Generation and detection of terahertz pulses from ... - OSA Publishing

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J. Opt. Soc. Am. B / Vol. 13, No. 11 / November 1996

Jepsen et al.

Generation and detection of terahertz pulses from biased semiconductor antennas P. Uhd Jepsen, R. H. Jacobsen, and S. R. Keiding Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark Received November 2, 1995; revised manuscript received April 26, 1996 We propose a simple model based on the Drude–Lorentz theory of carrier transport to account for the details of the ultrashort terahertz pulses radiated from small photoconductive semiconductor antennas. The dynamics of the bias field under the influence of the space-charge field from the accelerated carriers is included in the model. We consider in detail the optical system used to image the terahertz radiation onto the terahertz detector, and we calculate the frequency-dependent response of the detector. The proposed model is compared with several different experiments, each focusing on different parameters of the model. Agreement between experiment and model is found in all cases, supporting the validity of this simple and appropriate model. © 1996 Optical Society of America.

1. INTRODUCTION The use of ultrafast lasers to generate subpicosecond pulses of electromagnetic radiation, terahertz (THz) pulses, has evolved during the past 5 years into a very active research field. When an ultrafast laser interacts with a material to generate a dc polarization, the polarization acts as a source for the THz pulses. The dc polarization can have its origin in either a simple flow of free carriers in a photoconductive switch, P dc 5 * jdt, where j is the current density, or from a nonlinear optical x 2 process, P dc 5 x 2 (0, 2v , v )E( v )E * (2v ). The resulting THz pulses generally consist of only a few cycles of the electric field and consequently have a very high bandwidth. Centered around 1 THz with frequencies extending beyond 5 THz, the pulses have proven to be very useful in time-domain spectrometers and, equally important, as probes of fundamental carrier transport processes in the semiconductors from which they originate or as probes of phonon dynamics in crystals. Several reviews exist covering different aspects of the THz field. The generation of THz pulses from semiconductor surfaces and from large-aperture photoconducting antennas was reviewed by Zhang.1,2 Grischkowsky reviewed the high-bandwidth THz systems that used small photoconducting antennas and their applications to timedomain spectroscopy.3,4 The generation of THz pulses from semiconductor quantum wells was recently reviewed by Nuss and Lou,5,6 and applications of THz spectroscopy in atomic and molecular physics were discussed by us.7 Recently, a special issue of the Journal of the Optical Society of America B covering THz pulses was published.8 In this work we focus on a detailed description of all the elements in a standard time-domain spectrometer, from the THz emitter by means of the THz optical system to the THz detector. The THz setup is illustrated in Fig. 1, in which the emitter, the THz optics, and the detector are shown. The inset shows the detailed geometry of the emitter and the detector antennas. With a 100-fs laser pulse from a mode-locked Ti:Sapphire laser we generate 0740-3224/96/1102424-13$10.00

the THz pulse shown in Fig. 2(a), with its Fourier transform shown in Fig. 2(b). The weak oscillations on the trailing edge of the pulse are caused by free-induction decay of pure rotational transitions in the residual H2O vapor present in the optical path of the beam. The spectral resolution is 6 GHz (0.2 cm21 ). The sharp water lines and the moderate structure in the spectrum are caused by weak reflections and are completely reproduced from scan to scan. The characteristic of a THz system was previously described by van Exter and Grischkowsky.9 We have used very simple models to describe the different components of the spectrometer, and we show by comparison with different experiments that the system is very well described by these simple models. To describe the THz emitters, we use the simple Drude–Lorentz model, modified to take into account ultrafast changes in the bias field owing to screening effects. From the calculations and the experiments it is evident that the screening process is the key factor determining the properties of the radiated THz pulses. The THz pulses are imaged onto the detector by lenses and mirrors. Owing to the extreme bandwidth, a frequency range covering two orders of magnitude, the optical system plays an important role in shaping the THz pulse. At the detector the electric field of the THz pulse acts as a transient bias, polarizing free carriers generated by a femtosecond laser pulse. Describing the detection process thus requires a detailed understanding of ultrafast carrier transport dynamics. Furthermore, the active size of the receiver also plays a central role in evaluating the response function of the detector.

2. TERAHERTZ EMITTERS To illustrate the physical properties of a THz emitter system, we consider the THz antenna shown in the inset of Fig. 1. This particular antenna geometry was first introduced by Grischkowsky et al.10,11 and proved to be a very efficient THz source with high bandwidth. As the emitter chips, we use a high-resistivity chromiumcompensated GaAs substrate wafer (Sumitomo) and use a © 1996 Optical Society of America

Jepsen et al.

Vol. 13, No. 11 / November 1996 / J. Opt. Soc. Am. B

Fig. 1.

Schematics of the THz setup.

Fig. 2. (a) Time-domain shape of the THz pulse. The electricfield peak amplitude is of the order of 100 V/cm, corresponding to an induced average current in the antenna of 10 nA at the peak of the THz pulse. (b) Frequency-domain spectrum on a logarithmic scale demonstrating a spectroscopically useful bandwidth extending to 3 THz, corresponding to 100 cm21 .

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standard mixture of gold–germanium–nickel for the metallization. The spacing between the metal lines in the emitter is 50 mm. The I – V curve for the emitter chip is shown in Fig. 3. For bias voltages less than 18 V, the I – V curve is exponential, as for a forward-biased metal– semiconductor–metal diode. The current is caused by thermionic emission over the barrier, and from the intersection with the current axis we can estimate a barrier height of approximately 0.5 V. As the current density increases, the space-charge field from the current becomes comparable to the space-charge field responsible for the barrier, and the resistivity changes dramatically. This regime is characterized by a V 3 dependence of the current. The observed metal–semiconductor interface properties are in agreement with the observations previously published and explained by Ralph and Grischkowsky.12 The I – V curve and the bias dependence of the peak amplitude of the THz field are similar, and we do not observe significant changes in the shape of the THz pulse as we change the bias. The field distribution between the two metal lines is highly nonuniform, with the high fields concentrated near the positive electrode. The laser beam used to drive the THz antenna is focused in the high field region near the positive electrode. Since the spot size of the laser beam is smaller than the wavelength (l 5 45 mm at 2 THz in GaAs), we can regard the photocurrent driving the THz emitter as a point source. To simplify the source further, we also assume the spot size of the laser to be so small that we can neglect the spatial dependence of the bias field. Without light we can regard the THz emitter as a charged capacitor,13,14 with a stored electrostatic energy equal to E s 5 (1/2)CV 2 , where C is the capacitance of the gap, of the order of a few femtofarad, and V is the applied bias.13 For a 10 m m 3 10 mm spot size, the stored energy is approximately 2 pJ. This is the total amount of energy available to be distributed between carrier kinetic energy and radiation. With average laser powers of tens of milliwatts, the stored energy is depleted in less than 150 fs by the separating carriers. A consequence of this depletion is that the external field is completely screened by the space-charge field generated by the separating electron–hole pairs. With a completely depleted bias

Fig. 3.

I – V curve for the THz emitter.

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field the carrier velocity field will rapidly randomize owing to collisions, and the photocurrents will vanish. For high carrier densities the radiated THz pulse will thus contain information about the carrier acceleration and subsequent deceleration, since the radiated signal is proportional to the time derivative of the photocurrent. Although conceptually very similar to velocity overshoot, this effect is not the same as the velocity overshoot observed by several other groups, at lower carrier densities.15 The saturation velocity of the carriers in a semiconductor is given by v s 5 eE molt /m * , where E mol is the local field at the position of the carriers, t is the collision time, and m * is the effective mass. Velocity overshoot refers to the situation in which carriers are accelerated ballistically to velocities exceeding v s . To observe velocity overshoot, one needs low carrier densities to keep the scattering time t as large as possible. The phenomenon we observe is attributable to carriers that have reached the saturation velocity v s but then experience a 8 , E mol , leaving them with a drop in the bias field, E mol 8 t /m * . velocity exceeding the new value v 8s 5 eE mol To investigate the dynamics of the space-charge field in the THz emitter, we designed an experiment that directly measures the magnitude of the local field in the emitter as a function of the photogenerated carrier density.14 A different experimental technique, giving similar information, was introduced by Sha et al.13 This paper also contains a discussion of the dynamics of the screening process. Two femtosecond laser pulses are used to drive a THz emitter, and a third laser pulse is used to sample the radiated THz pulse. The electron–hole pairs generated by the first pulse (the pump pulse) drift apart in the bias field, and in doing so the space-charge polarization screens the bias field. This reduced local field is monitored by measuring the amplitude of the THz pulse emitted by the second laser pulse (the probe pulse). Since the amplitude of the THz pulse is proportional to the local field, this directly measures the magnitude of the local field in the THz emitter. The local field is the sum of the applied bias field and the space-charge field from the separating carriers. By changing the time delay between the pump and the probe pulses, we can follow the time evolution of the local field with subpicosecond time resolution. The local field dynamics is shown in Fig. 4. At t 5 0 the pump and the probe pulses overlap. Using low power in the pump pulse, we observe a linear decrease of the local field. At low density the stored energy in the THz antenna is sufficient to accelerate all carriers to the saturation velocity without a significant drop in the local field, and the space-charge field, proportional to the separation of the carriers, increases linearly in time. For high carrier concentrations the space-charge field generated by the carriers becomes comparable to the applied bias field, and the reduced local field causes deceleration of the carriers. Initially, the local field thus drops very fast, but at later times, when the carriers separate with lower velocity, the field decreases more slowly. The switch will recharge again when the carriers reach the metal electrodes or if they disappear by recombination. In the experiment shown in Fig. 4 we used a GaAs substrate with long carrier lifetime (t rec . 100 ps) and a rather large spot size of the pump laser. Consequently

Jepsen et al.

we do not observe recharging of the antenna. In the experiment,14 on which Fig. 4 is based, the spot size was made very small, and recharging was observed as the carriers reached the electrodes. In a similar experiment16 we used GaAs THz antennas grown on silicon substrates. This gives a very short recombination time (t rec 5 2 ps), and recharging of the THz antenna was observed with a time constant given by the recombination time. Recently a similar technique was used by Hu et al.17 to observe the initial ballistic carrier transport with 10-fs time resolution. The solid curves in Fig. 4 are calculated from the simple model to be described in this paper. To obtain this agreement, we use a carrier scattering time of 30 fs, and a carrier density of 1 3 1015 cm23/mW average laser power. The pump laser was slightly defocused, and consequently no recharging was observed within the first 20 ps. From the figure it is evident that, at high carrier densities, the bias field is screened on a time scale comparable to the duration of the THz pulse. In order to model the generation of THz pulses, we must therefore include screening in the model. To model the carrier transport, we use the simple onedimensional Drude–Lorentz model. Other authors have used this and related models to describe carrier dynamics in a THz antenna.18–20 The main difference between this and previous works is the explicit inclusion of dynamic screening in the Drude–Lorentz model. The current density is related to the velocity of the free carriers in the THz antenna as j 5 2en f v,

(1)

where n f is the density of free carriers and v is the carrier velocity averaged over the carrier distribution. For simplicity we neglect the minor current contribution from the holes. The time dependence of the carrier density is given by Eq. (2), where tc is the trapping time and G(t) describes the generation of free carriers by the laser pulse: dn f nf 5 2 1 G~ t !. dt tc

(2)

Fig. 4. Time evolution of the local field in the THz emitter as a function of laser power and time delay between the pump and the probe pulses. The solid curves are calculated local field dynamics based on the simple model described in the text.

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The time dependence of the average velocity is in the Drude–Lorentz picture, given as dv ~ t ! v e 52 1 E , dt ts m * mol

(3)

where ts is the momentum relaxation time, m * is the effective mass, and E mol is the electric field at the position of the carriers, given by E mol 5 E bias 2

P sc , he

(4)

where P sc is the space-charge polarization created by the carriers separating in the field. The geometrical factor h is equal to three for an isotropic dielectric material. The time dependence of the space-charge polarization can be represented by dP sc P sc P sc 52 1 j~ t ! 5 2 1 n f ev, dt tr tr

(5)

where tr is the recombination lifetime.21 In general, t s ! t c < t r , where typical values are t s , 100 fs and t c , t r ; 1 – 100 ps. If we insert Eq. (4) into Eq. (3) and take the time derivative, we can use Eq. (5) to obtain the following secondorder differential equation for the carrier velocity: d2 v dt 2

1

w 2p v 1 dv eP sc 1 5 . t s dt h m * h et r

(6)

In Eq. (6) we introduce the plasma frequency, v 2p 5 n f e 2 /m * e . The radiated electric field is proportional to the carrier acceleration, dv/dt, so solving the coupled Eqs. (5) and (6), together with Eqs. (1) and (2), gives full information about the radiated THz pulse and the dynamics of the local field. Before we start solving the equations, we note that, in spite of its simplicity, the Drude–Lorentz model is expected to work well for the THz emitter. The carrier density in the THz emitter is high (1016 –1018 cm23 ), and, at these high densities, scattering works very efficiently to restore thermal equilibrium among the carriers in less than 100 fs. At thermal equilibrium the carriers can be described by the Fermi–Boltzmann distribution function, and carrier dynamics can be described by the Drude– Lorentz model. The short scattering time also implies a very short optical dephasing time T 2 , effectively erasing all coherence built into the carriers by the laser pulse. We also note that recent theoretical work suggests that the main contribution to the radiated THz pulse, at high bias fields (we typically use more than 100 kV/cm), is the drift current j(t). The displacement current, which can be represented as the nonlinear optical x2 process, mainly contributes at low bias fields and at excitation energies close to the semiconductor band gap.22,23 If we make the simple assumption that 50% of all photons in the laser pulse are converted into electron–hole pairs, we can plot the resulting photocurrent in the THz antenna by solving the coupled Eqs. (5) and (6) and inserting the solutions in Eq. (1). There are three regimes, depending on the value of v p t s . For v p t s ! 1 the screening field is much smaller than the bias field, and the current is a simple linear function of the applied field. The current rises in a

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time given by the scattering time convolved with the temporal shape of the laser pulse, and it decays by recombination or trapping of the free carriers. If we increase the laser power to a regime where v p t s @ 1, then the screening field increases to the same magnitude as the bias field on a subpicosecond time scale. The current increases quickly but then overshoots when the molecular field decreases because of screening. We note that, for t s . t laser , we observe plasma oscillations. For v p t s ' 1 the situation is, not surprisingly, intermediate between the high and the low power regimes. In Fig. 5 we show the normalized current pulses in the three regimes. The model assumes a scattering time of 30 fs and a carrier trapping time of 20 ps. The curves represent the three limiting cases for the product v p t s . For v p t s ! 1 the current decays exponentially with a decay time given by the carrier trapping time. For v p t s ' 1 the decay is faster, since the bias field is being reduced by the carrier drifting apart. For v p t s @ 1 the bias field is completely screened on a subpicosecond time scale, resulting in an overshoot of the current, as evident in the negative value. For longer scattering times ts this would give rise to THz plasma oscillations. By adjusting the scattering time ts and the screening factor h, we obtained very good agreement between the measured internal field dynamics and the field dynamics obtained from Eq. (4) with P sc and v obtained from Eqs. (5) and (6). We found that a scattering time ts of 30 fs and a screening factor h of 900 reproduced the measurements. The conversion factor from laser power to carrier density was kept fixed in the calculation so that the model carrier density scaled with the experimental laser power. We note that the phenomenological screening factor, assumed to be of the order of unity, is of the order of 103. The laser generates a spatial carrier distribution in the semiconductor similar to the Gaussian distribution of photons in the laser beam. As the carriers start to drift apart in the field, the carrier distribution equals two slightly displaced Gaussians. Only the nonoverlapping part of the distributions contributes to the screening field, whereas the contributions from carriers in the overlapping part of the Gaussians cancel. This reduces the effective screening, and consequently h increases. Using the same model, we calculated the peak THz signal as a function of the incident laser power or equiva-

Fig. 5. Time-dependent photocurrent calculated from the Drude–Lorentz model with screening included.

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lently the carrier density. The detailed shape of the radiated THz pulse depends not only on the dynamics of the photocurrent but also on the optical system used to image the pulse onto the detector. We address this point in the following sections but now we just concentrate on the peak amplitude of the THz pulse radiated from the THz emitter. In Fig. 6 we show the peak amplitude of the emitted THz pulse, measured as function of the incident laser power over four orders of magnitude. Note that, with more than 60-mW average power focused to approximately 5 mm 3 5 mm, a fluence of more than 3 mJ/cm2 per pulse at a laser repetition rate of 86 MHz, the THz emitter is damaged. From the same model calculations as those used to describe the internal field dynamics, we calculated the peak THz amplitude by taking the time derivative of the photocurrent. As described later in the paper, the effects of the THz optical system and the THz detector were also taken into account. Again the model correctly reproduces the experimental observations. Ultrafast screening of the photocurrent on a subpicosecond time scale is the dominating effect in the determination of the current pulse under typical experimental conditions. With the introduction of new ultrafast detection schemes there is a growing awareness of the role of screening. In our own work,15,21 ultrafast screening is observed and discussed. In the paper by Sha et al.13 and recent papers by Keil and Dykaar24 and by Alexandrou et al.,25 screening, or collapse of build-up bias, is observed. Common for all these papers is a field-sensitive detection with very high time resolution. We use the radiated THz pulse as the probe,15 Sha et al.13 uses the quantum confined Stark effect, and Keil and Dykaar24 and Alexandrou et al.25 use an electro-optic probe tip to measure the field. In the usual way of measuring ultrafast currents in THz antennas, one measures the photocurrent induced by the probe laser as a function of the delay with respect to a pump laser pulse. The current is typically measured with a high-impedance current amplifier with a bandwidth in the kilohertz range. If we consider such a pump–probe experiment in the v p t s @ 1 limit, we would expect that the current signal owing to the probe pulse would decrease rapidly after the electric field is screened by the carrier injected by the strong

Jepsen et al.

pump pulse, similar to the situation shown in Fig. 5. We have, however, neglected two important concepts, namely, the recharging of the photoconducting switch and the integrating properties of the current detection system. The THz antenna recharges as the mobile carriers are collected at the metal electrodes in the photoconducting gap. The time constant for this process is of the order of 100 ps. As the switch recharges, the carriers, which are still present, will give rise to an increased current density, and for the slow current amplifier the ultrafast screening shortly after the pump pulse will remain undetected. In order to see the ultrafast screening, the carrier lifetime must either be shorter than the recharging time, as it is in radiation-damaged silicon on sapphire, or the field probe must have high time resolution. In a recent paper by Keil et al.,26 this was realized with the second harmonic light generated from the biased metal– semiconductor interface in the THz antenna. In a recent paper we have discussed the influence of screening in ultrafast current measurements.27

3. RADIATION PATTERN FROM THE EMITTER Before we can perform a direct comparison between the measured THz pulses and pulses calculated from the modified Drude–Lorentz model, we must consider the optical problem of imaging the THz pulse from the emitter onto the THz detector. The THz emitter consists of a point dipole located at a dielectric interface, radiating a pulse with a bandwidth covering more than two orders of magnitude in frequency through a small lens attached directly to the chip, as illustrated in Fig. 1. The aim of this section is to describe the radiation pattern from the THz emitter. In a previous paper28 we calculated and measured the distribution of radiation from a typical THz emitter. Here we briefly describe the calculation method and present results showing that the radiation pattern is well described by a Gaussian beam with waist at the lens tip on the emitter. A collimating lens is attached to the emitter chip to enhance the radiated power in the forward direction and hence improve the signal strength at the detector, which is placed several tens of centimeters away from the emitter. This lens is attached directly to the reverse side of the emitter substrate, following the design by Fattinger and Grischkowsky.29 The lens is a truncated, spherical lens, cut off at a length h from the lens tip, where h is given by h 5 RL

Fig. 6. Induced current corresponding to the peak amplitude of the radiated THz pulse as a function of the average laser power driving the THz emitter. The solid curve is the peak amplitude calculated from the Drude–Lorentz model.

n , n21

(7)

where R L is the lens radius and n is the index of refraction of the lens material. The distance is chosen so that, within the ray approximation, a parallel beam incident upon the lens will have a focus inside the lens at a distance h from the lens tip. A truncated, spherical lens with a relatively small radius is used in most THz emitters,29 although this lens has large spherical and chromatic aberration. However, by placing the lens in direct contact with the emitter substrate, in-

Jepsen et al.

terfaces between air and the lens material are avoided, minimizing reflection losses and modulations in the signal caused by multiple reflections. A well-suited lens material is high-resistivity crystalline silicon, which is transparent and dispersion-free in the THz range.30 The absorption of THz frequencies in this material is extremely low, and the index of refraction varies only a few parts per thousand over the frequency range of interest. Furthermore, the index of refraction matches the index of the most common substrate materials [sapphire (n o 5 3.07, n e 5 3.40), GaAs (n 5 3.60), and Si (n 5 3.42) 30] very well. Lenses of quartz or sapphire can also be used, but these materials have a higher absorption coefficient, and the dispersion is larger, limiting the bandwidth of the THz pulses propagating through the lenses. We present here a model for the radiation pattern emitted from a lens-coupled THz emitter. The model consists of five steps, which are illustrated schematically in Fig. 7. The steps are (1) modeling of the source of radiation, (2) inclusion of internal reflection from the substrate interface, (3) wide-angle interference between the part of the radiation emitted directly into the lens and the part of the radiation reflected at the substrate interface, (4) coupling of the radiation out of the lens, and (5) propagation through air to the detector. Points (1)–(4) are full vector calculations, and the last point incorporates a common scalar approximation to the full electric field. The radiation pattern is modeled in the frequency domain, and, for simplicity, the index of the emitter substrate and the index of the lens material are assumed equal, hence ignoring the interface between the two media. We use sapphire as the substrate material (oriented with the extraordinary axis parallel to the THz polarization). In the experimental situation the index difference is very small (n sapp,e 5 3.40, n si 5 3.42 at 1 THz), and the assumption does not limit the accuracy of the model. We also neglect the effect of the thin semiconductor layer, which is also index matched to the substrate. The source of the radiation is the excited electron–hole plasma in the photoconductive gap of the emitter. We model this source as an oscillating electric point dipole lo-


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cated at the dielectric interface between the substrate and the air above the stripline structure. This approximation is valid since the spot size of the electron–hole pairs at the tight laser focus is smaller than 10 mm, which is much smaller than the wavelength of the radiation. The electric field from a small electric dipole in a homogeneous medium is31 E hom 5

F

G

m 0c 2 k 2 ~ nˆ 3 p! 3 nˆ exp~ ikr ! , 4p r

(8)

where nˆ is a unit vector in the direction of observation, p is the dipole moment vector, and k is the magnitude of the wave vector. The near-field contributions are neglected. Since the dipole is located at a dielectric interface, Eq. (8) must be modified to describe the THz emitter. The part of the electric field emitted toward the dielectric interface is reflected in accordance to Fresnel’s reflection laws. Since the dipole is located directly at the interface, the reflection results in wide-angle interference between the reflected part of the field from the plane substrate/air interface and the part of the field emitted directly into the substrate.32,33 The internal reflection is the reason why most of the generated THz pulse is emitted into the substrate and not into the space above the strip lines. Adding the two contributions to the electric field gives the radiation pattern inside the lens as calculated earlier by Fattinger and Grischkowsky.34 The electric field inside the lens is transmitted into free space over the lens surface. The transmission depends on the incident angle according to Fresnel’s laws, and the refraction of the field is given by Snell’s law. A feature very important to note here is that, since the index of refraction of the lens material is larger than the index of air, there will be an angle of total internal reflection above which the electric field will not be coupled out of the lens. This angle is determined alone by the index of refraction of the lens. The size of this angle defines a circular aperture on the lens surface, which causes diffraction features in the final radiation pattern. We calculate the electric field outside the lens at a given point Q by applying the Fresnel– Kirchhoff diffraction integral to the problem. This integral is the mathematical statement of Huygen’s principle of superposition of secondary wavelets35: ik EQ 5 2 4p

E

A

E surf

exp~ ikr ! ~ cos u o 2 n cos u i ! dA. r (9)

Fig. 7. Schematic illustration of the fundamental steps in our calculation of the radiation pattern from the THz emitter.

The cosine term in parentheses is the inclination factor, and the angles represent the outgoing direction with respect to the surface normal and the incident direction with respect to the surface normal, respectively. The electric field E surf is the electric field just outside the lens surface, determined in the previous step of the calculation. The integration runs over the total lens surface, if we assume that no other radiation sources are contributing to the signal. This integral is the scalar approximation in the theory, since only a scalar representation of the field on the lens surface is used. The detector in the THz setup is sensitive only to the x component of the field, so we insert this component of the field on the surface of the lens into Eq. (9).

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Fig. 8. (a) Time-domain shape of the pulse from the experiment performed to measure the radiation pattern from the THz emitter. (b) Frequency-domain spectrum of the time-domain data in (a). The spectrum extends to 1 THz.

To test the validity of the model described above, we measured the frequency-dependent radiation pattern from a THz emitter equipped with a truncated, spherical silicon lens to collimate the radiation. We used a setup that generates and detects THz pulses in a fashion similar to the THz setup shown in Fig. 1. To measure the radiation pattern from the emitter, we removed the paraboloidal mirrors and placed the detector directly in front of the emitter. The detector was placed on a translation stage, allowing it to be moved in the plane perpendicular to the forward direction of the emitter. As opposed to Fig. 1, the detector was used without a collimating lens, since a high spatial resolution was needed. The method to obtain the radiation pattern at a single frequency from the time-domain data is to measure the THz pulse shape at various positions in a plane perpendicular to the emitter. Then the pulses are Fourier transformed, and from the frequency spectra the field strengths of the individual frequency components can be extracted. In Fig. 8(a) a typical pulse generated and measured by the setup is shown in the time domain. The frequency spectrum of the pulse is shown in Fig. 8(b). The pulse is truncated at 9 ps to avoid multiple reflections in the spectrum caused by multiple passes of the THz pulse through the thin detector chip. The spectrum of the pulse extends to approximately 1.5 THz and peaks at 400 GHz. Compared with other THz setups, the bandwidth is small, and the noise level is high, which is due to the full-gap

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excitation method of the emitter and the large size (20 m m350 mm) of the detector, which is considered later in the paper. Furthermore, a detector with collimating THz optics, as shown in Fig. 1, has a roughly 1000 times larger signal-to-noise ratio than does the detector used here. Consequently, we are able to obtain radiation patterns of up to approximately 1 THz. Figure 9 is a comparison between experimental data and numerical integration of Eq. (9), where all parameters mimic the experimental situation. The distance from the lens tip to the plane of the detector is 35 mm, and we show four sets of patterns at frequencies from 0.234 to 0.820 THz. There is good quantitative agreement between the experimental and the numerical data at all frequencies, and several features are worth noticing. We see that the experimental measurements reproduce the first of the predicted diffraction fringes caused by the total-internal-reflection aperture on the lens surface. The central lobe of the radiation pattern is fitted very well by a Gaussian-beam profile, and at larger distances the overall radiation pattern is represented very well by a Gaussian beam with a waist w 0 5 3.82 mm and a wavelength-dependent divergence u 5 83 • l (wavelength in millimeters gives the divergence in milliradians). These numbers are specific for a lens with diameter R L 5 5.0 mm cut in accordance to Eq. (7). We measured radiation patterns at other distances and found similar agreement with the model. The actual size of the total-internal-reflection aperture on our lens is 4.49 mm, with the difference between the calculated waist of the Gaussian beam and the aperture size probably being caused by the strongly inhomogeneous electric-field distribution over the aperture. The central part of the radiation pattern from the THz emitter can also be measured by use of the setup shown in Fig. 1. Here we mount a small silicon lens, identical to the lens on the emitter, on the detector with two transla-

Fig. 9. Experimentally determined radiation patterns compared with numerical results obtained by numerical integration of Eq. (7). Four cases are shown, all at the distance 35 mm from the emitter, at various frequencies from 0.234 to 0.820 THz.

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tion stages so that the lens can be slid along the x and the y directions indicated in the inset of Fig. 1. In principle the detector itself should also be moved together with the lens, but if the shift of the lens is small compared with the radius of the lens ( R L 5 5 mm), measuring the pulse shape as a function of x (or y) will give the distribution of radiation in the detector plane. This distribution is the original radiation pattern modified by the optics in the beam path, and because of the simple configuration of the optics, this modification is only a scaling of the emitted radiation pattern to a spot size determined by the wavelength of the frequency components. In Fig. 10 we plot the time-domain THz pulses obtained by scanning the lens 60.38 mm along the x direction, in steps of 20 mm between the successive curves. As evident from the figure, the most intense and the highest bandwidth THz pulses are observed only when the lens is precisely centered at the detector (trace marked with filled circles in Fig. 10) within a tolerance of only 620 mm. As with the radiation pattern shown in Fig. 9, we Fourier transformed the time-domain pulses in Fig. 10, and for each frequency component we plotted the amplitude as a function of the lens position. The results shown in Fig. 11 are in complete agreement with the results from Fig. 9, scaled with the demagnifying factor characteristic of the THz optical system described in Section 4. Again we see that, since the spot size of the highest frequencies is very small (1.927 THz → 56 mm), the lens must be very carefully aligned to obtain a high-bandwidth THz pulse. The spot size at the detector is used in the following to test our model for the THz optical system.

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Continuing the beam propagation to a distance d into the lens on the detector gives the following propagation matrix:

M tot 5

F

fd ~ n 2 1 ! 1 R L d ~ D 2 2 ! 21 R Ln f~ n 2 1 ! 1 R Ln~ D 2 2 ! R L nf

G

d 2 n . 1 2 n (11)

For a while we will concentrate on the D 5 2 situation. The matrix in Eq. (11) is then used to find the spot size w 1 (d) and the radius of curvature R 1 (d) inside the lens: w 1~ d ! 5

1 n p w 0R L

A@ d ~ 1 2 n ! 1 nR L # 2 p 2 w 04 1 d 2 R 2L l 2 , (12)

4. TERAHERTZ OPTICS Since the radiation pattern from the emitter is approximately a Gaussian beam, we will apply a Gaussian-beam formalism to describe the propagation of the different frequency components of the THz pulse through the optical components to the detector area. The initial spot size is w 0 , and the initial radius of curvature of the wave fronts R 0 is assumed infinite. The optical system guiding the THz pulse to the detector consists of the truncated, spherical lens on the emitter, two off-axis paraboloidal mirrors, and a focusing lens on the detector identical to the lens on the emitter. The distance between the two paraboloidal mirrors is D 3 f, where f is the focal length of the paraboloidal mirrors and D is the separation of the mirrors relative to f. The distance from the truncated, spherical lenses on the emitter and detector to the respective paraboloidal mirrors is f. By modeling the paraboloidal mirrors as thin lenses with focal lengths f, we can calculate the propagation matrix for the optics to the lens tip of the detector (i.e., the part of the optics that can be easily adjusted):

M air 5

F

21

0

D22 f

21

G

.

Fig. 10. Time-domain THz pulses obtained by sliding the silicon lens along the x direction, as indicated in Fig. 1. The pulse trace marked by filled circles corresponds to the best alignment of the lens, giving the highest bandwidth and the strongest amplitude. Each trace corresponds to a 20-mm movement along the x direction.

(10)

Here it is evident that the separation constant D should be equal to 2, since the optics then creates an undistorted image of the original electric field on the detector lens tip.

Fig. 11. Measured radiation patterns for four different frequencies, as indicated in the legend on the figure. The low signal strength at high frequencies is caused by the high-frequency rolloff of the THz signal, as seen in Fig. 2(b).

2432


R 1~ d ! 5

@ d ~ 1 2 n ! 1 nR L # 2 p 2 w 04 1 d 2 R 2L l 2

1 ! p 2 w 04

@ d ~ 1 2 n ! 1 nR L #~ n 2

2

dR 2L l 2

Jepsen et al.

.

(13) Finding the value of d, where R 1 (d) becomes infinite, or, equivalently, solving the equation ] w 1 / ] d 5 0, gives the position and the size of the focus inside the lens: R Ln~ n 2 1 !

d focus 5

~n 2 1! 1 2

w 1 ~ d focus! 5

pw0

A

S D R Ll

2

,

(14)

p w 02

lR L

~n 2 1! 1 2

S D R Ll

2

.

(15)

p w 02

These equations show that for long wavelengths the truncated, spherical lens has strong chromatic aberration. For wavelengths l ! p w 02 /R L , however, the position of the focus is at the position h predicted by Eq. (7), justifying the lens design: lim d focus 5 R L

l→0

lim w 1 ~ d focus! 5

l→0

n , n21

RL RL c l 5 , pw0 n 2 1 pnv 0 n 2 1 (16)

where n 5 c/l is the frequency. Since the lens is cut to place the detector gap at the distance h given by Eq. (7) from the lens tip, we now fix d to this distance, and explore the effect of changing the spacing between the paraboloidal mirrors. In Fig. 12 we plot w 1 versus frequency for various choices of D from D 5 2.0 to D 5 2.8. This is done by using the full propagation matrix in Eq. (11) and setting d 5 R L n/(n 2 1). Together with the calculated curves we show experimentally determined spot sizes for comparison. The experimental data were obtained by fitting the central lobe of the field distributions in Fig. 11 with a Gaussian-beam profile. The ex-

perimental data correspond to D ' 2, and the agreement with the calculations is good, indicating the neardiffraction-limited performance of the optical system. In the optimum case (D 5 2) the spot size varies in inverse proportion to frequency, whereas for the other situations illustrated, the spot size settles at a nonzero minimum value determined by the separation of the paraboloidal mirrors. A small spot size corresponds to a high field strength and consequently a large detector signal, so the situation where w 1 } 1/n is highly favorable when a good high-frequency response of the detector is needed.

5. TERAHERTZ DETECTOR RESPONSE This section describes the most important mechanisms involved in the detection of the THz pulses. The ideal detector should be able to provide exact information about the signal waveform, but different factors influence and limit the detection capabilities, resulting in reshaping of the signal, owing to finite bandwidth and sensitivity of the detector. Here we examine two important aspects of the signal detection. The geometry of the detector influences the speed and the sensitivity of the detector, mainly owing to the finite detection area. Since the detection is based on electron flow in the photoconducting material of the antenna, the carrier dynamics in this material must also be considered. First, we consider the effects of the interplay between the frequency-dependent spot size of the THz beam and the finite size of the detector area. The detector in the THz experiment detects the radiation by measuring a current flow caused by the average electric THz field over the detector area. This area has length L 5 L M 1 L S , where L M is the total length of the metal electrodes and L S is the length of the switch area between the two electrode tips. The width of the electrodes and the gap area is d (see the inset of Fig. 1). The signal from the detector is the current drawn by the THz electric field, and this current function is in the following designated as the geometrical response function (GRF). When the laser pulse shorts the photoconductive gap, the THz electric field drives the carriers over the conducting bridge formed by the two metal electrodes and the photoconductive area. Therefore the detector area is defined by the total length L and the width d. Normal antenna theory is not applicable in the determination of the GRF since the electric field amplitude can vary substantially over the spatial extend of the antenna. The electric field incident upon the detector is a superposition of Gaussian beams of different frequencies, and here we concentrate on only one of these components, which is E ~ x, y ! 5 E 0 exp@ 2~ x 2 1 y 2 ! /w 12 # .

Fig. 12. Calculated spot size versus frequency for different choices of the separation constant D. Shown on the same plot are experimental measurements of the spot size for D ' 2.

(17)

The frequency dependence enters by means of v1 . In Section 4 it was found that the spot size w 1 varies in inverse proportion to the frequency n of the radiation. High-frequency components are confined closer to the propagation axis than components of lower frequency. Owing to the design of the lens that focuses the THz beam onto the detector, we can safely ignore the curvature of

Jepsen et al.


the phase fronts of the THz beam and consider only the simplified beam profile in Eq. (17). The GRF is proportional to the current flowing across the detector gap. This current is determined by the potential and the resistance across the gap. The detector area can be thought of as two serially connected resistors. One resistor is formed by the two metal electrodes, of total length L M and resistivity rM . The other resistor is formed by the photoconductive gap (switch area) of length L S and time-averaged resistivity rS . The average resistivity rS is much larger than rM owing to the low duty cycle of the driving laser (100 fs/10 ns 5 1025 ). The total resistance over the detector is then

r ML M 1 r SL S r SL S ' , td td

R5

(18)

where d is the detector width and t is the thickness of the semiconductor layer covering the detection area. The resistivity rS depends on the photogenerated carrier density, which for homogeneous illumination of power P laser scales as L Sd , j P laser

rS 5

¯ ~L 1 L ! U E M S 5 td j P laser . R L 2S

(20)

The average electric field across the detector area is E ~ L,d ! 5 5

E E

1 Ld

L/2

d/2

2L/2

2d/2

E 0 p w 12 Ld

Erf

E ~ x, y ! dxdy

S D S D

L d Erf , 2w 1 2w 1

1 ce 2 0

EE `

`

2`

2`

E 2 ~ x,y ! dxdy 5

1 p w 12 c e 0 E 02 2 (22)

⇒E 0 5

2 w1

A

P THz . pce0

(23)

By inserting Eq. (23), together with Eq. (16), into the expression for the detector current, Eq. (20), we get I ~ n ! 5 j P laser

A F

3 Erf

cP THz

2R L t 2 p e 0 L Sd v 0~ n 2

I n →0 5 j P laser

A

cP THz 2 ~ n 2 1 ! pv 0 t ~ L S 1 L M !

pe0

L 2S

c 2R L

n (25)

by applying the approximation Erf ( x ) → 2 x / Ap for x → 0. This shows that the response increases in proportion to the frequency of the THz radiation. This type of response results in a signal differentiation, since, as is well known from Fourier theory, a multiplication of a signal by n in the frequency domain corresponds to a differentiation in the time domain (2i v ↔ ] / ] t). For small detectors this approximation is valid up to several THz, so a signal very close to the differentiated original signal is detected. In the other limit, where the wavelength of the THz radiation is much smaller than the detector dimensions, the GRF approaches

1 1! n

G F

I n →` 5 j P laser

A

cP THz

2R L t 2 p e 0 L Sd v 0~ n 2

1 1! n

,

(26)

where the approximation Erf ( x ) → 1 for x → ` is applied. Dividing a signal by n in the frequency domain corresponds to an integration of the signal in the time domain (1/i v ↔ * dt), and hence a large detector works as an integrator of the original signal. A finite-sized detector will for low frequencies work as a differentiator, and for high frequencies as an integrator, a fact that in some situations can complicate the interpretation of the detected signal. The approximate cross-over frequency between the two types of behavior can be found by equating Eqs. (25) and (26). The maximum of the GRF is also located approximately at this frequency:

(21)

where Erf( x ) 5 (2/A p ) * 0x exp(2u2)du is the error function. The peak strength of the electric field, E 0 , can be expressed in terms of the total power in the THz beam: P THz 5

In the low-frequency limit, where the wavelength of the THz radiation is much longer than any dimension of the detector (l @ L S , L M , d), the GRF simplifies to

(19)

provided that the response of the photoconductive material is linear with respect to the laser power (no saturation effects). Here j is a conversion factor between laser power and number of photogenerated carriers. The av¯ across the detector gives rise to a erage field strength E ¯ (L 1 L ), so the average potential difference U 5 E M S current is I5

2433

G

L ~ n 2 1 !pw0 d ~ n 2 1 !pw0 n Erf n . 2 cR L 2 cR L (24)

n max '

cR L ~ n 2 1 !v0

1

Ap d ~ L S 1 L M !

.

(27)

In Fig. 13 we plot the GRF for several dimensions of the detector as functions of the frequency n. The approximate positions of the response maxima, as calculated from Eq. (27), are indicated on each curve by a vertical line. In Fig. 13(a), three parameter sets are illustrated. The photoconductive gap spacing and the electrode spacing are kept constant (L S 5 10 mm, d 5 20 mm), and the metal electrode length L M is varied, as indicated in the figure legend. Since the gap area is unchanged, this figure corresponds to the situation with constant laser intensity in the photoconductive gap. As L M is increased, the maximum of the GRF shifts to lower frequencies, and the over-all sensitivity increases. It is interesting to note that the high-frequency response is unaffected by the variation of L M .

2434


Jepsen et al.

where T is the repetition time of the laser source (11 ns). If the conductivity g(t) is a delta function, the measured current j(t) directly represents the field of the THz pulse. If g(t) is a square function, with a width shorter than the repetition time of the laser but much longer than the duration of the THz pulse, the measured current represents the time integrated field of the THz pulse. We adopt a model for the detector response previously published by Grischkowsky and Katzenellenbogen,20 with a conductivity rise time given by a collision time ts and a fall time determined by carrier recombination and/or trapping. This model is completely analogous to our Drude–Lorentz model for THz generation, apart from the assumption that, in the detector, the current is strictly linear with respect to the THz field. Assuming this linearity, we can separate the spatial part of the detector response, given by the size of the antenna, from the dynamical response of the detector material.

6. TERAHERTZ SIMULATIONS Fig. 13. (a) GRF for three parameter sets. The photoconductive gap length L S and the electrode width d is kept constant, while the metal electrode length is varied. (b) The GRF for four different parameter sets. The metal electrode length L M and the electrode width d are kept constant, and the photoconductive gap length L S is varied.

In Fig. 13(b), four different parameter sets are illustrated. The metal electrode length and the electrode width are kept constant (L M 5 10 mm, d 5 10 mm). The photoconductive gap length L S is varied, so the concentration of photogenerated carriers is now varied, owing to the variation in laser intensity in the photoconductive gap. As L S is increased (top curve to lower curve), the response maximum again shifts to lower frequencies, but this time the overall sensitivity also decreases, mainly because of the lower laser intensity in the larger gaps. The two parts of Fig. 13 are normalized by the same normalization factor, so they can be compared with each other. The interesting feature is that, for frequencies higher than approximately 4.5 THz (with the present parameters), the smallest detector (L S 5 L M 5 d 5 10 mm) has the largest GRF of all the considered examples. This illustrates that, to obtain a large signal and high bandwidth, one has to use a very small detector area, which is also seen experimentally. The final part of the detection process takes place in the photoconductive detector, where the incident THz field, E(t), is sampled by the time-dependent conductivity, driven by the probe laser, as illustrated in Fig. 1. The resulting photocurrent as a function of the delay between the pump laser and the probe laser is the experimentally measured quantity in a THz experiment. Before we can compare the measured THz pulse with the simulated THz pulses, we must adopt a model for the time-dependent conductivity of the detector material. The photocurrent can be written as

j~ t ! 5

1 T

E

T

0

E ~ t ! g ~ t 1 t ! dt,

(28)

With simple models for each element of the THz setup, the THz emitter, the THz optics, and the THz detector, the THz pulses can be simulated with a small personal computer. The simulation proceeds as follows: A fourthorder Runge–Kutta routine was used to solve Eqs. (5) and (6), together with Eqs. (1) and (2). As we saw earlier, this gave excellent agreement with the measured field dy-

Fig. 14. (a) THz pulse simulation shown together with the experimental data. The simulation parameters were the same as those used to obtain the results shown in Figs. 4 and 6. (b) The corresponding frequency spectrum of the simulated pulse shown together with the experimental spectrum.

Jepsen et al.


2435

Katzenellenbogen and Grischkowsky, who used very small antennas and low-loss substrates.36 If we increase the scattering time in the simulation, there is a pronounced change in the pulse shape and spectrum. For a moderate carrier density, v p t s < 1, the pulse broadens, and the spectrum shifts to lower frequencies. The frequency down-shift was observed experimentally by Howells et al.37 and by Hu et al.38 From the simulation it is also evident that in the limit, where the scattering time is longer than the laser pulse duration, the radiated THz pulse is simply the plasma oscillation, and the peak frequency of the spectrum scales with the square root of the carrier density or, equivalently, the laser power.

7. CONCLUSIONS

Fig. 15. Simulated and experimentally determined frequency spectrum of a THz pulse measured under various conditions: (a) the gap spacing L of the detector is varied, (b) the detector rise time is varied, and (c) the optically induced carrier density is varied.

namics and the laser-power versus THz-amplitude characteristics. We took the THz optics into account by Fourier transforming the pulse to the frequency domain and multiplying by the response function given in Eq. (24). We note that neglecting the phase shift in Eq. (24) is equivalent to introducing a minor error in the response function, which is not strictly Hermitian, as it should be in linear response theory. The error, represented by a small imaginary part of the THz field E(t), when one transforms back to the time domain, is, however, very small. While still in the frequency domain, the THz field is corrected for the dispersion and the loss caused by the emitter and the detector substrate materials. The pulse is then transformed back to the time domain and convolved with the temporal response of the detector conductivity, Eq. (28), and the finite duration of the probe laser pulse. By using the same parameters for the THz emitter as in the calculation of the field dynamics shown in Fig. 4, assuming a paraboloidal mirror spacing of 2 f, a detector antenna of 30 mm length and 10 mm width, rise time of 100 fs and fall time 800 fs, we obtain the THz pulse and the corresponding spectrum shown in Fig. 14. The agreement between simulation and observation is in general very good. If we follow the pulse simulation step by step, we notice that the bandwidth is mainly limited by the size of the detector antenna and the dielectric losses in the GaAs substrate material. In Fig. 15 we show the THz spectrum obtained from the model for different values of the detector rise time, gap spacing L, and laser power. If we make the antenna gap smaller and remove the substrate losses, we see that the bandwidth increases, and a pronounced ringing develops after the pulse. The simulated pulse shape and spectrum are qualitatively in agreement with the results obtained by

In conclusion, we have performed a series of experiments that illustrate the properties of a standard THz-pulse setup. Each element of the setup, from the THz emitter by means of the THz optical system, to the THz detector, was modeled by application of the simplest possible theories. Combining all these elements into a simulation program allowed us to calculate the measured properties of the THz system, and we have observed good agreement with experimental data, using only a limited number of free parameters. From experiments and simulation, three important conclusions emerge: (I) The THz source can be described simply as a plasma oscillation of the optically injected electron–hole plasma. However, at room temperature and high carrier concentrations, the characteristic features of a plasma oscillation are masked by the very short scattering time, t s < 50 fs. (II) Since the energy stored in the THz emitter is very small, the local bias field is rapidly screened by the carriers, and thus the dynamics of the local field becomes an important parameter in the pulse-shaping process. (III) Finally, we have shown that the optical system used to image the THz radiation onto the detector and the size of the detector itself are the main bandwidth-limiting factors in the present setup. With diffraction-limited imaging, the spot size of the THz radiation becomes comparable to the detector dimensions at higher frequencies, thereby limiting the bandwidth. Based on the simple models introduced here, we believe it is simple and straightforward to optimize specific THz systems intended for high time resolution, high sensitivity, or imaging THz system.39

ACKNOWLEDGMENTS We acknowledge the many discussions with M. van Exter and D. R. Grischkowsky concerning the details of the THz pulses. Most of the work presented here originated in these discussions. We also acknowledge D. R. Grischkowsky for the generous donation of a THz antenna in the initial stages of this work. The excellent chip mounts were constructed by P. Strand.

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