thz spectroscopy and imaging

8/12/2019 THz Spectroscopy and Imaging

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50 3 THz Spectroscopy and Imaging

Fig. 3.1 Spectrum of THzpulses generated from anInAs emitter. Backgroundnoise is shown as acomparison

THz-pulse-covered bandwidth. Since the above spectroscopic measurement is car-ried out recording the THz waveform in the time-domain, this technique is calledtime-domain spectroscopy (also named TDS) [ 1]. Figure 3.1 shows the spectrum of a THz pulse compared to the background noise.

The spectral resolution of THz-TDS, , is determined by the temporal scan-ning range T . The frequency range of the spectrometer is limited by the responseof the THz source and detector, while mathematically the spectrum is signicantwithin a bandwidth , which is related to the temporal sampling interval t . The

bandwidth and spectral resolution of THz-TDS are given by

=2T

,

=2t

.(2)

When a fast Fourier transform is used, the spectrum is symmetric about =0. Therefore, the frequency range spans from 1/(2 ) to 1/(2 ). In order to havea smooth calculated THz spectrum, 0 padding technique may be applied. The 0padding method adds several 0 values on one side or both sides of the THz wave-form. Padding with 0s mathematically increases the temporal scanning range T , andthus gives more intense data points in the THz spectrum. However, padding with0s does not provide any additional information and does not improve the spectralresolution.

To measure the spectral response of a target, one should rst record the THzwaveform of a reference sample, i.e. free space. The THz waveform of the referencesample is called the reference waveform. The THz waveform transmitted throughthe target is called the signal waveform. Fourier transform of the reference and sig-nal waveforms gives the reference spectrum, A R( )ei R() and the signal spectrum, AS ( )eiS ( ), respectively. The spectral properties of the target can be extracted bycomparing the signal spectrum with the reference spectrum


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THz Time-Domain Spectroscopy 51

=1d

ln A R AS

n

=1

+

[S ( ) R( )]cd

,(3)

where is absorption coefcient, n is refractive index, d is thickness of the tar-get and c is the speed of light in vacuum. In a real measurement, one can use twosamples with the same composition but different thickness as the sample and the ref-erence. In this case, the Fresnel loss at the surfaces of the samples will be canceled.Equation (3) gives the result in transmission spectroscopy. The complex refractiveindex of samples can also be measured by other types of spectroscopies, such asreection spectroscopy or diffuse scattering spectroscopy. The exact mathematicexpression may differ from Equation (3). THz-TDS measures electric eld of the

THz pulse, which has amplitude and phase information; therefore, it solves both theabsorptive and refractive properties of the target. THz-TDS directly measures thecomplex refractive index of the target, then obtains its complex permittivity withoutusing the KramersKronig (KK) relationship.

THz-TDS has certain advantages, compared to other spectroscopies. THz-TDSprovides coherent spectroscopic detection in a wide range at THz, which is difcultto access using other methods. The THz pulse has ps pulse duration, thus it hasintrinsic high temporal resolution. It is thus very suitable for measuring dynamicspectroscopy. THz-TDS utilizes coherent detection methods, which can be used to

measure coherent processes of carriers. Additionally, THz-TDS uses time-gating insampling the THz pulses. This method dramatically suppresses background noise.As a result, THz-TDS usually has a very high signal-to-noise ratio. It is especiallyuseful to measure spectroscopy with high background radiation which is comparableor even stronger than the signal [ 1].

THz-TDS also presents challenges. THz-TDS is usually slow, due to the tempo-ral sampling of the THz pulses. Techniques have been developed in order to improvethe speed of THz-TDS measurements. Today, THz-TDS can perform a single mea-surement in less than one second with a fairly high SNR. THz-TDS also suffers from

poor spectral resolution due to the limited temporal scanning range T in real mea-surements. In principle, one can scan a THz pulse as long as it is needed. However,a longer scan not only takes more data acquisition time, but also reduces dynamicrange of the spectrometer. The relationship between dynamic range and scanninglength will be discussed in detail in the next section. The limited spectral resolu-tion is not a problem when measuring the spectrum of a target in a condensed state,whose spectral features often have a bandwidth of a few tenths of THz. The spectralresolution could be insufcient when gases are measured. For such a measurement,a cw-THz source with narrow line width is preferred.

Both THz-TDS and FTIR are spectroscopic techniques in the mid- and far-infrared. THz-TDS uses THz pulses, which are generated using a fs laser, as thelight source, while FTIR typically uses a thermal light source. Both use broad-band sources, and rst record signal based on time delay and present it in thetime-domain, and then convert the signal into the frequency domain using Fourier


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transform. However, there are clear differences between these two spectroscopytechniques.

A standard THz-TDS system covers the frequency range 0.13 THz. It can alsoreach beyond 10 THz if a shorter laser pulse is used, while a 100 THz bandwidthcan also be achieved if an even shorter laser pulse is used with a thinner EO crystalas emitter and sensor. However, the measurement of the dynamic range is oftenlimited for a THz-TDS system with a very broad bandwidth. The light source usedin an FTIR system can cover a broader band than common THz sources. The bandlimitation of an FTIR spectrometer depends on its detector, which usually has alower response for low frequency radiations, and the optics in the FTIR spectrometerhave also limited the usable spectral range. Based on current technologies, FTIRspectroscopy usually gives better results with frequencies in excess of 10 THz, whileTHz-TDS is preferred when the relevant frequencies are below 3 THz. These two

techniques give a comparable performance between 3 and 10 THz.The light source for THz-TDS is a THz pulse; therefore, it is very suitable for

time-resolvable spectroscopy, with a temporal resolution in ps scale. The FTIRspectrometer can also measure time-resolvable spectroscopy; however its temporalresolution can hardly reach beyond the ns scale, which is limited by the speed of theash light source or optical modulator. THz-TDS directly measures the electric eldof the THz pulses; therefore, it directly measures the absorption and refraction indexof the sample. FTIR measures intensity of the light, and can only obtain amplitudeinformation. The Kramers-Kronig transform is required in order to obtain the refrac-

tive index of the sample. Table 3.1 compares THz-TDS and FTIR spectroscopy.

Table 3.1 Comparing of THz-TDS and FTIR

THz-TDS FTIR

Bandwidth 0.1100 THz Full spectrumAdvanced range 0.110 THz >10 THzMeasurable Electric eld IntensityTemporal resolution ps ns

Coherent Yes No

Dynamic Range of THz-TDS

The measurement of dynamic range D( ) is essential in spectroscopic measure-ments. If the THz source in a spectrometer gives an electric eld E( ) and thenoise equivalent eld of detector is N( ), the measurement dynamic range of thisspectrometer is D( ) = E( )/N( ). When this spectrometer is used to measure thespectrum of a target with a thickness of d , the reduction of the THz wave caused bythe target can be divided into two categories, according to whether or not the lossis associated with the spectral features of the target. To simplify the discussion, we


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Dynamic Range of THz-TDS 53

assume that the loss associated with the spectral feature of interest of the target isfrequency independent. The spectral-feature-associated loss can be described usingthe absorption coefcient of the target, ( ). Since the THz TDS usually directlymeasures electric eld rather than power of the THz beam, here we dene a(w) asthe absorption coefcient of electric eld. To further simplify the discussion, weassume the target has only one absorption line, and this absorption line is a deltafunction located at =0. The detected THz eld of the spectrometer is:

E D( ) = E ( ) L =0 E D( ) = E ( )e d L =0

, (4)

where L denotes the frequency-independent loss. In order to identify the absorptionfeature, modulation of the THz eld caused by this absorption line must be higher

than the noise-equivalent eld of the system.

E (0) L 1 e d > N (0)]. (5)

When d 1

d 1 L

. (6)

Equation (6) shows the importance of the measurement dynamic range. Only withsufcient dynamic range is the THz wave imager able to identify spectral featuresof the target.

Another critical parameter of THz-TDS is its signal-to-noise-ratio (SNR), whichis dened as the maximum amplitude of THz waveform over noise in the detectionsystem. As we discussed several times already, by using the time-gating technique,THz-TDS usually has very high signal-to-noise-ratio (SNR). However, the SNR intime-domain measurements is not always equal to the actual dynamic range of theTHz-TDS system. It is important to understand the relationship between SNR in the

time-domain measurement and the dynamic range of the spectrometer.There are two primary noise sources in the pulsed THz system; those that orig-

inate in the probe laser beam N b and those that originate in the THz pulses N THz . N THz is proportional to THz eld E(t) and can be written as:

N THz (t ) = R(t ) E (t ), (7)where R(t) is a unitless factor giving the relationship between the THz wave relatednoise and the THz eld. Both N b and R(t) can be considered as random functions of time. Standard deviation of N b and R(t) denoted as respectively, B and R , denethe noise level in the THz time-domain measurement. To distinguish the backgroundnoise and the noise carried by THz wave, we call the ratio between the THz ampli-tude A and background noise standard deviation B the time-domain measurementdynamic range ( D), and we call 1/ R the measurement SNR. It is worth to notice that


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the denition of SNR differs from what was dened at the beginning of the previousparagraph. In a common pulsed-THz system, the noise carried by THz wave usu-ally dominates the background noise. Therefore, both denitions give very similarresults.

If the temporal resolution in THz-TDS is t , and the entire scanning range is T ,then the spectral range and resolution can be calculated based on Equation (2). Inmost cases, both B and R are not dependent on the THz spectrum. Those noises,as functions of frequency in the extracted spectrum, are solely due to the measure-ment. Most likely, they are a combination of white noise plus 1/f noise. Since thedetailed distribution of noise does not affect the basis of the discussion, we considerthat both of these noise sources are frequency independent. The THz source has apulse width of T and a bandwidth of . In a TDS measurement, one always hasT

T and

, and consequently the noise level in the THz spectrum is

THz = t 2 R A , B = tT 2 B.

(8)

Here = A1 | E (t )|2 dt is the root mean square of the THz eld normal-

ized by its amplitude. Equation (8) indicates that, in the frequency domain, the

THz wave-carried noise is not related to the temporal scanning range T , while thebackground noise is proportional to the square root of T .

The measured dynamic range of the THz spectrometer D( ), is written as:

D( ) =k ( )

t 2 S 2 + T t 2 1 D2, (9)

where k ( ) = E ( )/ A is the normalized spectrum of the THz pulse. Table 3.2gives the expression of k( ) and

2

which can be used to present several typicalTHz waveforms. Figure 3.2 shows the measured dynamic range of the spectrometer

Table 3.2 k() and 2 for typical THz waveforms

Emitter Description Waveform k() 2

PC antenna Mono-polar 2 A t 2

et 2 / 2 2 e2 2 / 4 2

2

Surface eld Bi-polar 2 A

2 et 2 / 2

4 A t

2

4 et

2 / 2 2

2 e2 2 / 4 3 2

2 3

Optical recti-cation

Dampedoscillation

A sin(0t )eat t > 00 t < 0 20[a 2+(0)2][a 2+(0+)2 204a (a 2+20 )


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Detection of a THz Waveform Using a Single Laser Pulse 55

Fig. 3.2 Dynamic range of aTHz spectrometer as afunction of its spectralresolution

as a function of spectral resolution. Clearly, a higher spectral resolution leads to alower dynamic range. Equation (9) also indicates that the measured dynamic rangeis related to the sampling step width in the time-domain. Equation (2) shows that thesampling step width t decides the total frequency range in the spectrum. When 1 / t is larger than the bandwidth of the THz pulses, a narrower sampling step width willnot lead to a more useful broad bandwidth; however, it does provide higher dynamicrange in the frequency domain.

Equation (6) suggests that to identify a sample through its spectral features,the THz wave spectrometer has to give sufcient dynamic range in the frequencydomain. Combining Equation (6) with Equation (9) leads us to conclude that thepossible spectral resolution, which a THz-TDS system may provide, is limited bythe dynamic range of the time-domain measurement,

1

2 1

dk ( ) De d 2

. (10)

The relationship between the measured dynamic range and possible spectral reso-lution can be easily understood. The THz pulse energy is distributed across its entirebandwidth. A higher spectral resolution means that one needs to detect energy con-tained in a narrower bandwidth, thus containing less energy. As a result, the dynamicrange of a spectrometer is inversely proportional to its spectral resolution. Since itis the electric eld, rather than intensity, that is measured in THz-TDS, the possi-ble spectral resolution is thus inversely proportional to the square of the dynamicrange in the time-domain measurement. A ner sampling step-width acts to averagethe measurement, thus it enhances the measured dynamic range in the frequencydomain.

Detection of a THz Waveform Using a Single Laser Pulse

Sampling a THz waveform usually requires a series of THz pulses and probe pulseswith various temporal delays. However, if the THz eld is strong enough, one cancatch the entire THz waveform using a single laser pulse. Various methods could


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Fig. 3.3 Optical setup of achirped-pulse detectionsystem

be applied to detect the THz waveform using a single laser pulse, which includeevaluating the THz waveform by the spatial distribution of the probe pulse or bythe spectral distribution of the probe pulse. The single laser pulse detection tech-nique provides a signicant reduction in the acquisition time and greatly extends theapplicability of THz systems in situations where the sample is dynamic or moving.

Figure 3.3 exhibits the concept of measuring the THz waveform using a chirped

probe pulse. The setup is similar to a standard THz-TDS measurement, where anultrafast laser beam is split into a pump and probe beam, while the pump beam isused to generate THz pulses and the probe beam is used to detect the THz pulses.However, there is no delay scanning instrument in the chirped pulse detection sys-tem. Instead, the optical probe pulse is frequency-chirped and time-stretched with agrating pair from sub-picoseconds to a few tens of picoseconds. The negative chirpof the grating makes the blue component lead the red component. The output fromthe grating is a pulse with a longer pulse duration and a wavelength that varies lin-early with time. The chirped probe pulse is modulated by the THz pulse when they

are mixed in the EO crystal. Conceptually, the chirped probe pulse can be seen asa succession of short pulses each with a different wavelength. Each of these wave-length components encodes a different portion of the THz pulse. A spectrometerspatially separates the different wavelength components and thus reveals the tem-poral THz pulse. The spatial signal output from the spectrometer is measured usinga CCD. Figure 3.4 gives the CCD recorded probe pulses with and without the THzpulse. The difference between those two signals gives the waveform of the THzpulse. For maximum image acquisition speed, the THz pulse and probe pulse maybe expanded in the vertical dimension using cylindrical lenses. The CCD is thenable to capture both the THz temporal waveforms and several hundred vertical pix-els simultaneously and only a single translation stage is required for spectroscopicimage acquisition.

It is assumed that the probe pulse originally has a Gaussian distribution inboth the time and frequency domains, and its central frequency is 0. If only


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Detection of a THz Waveform Using a Single Laser Pulse 57

Fig. 3.4 THz waveformextracted using achirped-pulse measurement

second-order dispersion is taken into account, while ignoring third and higher orderdispersion of the grating pair, the electric elds of the probe pulse before and after

chirping are:

E 0(t ) =exp t 2

T 20 i0t ,

E C (t ) =exp t 2

T 2C i t 2 i0t ,

(11)

where T 0 and T C are the pulse widths of the probe pulse before and after chirping,respectively, and is the so-called chirping rate of the pulse. Once the chirped pulseis modulated by THz pulses, its electric eld is

E m(t ) = E C (t )[1 +kE THz (t )], (12)k is a modulation factor, which indicates electric eld of the probe pulsebeing affected by the THz eld. denotes the relative temporal delay betweenthe THz pulse and the probe pulse. In most cases kE THz is much smaller than 1. Themodulation of the electric eld in the different frequency components of the probebeam is

N ( ) I ( )|THzOn I ( )|THzOff

I ( )|THzOff = g( )2kE (t )exp( 2t

2 / T

2C )d

g( )exp( 2t 2 / T 2C )d

2kE THz (t ).

(13)

Here I( ) is the intensity of the probe pulse in the frequency domain, which canbe dened as intensity with or without THz eld as presented in the foot notes.g( ) is the spectral function of the spectrometer. t is dened as t (0 )/ 2 . Equation (13) indicates that the modulation of the probe pulse intensity in


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the frequency domain is linearly proportional to the electric eld of THz pulse inthe time domain.

According to Heisenbergs uncertainty principle, higher temporal resolutionrequires more bandwidth. When the probe pulse is chirped and only a part of thefrequency component is used to sample a temporal window in the THz waveform,the temporal resolution will not be as high as the original pulse. If the original pulseis transform-limited, its time-bandwidth production should be a constant T 0 0 =k , whose value depends on the actual pulse shape, i.e. 0.44 for a Gaussian pulse.Once the pulse is chirped, its pulse width becomes T C while keeping the samebandwidth. The best temporal resolution can be obtained if all frequency compo-nents used in the THz waveform measurement remain transform-limited, which isthe component with bandwidth providing a temporal resolution of T =k/ .In this case, the temporal resolution T is:

T =T C /

= T 0T C .(14)

Equation (14) gives the nest temporal resolution possible for the chirped pulsemeasurement method. The temporal resolution is linearly proportional to the squareroot of the pulse width for both the original pulse and chirped pulse. A larger chirpedpulse width results in lower temporal resolution.

THz Differential Spectroscopy

Lock-in technology is usually used in THz-TDS measurement in order to suppressthe background noise. To use lock-in technology, the detected signal is modulatedat a certain frequency. The modulation frequency is used as the reference fre-

quency of the lock-in amplier. Only a certain component within the input, whichhas the same frequency and certain phase shift with the reference is amplied andrecorded. All other components are blocked from recording. Thus tremendous noiseis screened due to frequency and phase selection. Using the lock-in technique maysuppress noise level down to 10 6 of the original noise level. Employing lock-intechnology into the THz-TDS measurement by modulating the THz source, onecan efciently reduce the background noise generated in the detection system, suchas that which is associated with the probe laser beam. However, such a techniquecannot reduce noise associated with THz waves since noise is also modulated at thesame frequency. When the target is a thin lm or tracing material, it only gives weak modulation to the THz eld. This weak modulation may be buried under the noiseassociated with the THz eld. In this condition, one can use the differential spec-troscopy method, which directly measures the difference between the target and thereference.


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THz Differential Spectroscopy 59

Fig. 3.5 Experimental setupof a THz time-domaindifferential spectrometer

Figure 3.5 shows the concept of THz time-domain differential spectroscopy,where a double lock-in technique is used. The two lock-in ampliers are seriallyconnected. One of them uses a higher reference frequency H , with a shorter inte-gration time constant T H . The output of this lock-in amplier is used as the input of the other lock-in amplier, which uses a lower reference frequency L and a longerintegration time constant T L. Figure 3.6 illustrates the process of double lock-intechnology. To make the double lock-in technique work, the following relationshipmust be satised,

H >> 1/ T H >> L >> 1/ T L. (15)

In differential spectroscopy measurements, the THz beam alternately passes throughthe sample and reference with a frequency of L. For instance, a galvanometercan be used to shake the sample in and out of the THz beam. The THz source ismodulated, i.e. using an optical copper, with a frequency of H . The rst lock-in amplier with higher reference frequency was used to suppress the background

Fig. 3.6 Concept of doublelock-in technique.(a ) Frequency distributionof a signal modied by twofrequencies, ( b ) frequency of signal is shifted by H afterthe rst lock-in amplier, and(c) frequency of signal isshifted by L after thesecond lock-in amplier


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noise associated with the detection system, i.e. power uctuation of the probe laserbeam, while the second lock-in amplier with lower reference frequency is used tosuppress the noise associated with the THz wave.

THz Wave Imaging

Just like the adjacent bands, microwaves and infrared radiation, THz waves can beused as imaging media in a variety of applications [2]. Terahertz wave imaging isattractive for several reasons: the radiation is non-ionizing and poses very few safetyrisks, it is capable of submillimeter spatial resolution and signicantly, and a num-ber of materials, including paper, plastics, and cardboard are relatively transparentin this frequency band. Figure 3.7 shows the concept of raster scanning THz waveimaging. The THz wave is focused by a lens or parabolic mirror. The target is placedacross the THz beam at its focal spot. It is then raster scanned in a plane perpendic-ular to the THz beam. THz wave transmission through or reected from each spotof the target is recorded, which forms the THz wave image of the target.

Different from a common optical image or X-ray image, each pixel in a pulsedTHz wave image contains the entire THz waveform rather than just the intensityof the beam. Fourier transform of the THz waveform extracts the spectral informa-tion of that pixel. Therefore, THz-wave imaging not only identies the target byits prole but also obtains composite information of the target. Besides the reec-tion distribution, pulsed THz wave imaging can also prole the target dependingon its refractive index distribution, which causes a phase change of the THz pulse.Figure 3.8 shows a THz wave image of the water mark in a 100 dollar bill, whichwas made by the phase change of the THz pulses.

Fig. 3.7 Concept of rasterscanning THz-wave imaging

Fig. 3.8 THz-wave image of a water mark in a $100 bill


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THz Wave Imaging 61

Imaging with THz pulses requires scanning in three dimensions, including space(2-D) and temporal (1-D), thus it is usually very time consuming. If spectral infor-mation is not required, one can take the THz wave image of a target at a xedtemporal delay, i.e. at the peak of the THz waveform. Only amplitude of the THzeld is recorded while scanning the target across the THz beam. A similar imagecan also be taken using a cw-THz source, by recording the intensity of the transmit-ted or reected THz beam. Figure 3.9 shows cw-THz wave transmission images of a tea pot when it is empty and when it is half-full with water. Table 3.3 comparespulsed and cw-THz wave imaging.

Fig. 3.9 cw THz-waveimages of a tea pot. Left , anempty tea pot, and right , a

half full tea pot

Table 3.3 Comparing of cw and pulsed THz wave imaging systems

cw-THz wave imaging Pulsed THz wave imaging

Cost $50,000$150,000 $200,000$1,000,000System

complicityLow High

Weight 3 kg 10 kgSpeed 100,000 point/s < 4,000 point/sData complicity Low HighSpectral

informationNo Yes

Depthinformation

No Yes

Refractive index No Yes

Besides being time consuming, THz wave imaging also faces other challenges.THz waves are highly reected by metal surfaces, thus it cannot see through metalcontainers. THz waves are highly absorbed by water, so they cannot penetrate intoa material which contains a lot of water. THz wave imaging cannot be used formedical diagnostics of organs inside the human body except when the endoscopetechnique is being used. Additionally, since THz waves have longer wavelengthsthan visible and IR waves, spatial resolution of THz wave imaging is limited to sub-millimeter in the far-eld. Near-eld imaging must be used in order to break thediffractive limitation for even higher spatial resolution.


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2-D Focal Plane THz Wave Imaging

The Raster scanning method of THz wave imaging utilizes the entire THz radiationgenerated from the emitter to investigate each pixel of the image, so that it providesa very high signal-to-noise ratio. However, the speed of linearly transporting thetarget with nite mass back and forth is quite limited. The low frame rate is oneof the major technical obstacles blocking THz wave imaging in its approach to realworld applications. One can use a similar method that is used in a traditional opticalimaging system, THz wave imaging optics, which can be a lens or a concave mirrorthat directly images the THz wave transmission or reection from the target ontoan extended THz wave sensor. The THz wave image of the target is simultaneouslyrecorded using the extended sensor. Since no scanning is required, 2D THz waveimage highly reduces the acquisition time. The extended THz wave sensor could be

THz wave detector array, such as pyroelectric detector array, micro-bolometer array,heterodyne detector array, et. al. When using EO sampling to detect THz wave, theextended sensor could be an EO crystal with sufcient aperture.

Figure 3.10 presents the concept of taking THz wave 2-D imaging system usinga large aperture EO crystal. A THz wave imaging optic, such as a polyethylenelens, is used for forming the THz wave image of the target, while the target andEO crystal locate the object and image plane, respectively. Thus, the THz wavedistribution on the EO crystal carries spatial information of the target. An extended,linearly polarized probe beam, which covers the entire THz wave distribution area,

is collinearly propagated with the THz beam through the EO crystal. The spatialdistribution of the THz wave is printed onto the probe beam via the EO process. Across-detection method is used to detect THz wave modulation of the probe beam,where an analyzer with cross-polarization is placed after the EO crystal and theleaking of the probe beam is imaged onto a CCD camera. Through a serial imagingprocess, the optical image in the CCD camera reects the THz wave image of thetarget.

Fig. 3.10 Concept of 2D THz wave imaging using a large aperture EO crystal as extended focalplane detector


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2-D Focal Plane THz Wave Imaging 63

The spatial resolution of a far-eld imaging system is limited by diffraction of the carrier wave.

=1.22l

D , (16)

where is the wavelength of the carrier wave, l is the distance from the target to theimaging lens, and D is the aperture diameter of the lens. The depth of eld plays animportant role in an imaging system, which determines a range, within which a tar-get can project a clear image on a xed imaging plane. Complicated formulas havebeen developed for photographers to calculate the depth of eld of their cameras.The exact formation of depth of eld could be different for different lenses or imagetaken conditions. To get a brief idea without involving detail lens parameters, wecan consider an imaging system using an ideal lens. In such a system, a point sourcewhich does not locate on the object plane projects an extended spot on the imageplane. One can consider a point within the depth of eld if its extended image size issmaller then the dened spatial resolution on the image plane. When depth of eldis much smaller than the object distance, it can be described as

L = Dl/ ( D D), (17)

Here D is the required spatial resolution on the target and D is the required res-olution on the imaging plane, while the sign indicates the depth of eld at differentsides of the target. D and D have the following relationship: D = D(l /l), wherel is the image distance, which approximately equals to the focal length of the imag-ing lens for a far eld object. A wave with 1 THz frequency has a wavelength of 300 m, which is much longer than the optical wavelength. Thus, the THz waveimage usually has a much lower spatial resolution than an optical image. Using a40 cm diameter lens to image a target at 10 m away, the spatial resolution is 9.15 mmif the frequency of the carrier wave is 1 THz. If the required spatial resolution equalsto the diffraction-limited resolution, then the depth of eld is 45.8 cm.

The size of the EO crystal is determined by the target size and focal length of theimaging optics. The object distance in an imaging system is usually much longerthan the focal length of the imaging optics. Therefore, the dimension of the EOcrystal is dened as the image size.

DS DT f l, (18)

DT is the dimension of the target, and f is the focal length of the imaging optics. Inthe previous example, if the target is a circle with 1 m diameter, and focal lengthof the imaging lens is 44.7 cm (NA of the lens is 0.5), then the diameter of the EOcrystal is 4.47 cm. The thickness of the crystal can be estimated with two-times of the focal depth, which is:


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L = D f D

D2N.A.

, (19)

where D denotes the required resolution on the imaging plane, which can be set

by the diffraction-limited spatial resolution. N.A. is the numerical aperture of thelens. In the previous example, the maximum thickness of the crystal will be 0.7 mmin order to maintain the spatial resolution. The above discussion does not take intoconsideration the refractive index of the crystal. Since commonly used EO crystalshave fairly large refractive index, the EO crystal could be a few mm thick withoutaffecting the spatial resolution.

Telecentric Beam Scanning THz Wave Imaging

Since the THz wave is diluted onto the extended sensor in a 2-D imaging system,the detection signal-to-noise ratio is usually reduced. A strong THz source is desiredin such a 2-D imaging system. An alternative way to perform the THz wave imagewith high speed is to scan the THz beam with ying or shaking optics rather thanthe target. Since less mass is associated with scanning the THz beam, a much fasterimaging process is expected. Unlike optical imaging, where scattering light is themajor information carrier, with THz waves, due to their longer wavelength, thescattering or diffusion are usually less important in the imaging process than trans-mission and specular reection. As a result, collection of transmitted or reectedTHz waves is essentially important in a beam-scanning THz wave imager. A tele-centric beam-scanning technique can be used to ensure a high collection coefcientin the THz wave beam-scanning imaging process.

The concept of a telecentric beam-scanning imager is presented in Fig. 3.11. Thecollimated THz beam is guided into the imaging system and steered by bendingabout two orthogonal axes using a pair of shaking mirrors. The output beam fromthe shaking mirror pair is then guided into a telecentric lens. The telecentric lenscould be a single spherical lens, while the shaking mirror locates at one focal spotof the lens and the target locates at its focal plane on the opposite side. THz beam

output from the telecentric lens is normal to the focal plane and is focused ontothe target. Shaking the mirror pair scans the THz beam across the target. The THz

Fig. 3.11 Schematic of atelecentric beam scanningimager


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Time-of-Flight Imaging 65

wave reected by the target is collected by the same lens, and fed back followingthe same direction. It is picked up by the same shaking mirror pair and counterpropagates with the input beam. One can use either a THz wave transceiver, whichemits and detects THz radiation, or use a beamsplitter to guide the returning THzbeam into the detector.

A common understanding is that 2-D focal plane imaging, which is a parallelprocess, should be faster than a raster scanning imaging, which is a serial process.This statement is true for passive imaging, as well as active imaging with sufcientdynamic range. In THz-wave imaging, however, once the processing speed is fastenough, the dynamic range becomes a bottleneck due to the low-intensity sourceand less sensitive detector being used. In a raster scanning imaging process, everypixel uses the entire THz radiation, but shares data acquiring time with others. Thedynamic range of the measurement limits how long the THz beam should stay on

one pixel of the image. In a 2-D imaging process, each pixel fully uses the dataacquisition time, but shares THz power with others, which leads to lower dynamicrange for all pixels. If the overall dynamic range of the imager is limited, the speedof a 2-D focal plane imaging process cannot be faster than a raster scanning imagingprocess.

Time-of-Flight Imaging

The entire THz waveform is recorded for each pixel when pulsed-THz-wave-imaging is taken. If two THz pulses are reected from two surfaces located atdifferent depths, the reected THz pulses have different time-delays due to differentoptical paths. From time-delay, one can retrieve depth information of each pixel, andthus present a topographic prole of the target. This imaging method is called THzwave time-of-ight imaging. Figure 3.12 shows concept of time-of-ight imaging.

Fig. 3.12 Concept of THz-wave time-of-ightimaging. Inset shows timedelay between two THzwaveforms


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The depth at resolution in THz-wave time-of-ight imaging is limited by tempo-ral resolution in measuring the THz waveform, which is in principle related to theTHz wavelength. The actual depth (temporal) resolution one can get experimentallyis usually shorter than the THz central wavelength. When THz pulses with a 2 THzcentral wavelength are used, the smallest depth resolution can be a few microns.Although time-of-ight imaging provides a 3-D topographic prole or layer struc-ture of the target, it is usually not considered a full 3-D imaging technique, since itcannot provide interior information of the target if there is no layer structure pre-sented. Figure 3.13 gives a THz wave time-of-ight image of a spark plug, whichshows the surface topographic prole and layer structure inside the spark plug.

Synthetic Aperture and Interference Imaging

Equation (16) gives the spatial resolution of THz wave imaging. To obtain highspatial resolution, especially when imaging a target at a stand-off distance, one needsto use an imaging optic with large aperture. An imaging optic with a large aperture isnot only expensive but also not easy to operate and transport due to its bulky size andheavy weight. Synthetic aperture and interference imaging uses either one mobiledetector or detector array with discrete spatial distribution to create an image of thetarget. In these techniques, the spatial resolution of the image is no longer limited bythe aperture of each individual detector but is limited by the movement or locationof the detector(s). As a result, using the synthetic aperture and interference imagingmethod one can get high spatial resolution with small imaging elements.

The synthetic aperture and interference imaging technique was originally devel-oped for RF and microwave. Figure 3.14 gives a schematic diagram of the syntheticaperture imaging technique with airborne radar. The pulsed radar carried by an air-craft illuminates a carrier wave onto the ground to one side of the air-craft. Assumethat the speed of the air-craft is v, its ying attitude is H , the distance to the detec-tion spot is R, the radar pulse width is T , the aperture is D, and the wavelengthof the carrier wave is . The aperture of radar is = /D viewed from the far-eld. The synthetic aperture radar receives a back-scattering signal from objects onthe ground. Its lateral resolution is determined by time-delay between radar pulsesreected back from different spots

r g =Tc

2sin . (20)

Here is the angle presented in Fig. 3.14. Along the ight direction, the spatialresolution is dened by the optical path difference from an individual spot on theground to a different location on the plane. Determined by the aperture of the radar,the carrier wave covers a length of L = R on the ground. From a different view,the radar wave can cover the same spot within a ight distance L. As a result theimaging aperture is extended from the aperture of radar, D, to the ight distance L.Spatial resolution in the ight direction can be estimated using Equation (16)


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Synthetic Aperture and Interference Imaging 67

Fig. 3.13 THz-wavetime-of-ight image of aspark plug. ( a ) photo of thespark plug, ( b ) THz waveimage of the external shell,and (c) THz-wave image of the inner metal bar

r a

=

L

R = D. (21)

Here the factor of 1.22 is dropped. Typically, the spatial resolution at the ightdirection is dened as D/2 . Equation (21) shows that, due to the relationships among , D, and , the spatial resolution along the ight direction is not dependent on


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Fig. 3.14 Concept of an airborne synthetic-aperture radar imaging

either the wavelength of the carrier wave or the distance between the target andthe radar. In addition, the smaller the radar aperture, the ner the spatial resolution.This is different from a common optical imaging setup, where a larger apertureresults in ner resolution. It is worth noting that the above discussion is based onthe aperture of radar being much larger than the wavelength of the carrier wave.Therefore, Equation (21) does not lead to the following conclusion: by reducingthe size of the radar, the synthetic aperture image can have sub-wavelength spatialresolution.

A similar technique can be used in pulsed THz-wave synthetic-aperture imag-ing. The pulse width is about one picosecond for the THz pulses, which leads to alateral resolution of hundreds of microns. In reality, the temporal resolution in themeasurement of the THz pulses can be much smaller than the THz pulse width,which results in a lateral resolution on the sub- m scale. The spatial resolution inthe ight direction can be as small as the wavelength level. This is different fromcommon synthetic aperture radar imaging. A common radar pulse width is usually

much longer than the oscillation period of the carrier wave. As a result, it givesa better spatial resolution along the ight direction, while the THz wave syntheticaperture image gives a better lateral spatial resolution.

If the relative phase shift between the various sensor locations is recorded, thesynthetic aperture imaging system can take an interference image. In an interfer-ence imaging setup, any pair of detectors, i and j, form a base line. Each base linecorresponds to a point in phase space. It is assumed that all the detectors are dis-tributed within an x-y plane. The coordinate of base line ij in the phase space is

uij = x i x j

vij = yi y j

, (22)


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References 69

where u and v are coordinates in phase space, and x and y are those in the real space.The signal amplitude at that point is a product of the signal amplitudes of those twodetectors, which is Aij = Ai A j, while the phase is dened as the phase differencebetween these two detectors: ij

= i j. If the total number of detectors is N ,

then there will be N ( N 1) points in the phase space. The target image is extractedfrom the Fourier transform of the signal in the phase space, which is

I ( x , y ) = A(u,v)ei (u, )ei2 pxuei2 y dud ; (23) x and y are coordinate units in the imaging space. Spatial resolution of theinterference imaging setup is determined by the length of the longest base line.

References1. D. Grischkowsky, S. R. Keiding, M. P. van-Exter, and C. Fattinger, Far-infrared time-domain

spectroscopy with terahertz beams of dielectrics and semiconductors, J. Opt. Society Am. B 7,10, 20062015 (1990).

2. B. B. Hu, and M. C. Nuss, Imaging with terahertz waves, Opt. Lett . 20 , 16, 17161718 (1995).

thz spectroscopy and imaging

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