High-bandwidth terahertz radiation from ponderomotively accelerated

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High-bandwidth terahertz radiation from ponderomotively accelerated ... Terahertz generation from high intensity excitation of a semiconductor ... When a fem-.
APPLIED PHYSICS LETTERS 86, 064106 共2005兲

High-bandwidth terahertz radiation from ponderomotively accelerated carriers using Bessel–Gauss femtosecond pulses K. J. Chau, A. L. Dechant, and A. Y. Elezzabia兲 Ultrafast Photonics and Nano-Optics Laboratory, Department of Electrical and Computer Engineering, University of Alberta, Edmonton T6G 2V4, Canada

共Received 5 August 2004; accepted 17 December 2004; published online 4 February 2005兲 Terahertz generation from high intensity excitation of a semiconductor plasma by a radially polarized Bessel–Gauss femtosecond pulse is modeled. The results are compared with Gaussian pulses of equivalent fluence. Due to carrier confinement, a radially polarized Bessel–Gauss pulse exerts a significantly stronger ponderomotive force on photocarriers than a Gaussian pulse, resulting in an order of magnitude increase in the THz emission power. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1861135兴 The generation of free space terahertz 共THz兲 radiation is important for spectroscopic and imaging applications. Using femtosecond laser pulses, a variety of coherent THz generation methods have been explored utilizing electron-hole 共e -h兲 acceleration in semiconductors via an externally applied1 or built-in2–6 surface field and optical nonlinearities.7,8 Alternatively, the interaction of intense laser light and plasma has attracted much attention as a source of high power, broadband THz radiation. Hamster et al.9,10 have demonstrated THz emission resulting from high intensity 共⬃1019 W / cm2兲 photoexcitation of gas and metallic targets. The THz emission was attributed to ponderomotive forces, within the focus of a pulse, acting on the photoionized plasma. Ponderomotively induced space charge fields in excess of 108 V / cm were produced. However, very high threshold intensities, required to observe significant ponderomotive effects in the gaseous and metallic media, limit its feasibility for practical application. Semiconductor e-h plasmas, on the other hand, are attractive THz sources due to the low excitation energies and low effective electron mass. Ponderomotive acceleration of e-h plasmas in a semiconductor can be achieved by producing either extreme spatial field gradients or by high fluence excitation. The damage threshold of the semiconductor restricts the latter requirement. However, the polarization state and intensity distribution of the laser beam can be manipulated to produce the spatial gradients necessary to achieve strong ponderomotive interaction. In this letter, we examine THz generation based on the ponderomotive acceleration of carriers by an intense, radially polarized, Bessel–Gauss, femtosecond laser pulse and compare it with a linearly polarized, Gaussian pulse. Radially polarized pulses have attracted recent attention for applications in free electron bunch acceleration.11 The advantage of these pulses over linearly polarized Gaussian pulses is that subdiffraction spot sizes are attainable,12,13 and at the focus, extreme radial and longitudinal field gradients can accelerate electrons ponderomotively with high efficiency. When a femtosecond pulse is focused onto a high mobility semiconductor surface, the laser field acts to both generate and accelerate 共near ballistic兲 charge carriers. The ponderomotively induced transient current densities result in the creation of an instantaneous Hertzian dipole that emits THz radiation. a兲

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As shown in Fig. 1 共top兲, the radially polarized beam is focused onto a semiconductor slab with its surface placed at z = 0. The fields in the semiconductor half-space are obtained by angular summation over a Bessel–Gauss weighted amplitude function.14 Using Maxwell’s boundary conditions at the surface, the following analytical forms of the time-varying electric field components: Er共r,z,t兲focus = Eo





t共␪兲A共␪兲␰共␪兲cos共␪兲J1共rk1 sin ␪兲

0

⫻e关−i␻t+ik2␰共␪兲z−␣z−t Ez共r,z,t兲focus = Eo

k1 k2





2/␶2兴 o

d␪ ,

共1兲

t共␪兲A共␪兲sin共␪兲J0共rk1 sin ␪兲

0

⫻e关−i␻t+ik2␰共␪兲z−␣z−t

2/␶2兴 o

d␪ ,

共2兲

are obtained, where Eo is the amplitude of the electric fields

FIG. 1. Schematic of the THz emitter 共top兲. Comparison of C-FDTD 共bottom-left兲 and the analytical 共bottom-right兲 descriptions of a focused radially polarized beam inside a GaAs semiconductor.

0003-6951/2005/86共6兲/064106/3/$22.50 86, 064106-1 © 2005 American Institute of Physics Downloaded 31 Mar 2007 to 129.128.216.34. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

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Appl. Phys. Lett. 86, 064106 共2005兲

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at the focus, ␶o is the temporal full width half maximum 共FWHM兲 of the pulse, 0 艋 ␪ 艋 ␤ is the focusing angle measured from the z axis defined by the numerical aperture of the lens, t共␪兲 is the Fresnel transmission coefficient, ␰共␪兲 = 共1 − k21 / k22 sin2 ␪兲1/2, k1 and k2 are the wave vectors in vacuum and semiconductor, respectively, ␣−1 is the absorption depth in the semiconductor, J0 and J1 are the Bessel function of zeroth and first order, respectively, and A共␪兲 is the effective Bessel–Gauss weighted amplitude function given by A共␪兲 = exp共−␨2兲J1共2␨兲sin ␪ cos1/2 ␪, where ␨ = ␹ sin ␪ / sin ␤ and ␹ is the lens beam fill factor. When an above-band-gap Bessel-Gauss femtosecond laser pulse is focused onto the intrinsic semiconductor surface, an e-h plasma is photogenerated. Within a few field oscillations, the quiver motion of charge carriers is directed along the spatial field gradients from high to low field regions. The potential responsible for this effect is described by e2 U p共r,z,t兲 = ⵜ 关E共r,z,t兲focus · E*共r,z,t兲focus兴 , 2m*␻2

共3兲

where e and m* are the carrier charge and effective mass, respectively. At high carrier densities 共⬎1018 cm−3兲 and in the time scales of interest where carrier scattering rates are high, the Boltzmann transport picture can be used to describe carrier motion. In this formalism, carrier transport is given by an effective electron current density



je共r,z,t兲 = − e␮en共r,z,t兲ⵜ៝ ␺共r,z,t兲 +



U p共r,z,t兲 , e

共4兲

where the holes are taken to be stationary, ␮e = 8500 cm2 / Vs is the electron mobility, and n共r , z , t兲 is the electron density. The temporal evolution of the carrier density is described via the continuity equations

⳵ n共r,z,t兲 ␣ 1 = I共r,z,t兲 + ⵜ៝ · ៝j e共r,z,t兲, ⳵t ប␻ e

共5兲

⳵ p共r,z,t兲 ␣ = I共r,z,t兲, ⳵t ប␻

共6兲

and

where p共r , z , t兲 is the hole density, and I共r , z , t兲 is the intensity of the laser pulse. Charge separation induced by ponderomotive carrier transport is coupled self-consistently to the electrostatic potential through the application of Poisson’s equation: ⵜ2␺共r,z,t兲 =

e 关n共r,z,t兲 − p共r,z,t兲兴, ␧ 2␧ o

共7兲

where ␧2 is the relative permittivity of the semiconductor and ␧0 is the free space permittivity. Focusing at high numerical aperture 共NA⬎ 0.7兲 is essential for effective field spatial localization and creation of large ponderomotive potentials. The spatial distribution of the radial Ir共r , z , t兲focus and longitudinal Iz共r , z , t兲focus intensities inside an intrinsic GaAs semiconductor 共NA= 0.9, ␧2 = 13.4, ␣−1 = 0.8 ␮m兲 surface are shown in Fig. 1 共bottom兲 for a ␭ = 800 nm laser beam. A plot of a cylindrical finite difference time domain 共C-FDTD兲 simulation of the fields using the above parameters demonstrates excellent agree-

FIG. 2. Spatiotemporal evolution of the radial 共left兲 and longitudinal 共right兲 current densities within the semiconductor at times ⫺5, 0, and 5 fs relative to the peak of the radially polarized Bessel–Gauss pulse.

ment with the analytical field description. The profile of Ir共r , z , t兲focus is toroidally symmetric, while the profile of Iz共r , z , t兲focus is axially symmetric. However, due to the continuity of the displacement vector at the semiconductor–air interface, the peak amplitude of Ir共r , z , t兲focus ⬇ 100 Ir共r , z , t兲focus inside the semiconductor. For an energy fluence of 78.3 mJ/ cm2, extreme intensity gradients, 兩ⵜrI共r , z , t兲兩 = 1021 W / m3 and 兩ⵜzI共r , z , t兲兩 = 1022 W / m3 are established in the focal region. The system of Eqs. 共4兲–共7兲 are solved for all time using a predictor corrector method where at each step, ៝j e共r , z , t兲 is determined self-consistently. At r = 0 and z = 0, Neumann boundary conditions are imposed on ៝j e共r , z , t兲 and at r ⬎ wo, and z ⬎ ␣−1, ៝j e共r , z , t兲 is set to zero. The emitted THz electric field originating from the focal volume V, is evaluated from tTHz共␸兲 d Eជ 共D, ␸,t兲THz = 4␲␧2␧oc2 dt



ជje共r⬘,z⬘,t兲 dV, 兩D − r⬘兩

共8兲

where D is the distance of the detector from the origin, ␸ is the detection angle with respect to the z axis, defining the emission cone, c is the speed of light, and tTHz共␸兲 incorporates the transmission coefficient due to losses at the air– semiconductor interface and far-field cancellation of the emission from the current density component along aˆr. The transmission coefficient is expressed as tTHz共␸兲 2 −1 2 冑1 − ␧−1 = ␳共␸兲共sin 2␸ + 2␧−1/2 where 2 2 sin ␸ sin ␸兲 sin ␸ −1 −1/2冑 −1 2 2 ␳共␸兲 = 4共1 − ␧2 sin ␸兲sin␸ or ␳共␸兲 = 4␧2 1 − ␧2 sin ␸ for the aˆr or aˆz current density components, respectively.5,15 Figure 2 depicts the spatiotemporal evolution of the radial and longitudinal current density components, jr共r , z , t兲 and j z共r , z , t兲, near the surface when a GaAs slab is excited with a 50 fs, 800 nm, radially polarized, Bessel–Gauss pulse. The images are chosen to illustrate the temporal window where the current densities show maximum change. Beyond this temporal window, the current density spatial profiles change slowly and reduce in amplitude. In the radial direction at t = −5 fs, photoexcited electrons are bunched and ejected from the high intensity lobe of the pulse. Electrons bunching near r = 0 results in a high electron density and a large carrier screening potential. Complete current density

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Appl. Phys. Lett. 86, 064106 共2005兲

Chau, Dechant, and Elezzabi

FIG. 4. THz emission power vs energy fluence for Bessel–Gauss and Gaussian fs pulses.

FIG. 3. THz pulse components originating from the 共a兲 radial and 共b兲 longitudinal current density components detected 1 cm away from the semiconductor surface for fluence values of 共i兲 38 mJ/ cm2, 共ii兲 78 mJ/ cm2, and 共iii兲 99 mJ/ cm2. The emitted radiation pattern is depicted in the inset. Maximum THz emission is obtained at ␸ = 11° and ␸ = 14° relative to the normal for pulse components originating from jr共r , z , t兲 and jz共r , z , t兲. The bottom inset in 共b兲 depicts the THz spectrum.

screening is evident near the origin at t = 0 fs, where jr共r , z , t兲 has reversed polarity. Charge screening effects on jr共r , z , t兲 in the region r ⬎ 0.3 ␮m are weaker and thus, screening is incomplete. After the peak of the pulse, screening forces dominate and jr共r , z , t兲 is reduced. In the longitudinal direction, the leading edge of the excitation pulse induces a highly localized electron current density near the surface, as shown in Fig. 2 at t = −5 fs. With increasing ponderomotive force, a large current density is attained at t = 0 fs, while the electrostatic screening contributions to j z共r , z , t兲 also increase. Screening effects are evident where jz共r , z , t兲 near the surface has been reduced significantly at t = 5 fs. The net result is a time-varying dipole that oscillates in both aˆr and aˆz directions, emitting radiation in the THz regime. The emitted THz radiation for optical fluences varying from 38 to 99 mJ/ cm2 共corresponding to peak n共r , z , t兲 ⬃ 1021–1022 cm−3兲 is evaluated 1 cm away from the surface, at ␸ = 14°. Figure 3 illustrates the detected THz pulse components originating from the radial and longitudinal current density components for various fluence values. THz pulses with FWHM values of less than 22 fs and ultrahigh bandwidths greater than 25 THz are generated. For the highest fluence, peak-to-peak THz field amplitudes exceed 1 V/cm at the detector location. Due to the vector far-field cancellation, the amplitude of the THz emission from the radial current density component is 105 times lower than the emission from the longitudinal current density component. As a result, the net THz pulse in the far-field predominantly originates from jz共r , z , t兲. The positive peak of the THz pulse is attributed to ponderomotive electron acceleration, and the negative peak is attributed to screening and electron deceleration. In the trailing portion of the excitation pulse, electron motion is

driven to steady-state values. Since n共r , z , t兲 ⬀ I共r , z , t兲 and U p共r , z , t兲 ⬀ 兩 ⵜ I共r , z , t兲兩 ⬇ I共r , z , t兲ᐉ−1 where ᐉ is an effective ponderomotive interaction length, the bipolar pulse shape is maintained at all excitation fluences. Figure 4 compares the THz emission power as a function of the average energy fluence for radially polarized Bessel– Gauss and linearly polarized Gaussian pulses focused to similar spot sizes. For fluence values from 5 mJ/ cm2 up to the damage threshold of GaAs 共100 mJ/ cm2兲, THz emission driven by the radially polarized Bessel–Gauss pulse is an order of magnitude greater than with a Gaussian pulse. Clearly, in this fluence regime the Bessel–Gauss pulse interacts with a significantly higher efficiency than a Gaussian pulse. This increase is attributed to carrier confinement and increased ponderomotive interaction at the focus of the Bessel–Gauss pulse. In conclusion, we find that radially polarized pulses are advantageous over linearly polarized Gaussian pulses for THz generation from the high intensity excitation of semiconductors. THz generation using radially polarized Bessel– Gauss pulses may find initial applications in ultrahigh-power THz generation from gaseous plasmas, where damage thresholds do not limit the pump pulse intensity and near relativistic electron acceleration is possible. A. L. Dechant’s contributions were limited to the C-FDTD simulations. 1

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