Experimental setup with the Wide Bandwidth Soleil-Babinet Compensator The silicon prism/wb-sbc was placed where an experimental sample is typically placed in a traditional terahertz system as described in chapter 2. The silicon prism was placed in the terahertz system so that the incoming terahertz pulse train was perpendicular to the face of the prism. This alignment produced an angle of 45 on the backside of the silicon prism. In order to accommodate for the change in direction of the terahertz pulse train the second parabolic mirror and receiver were reoriented so that they could receive the terahertz pulse train. The modified experimental setup of the terahertz system can be seen in figure 4-1. 33

Figure 4-1 Experimental setup. The terahertz transmitter was rotated to a -45 from the vertical position so that a terahertz pulse has equal magnitude components for the S and P polarization. The polarizer placed in front of the transmitter is also at -45 and it is there to clean up the terahertz pulse. The receiver was used to collect phase and polarization information for different grating orientations. The grating orientation ranged from vertical to horizontal and back to vertical in five-degree increments. The receiver was at a fixed orientation, which was at 45, rotated from vertical. Though the receiver was attached to a fiber optic line and could be easily rotated it was held stationary. Frequent attempts to rotate the receiver would cause it to rotate off-axis and cause an orientation dependent change in 34

phase and polarization amplitude. Therefore a polarizer analyzer was placed in front of the receiver and its orientation was set to vertical to measure the S-polarized field or it was set to horizontal to measure the P-polarized field. The receiver, positioned at 45, was equally responsive of an incoming S or P-polarized pulse. The steps to achieve proper alignment are described in the final section of this chapter. 4.2 Gap Induced reflections between the silicon prism and the Wide Bandwidth Soleil-Babinet Compensator Due to the fact that an optically polished surface was placed against a polished surface there arose a small air gap in between the two surface of the silicon prism and the WB-SBC as described in section 2 of chapter 3. The small air gap caused reflections from the THz pulse to occur. These reflections were small compared to the main pulse and were numerically attenuated out before analysis of the data was done. An example of a main pulse and the reflected pulses can be seen in figure 4-2. 35

Figure 4-2 A measured experimental pulse. The reflections due to the gap can be seen located at 10ps and 24ps. Figure 4-2 shows a main pulse located at about 16ps and two reflected pulses about one-tenth the size of the main pulse. One of the reflected pulses is located before the main pulse at about 10ps with the other reflected pulse located after the main pulse at about 24ps. An estimation of the size of the air gap can be found by applying the theory behind Frustrated Total Internal Reflection (FTIR). When an electric field is incident on a boundary beyond the critical angle an evanescent field is created on the other side of the boundary. If another material is brought in close enough to the first boundary amount of the electric field that is propagates away is a function of the distance between the two materials. Continuing the derivation found in Novotny and Hecht [24] a plot of the transmitted electric field can be seen in figure 4-3. 36

Figure 4-3 Theoretical transmitted electric field as a function of gap distance for an air gap between two pieces of silicon. From figure 4-3 it is estimated that the size of the air gap between the silicon prism and the WB-SBC is between 4.5µm and 45µm wide corresponding to the terahertz frequencies between 0.2 and 2.0 THz. The range in the gap distance arises because the transmitted electric field through the gap is a function of wavelength. Knowing that the terahertz spectrum covers a broad bandwidth an analysis was done to determine if any phase information was lost from the attenuation of the reflected pulses. First up is a look at a plot of the change in the frequency spectrum due to the attenuation of the reflected pulses. Figure 4-4 is a graph of the frequency spectrum of figure 4-2 before and after the reflected pulses were numerically attenuated. 37