A. Gurtler et al.
SOQUE LASERS '98 Proceedings
EXPERIMENTAL AND NUMERICAL STUDIES OF THZ PULSE PROPAGATION IN THE
NEAR AND FAR FIELD
A. Gurtler, C. Winnewisser, H. Helm, and P. Uhd Jepsen
Department of Molecular and Optical Physics
University of Freiburg
Hermann-Herder-Str. 3, D-79104 Freiburg, Germany
e-mail: [email protected]
Abstract
In this paper we present a detailed investigation of the propagation properties of beams of ultrashort THz pulses
emitted from large aperture (LA) antennas. The large area of the emitter is demonstrated to have substantial inuence
on the temporal pulse prole both in the near eld and the far eld. We perform a numerical analysis based on scalar
and vectorial broadband diraction theory and are able to distinguish between near eld and far eld contributions to
the total THz signal. We nd that the THz beam from a LA antenna propagates like a Gaussian beam, and that the
temporal prole of the THz pulse, measured in the near eld, contains information about the temporal and spatial
eld distribution on the emitter surface. As result of pulse reshaping, focusing of the THz beam leads to a reduced
relative pulse momentum, with implications in THz eld ionization experiments.
Introduction
For the last decade a common method to generate sub-picosecond, single-cycle electromagnetic pulses (THz pulses)
for spectroscopic purposes (THz time-domain spectroscopy, THz-TDS) has been to excite a biased semiconductor or
an electro-optic crystal with a femtosecond light pulse. In the case of a biased semiconductor, the acceleration of
photogenerated carriers results in emission of the THz pulse, and in the case of an electro-optic crystal, optical rectication
of the excitation pulse generates the THz pulse.
The description and understanding of the propagation of THz pulses in optical systems has recently attracted considerable research eorts. P. U. Jepsen et al[1, 2] measured and calculated the distribution of the THz eld in front of
a lens-coupled point-source dipole emitter. R. P. Bromage et al[3, 8] made thorough investigations of the spatiotemporal shaping of THz pulses by the diraction through conductive apertures. D. You et al[4] calculated the propagation
characteristics of unipolar THz pulses through optical systems, and S. Feng et al[5] calculated properties of the temporal
prole and spectrum of a THz pulse in the vicinity of a focus. C. Winnewisser et al[6] measured and discussed spectral
and temporal pulse shaping by frequency-selective surfaces. E. Budiarto et al[7] modeled and measured the temporal
prole of a THz pulse in the near eld of a LA antenna.
One reason for the interest in propagation characteristics of THz pulses is that the temporal shape of the THz pulse
can be measured with great accuracy on a femtosecond time scale. Therefore numerical models of ultrashort pulse
propagation can stringently be tested. Furthermore, results obtained with single-cycle THz pulses can be scaled in
frequency to predict the behaviour of single-cycle pulses in the visible, which have not yet been experimentally realized.
In the work presented here we perform numerical and experimental investigations of the propagation of THz pulses
from a biased, LA antenna with dimensions larger than the characteristic wavelength of the radiation. Because of the
general nature of the numerical calculations the main conclusions drawn are also valid for other types of LA antennas,
e.g. unbiased semiconductor surfaces and electro-optic crystals.
Pulses emitted from a biased LA antenna tend to have a non-zero area, and consequently possess a transverse
momentum, and are therefore of particular importance in eld ionization experiments, pioneered by the groups of P. H.
Bucksbaum and R. R. Jones (see, for instance, [9, 10] and references therein). The apparent non-zero area originates
from from a highly asymmetric pulse shape, resembling that of a unipolar pulse.
The paper is organized as follows. First the results from scalar and vectorial diraction theory are presented. Then
we present an experimental measurement of the frequency-dependent spatial distribution (the radiation pattern) of the
electric eld emitted from a LA antenna, and compare the experimental radiation patterns to results from diraction
theory. Complementary to the experimental results in the frequency domain we then present measurements of the
temporal prole of the THz pulses as the detector is moved from the extreme near eld into the far eld of the emitter.
Again the experimental data are compared to results from diraction theory.
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Theoretical background
In this section we give an overview of the theoretical models we have applied in order to describe the propagation of THz
pulses emitted from a LA antenna. The calculation of the temporal and spectral radiation pattern from a THz emitter
is basically a diraction problem. A phase-sensitive integration of the electric eld distribution on the emitter surface
determines the electric eld in front of the emitter. The distribution of the electric eld on the emitter surface is, in
turn, determined partly by carrier dynamics in the semiconductor substrate of the emitter, and in part by the geometry
of the emitter structure.
All calculations presented here are performed on a model emitter system with dimensions identical to the emitters
used in the experiments. The emitter has a gap length l = 10 mm and height h = 10 mm. The emitter is excited by a
pump beam with a 1=e-radius of 3.5 mm. We use a cartesian coordinate system with x-axis along the direction of the
applied bias eld and z -axis along the propagation direction of the THz beam. The origin of the coordinate system is
taken as the center of the LA emitter.
Scalar and vectorial diraction theory
Due to the extremely large spectral contents of a typical THz pulse it is necessary to employ broadband versions of
the standard diraction integrals to give a full description of the radiation pattern from a THz emitter. It is useful to
study the radiation pattern both in the frequency domain and in the time domain. By doing so it is possible to see how
individual frequency components propagate, and to see how the temporal prole of the THz pulse is modied during
propagation.
The calculation of the radiation pattern in front of the THz emitter can be performed by the application of Huygens'
principle of superposition of spherical wavelets. If a monochromatic wave is emitted from a plane surface A (the THz
emitter surface) the scalar representation of the electric eld in the space in front of the emitter can be written as[11]
Z
0
U (~x) = 1 U (~x0 ) exp(ikr ) cos(^n;~r0 ) df 0 ;
(1)
i A
r0
where ~x; ; n^ ; and ~x0 are the observation point, the wavelength of the wave, the inward normal of the emitter surface,
and an integration point on the emitter surface, respectively. r0 is the distance between ~x and ~x0 . The cosine factor is
known as the inclination factor. The integral (1) is known as the Rayleigh-Sommerfeld Diraction Integral. The general
principle of the integration is simple: An emsemble of spherical waves (exp(ikr0 )=r0 ) with amplitudes U (~x0 ) are emitted
from the antenna surface A, and adds up to form the electric eld at ~x.
Obtained by inverse Fourier transformation of Eq. (1), the time domain version of the Rayleigh-Sommerfeld diffraction
integral, also known as the broadband Huygens-Fresnel diraction integral, is[11]
Z
n;~r0 ) d u(~x0 ; t ; r0 =c) df 0 :
u(~x; t) = cos(^
(2)
A 2cr0 dt
It shows that in the far-eld, the scalar representation of the eld at a point ~x is proportional to the time derivative of
the surface eld on the THz emitter at the retarded time t ; r0 =c. The two integrals (1) and (2) both represent a scalar
approximation of the full electric eld vector. In the near eld of the emitter the electric eld cannot be treated as a
scalar eld due to the large curvature of the wavefront. Diraction integrals for the full electric eld vector exists, and
for a monochromatic wave the electric eld vector E~ (~x) is given by[12]
Z 0) 0
~E (~x) = 1 r n^ E~ (~x0 ) exp(ikr
(3)
2
r0 df :
A
This integral is known as the Smythe-Kirchho Diraction Integral. An inverse Fourier transform of Eq. (3) gives the
broadband version, applicable in the time domain,
Z
1 n^ E~ (~x0 ; t ; r0 =c) df 0 :
~E (~x; t) = r (4)
A 2r0
The derivation of this integral is outlined in App. A. We show in App. B that in the far eld, near the propagation axis,
the scalar and vectorial broadband diraction integrals merge into the same expression, namely that the electric eld is
polarized parallel to the emitter surface, with an amplitude proportional to the time derivative of the surface eld on
the emitter.
All four diraction integrals presented here are valid for r0 >> , which at 1 THz is r0 >> 0:3 mm. All numerical
simulations and experiments are performed for z > 5 so that diraction theory is expected to be valid.
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Surface eld of the emitter
The time-dependent electric eld at the surface of the emitter is needed in the diraction integrals described in the
previous section. A widely accepted model describing the behaviour of the surface eld is the Current Surge model.
Within this model, the THz eld on the LA emitter is given by[13]
(5)
E~ THz (t) = ;Eb (t)s (+t)10+ p :
s 0
The surface conductivity is dened as
Zt
e
(1
;
R
)
( t) =
(t ; t0 )I (t0 ) exp(;(t ; t0 )= ) dt0 ;
(6)
s
h
;1
opt
car
which is a convolution of the time-dependent mobility , the intensity of the optical beam Iopt , and the carrier decay.
the time-dependent mobility can be expressed as
(t ; t0 ) = DC ; exp(;;(t ; t0 )) ;
(7)
where ; is the inverse of the collision time in the Drude model. M. C. Nuss et al[14] measured the time-dependent
mobility of carriers in GaAs following excitation by a 625 nm femtosecond laser pulse, and found that the mobility
increases with a time constant of several ps from a low, initial value, to the much higher, stationary value.
The temporal shape of the surface eld, when calculated from Eq. (5), is a step-like function, where the step steepness
is determined by the rise time of the optical excitation and the rise time of the mobility. In the case of excitation at
800 nm with a 100 fs pulse, the mobility rise time will be the limiting factor. We expect the mobility to increase faster
than in Ref. [14] since we excite electrons to a lower position in the conduction band of GaAs (100 meV for 800 nm as
opposed to 500 meV for 625 nm).
The decay of the electric eld follows the decay of the carrier density, which is slow (several hundred ps) on the
timescale considered here. Hence in the numerical simulations performed we assume an electric eld on the emitter
surface with a rise time of approx. 600 fs, which ts very well with experimental observations.
The electric eld from the bias applied across the emitter gap is not uniform. In the planar geometry of the emitter
the eld is enhanced near the electrodes, which can be seen by solving Poisson's equation (r E~ = =) in the given
geometry. The electric eld distribution is also modied if the contact between the electrodes and the emitter substrate
is non-ohmic. This will result in a depletion zone, and hence a strong eld enhancement, near the electrodes. In the
numerical modeling presented in the next section we apply a phenomenological eld distribution of the form
Eb (x) = Ec + (Ee ; Ec )(x=l)n ;
(8)
where Ec and Ee is the electric eld at the center of the emitter area and at the electrodes, respectively. The exponent
n determines how fast the eld drops from the value at the electrodes to the value at the center.
Results from Gaussian beam propagation theory
The integration of the Rayleigh-Sommerfeld diffraction integral shows that if the spatial intensity prole of the exciting
laser beam is Gaussian, the THz far eld U (x; y; z ) accurately reproduces the spatial prole of a Gaussian beam with
beam waist w0 given by the spot size of the pump beam located at the emitter surface. If the pump beam is blocked by
the electrodes, the THz beam is diracted by the resulting aperture. Based on these observations we can describe the
propagation of the THz pulse in the far eld region by normal Gaussian beam propagation theory.
If the THz beam propagates without focusing, the spot size (e;1 radius) at a distance L from the emitter is
cL q1 + (w2 =cL)2 :
wunfocused ( ) = w
0
0
(9)
If a lens with focal length f is inserted in the THz beam at a distance f from the emitter, a focus is formed at a distance
2f from the emitter. The spot size at this focus is
wfocused ( ) = fc=w0 :
(10)
Focusing the THz beam clearly modies the spatial distribution of the frequency components. Hence, on the propagation
axis the spectral and temporal shape of the THz pulse is modied by focusing. For a given pulse energy the on-axis eld
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SOQUE LASERS '98 Proceedings
strength of a Gaussian beam is inversely proportional to the spot size. Therefore, at a distance L = 2f from the emitter,
the ratio of the focused eld strength to the unfocused eld strength is given by
q
Efocused =Eunfocused = 2 1 + (w02 =2cf )2 :
(11)
All frequency components are concentrated on a smaller area by focusing, leading to a higher total eld strength at the
focal point.
In Fig. (1) the frequency-dependent spot size of an unfocused and a focused THz beam, determined by Eqs. (9) and
(10), and the eld enhancement factor given by Eq. (11) are shown. The initial spot size on the THz emitter, which is the
2 0
2 0
F o c u s e d s p o ts iz e
U n fo c u s e d s p o ts iz e
E fo c u s e d / E u n fo c u s e d
1 5
1 0
5
5
b e a m
1 0
0
0
1
2
3
4
5
6
7
8
9
1 0
R a t io
r a d iu s [m m ]
1 5
0
F re q u e n c y [T H z ]
Figure 1: Spot sizes of an unfocused and a focused THz beam with an initial spot size w0 = 3:4 mm, and the eld
strength enhancement due to focusing
only free parameter, is adjusted to mimic the experiments presented later in the paper, w0 = 3:4 mm. The unfocused spot
size (long dashes in Fig. (1)) shows a strong divergence at low frequencies. At high frequencies the spot size approaches
the initial spot size, as expected from ray optics. The focused spot size (short dashed in Fig. (1)) shows a less pronounced
divergence at low frequencies. At high frequencies the spot size is limited only by the wavelength of the radiation. At high
frequencies, the eld ratio (solid line in Fig. (1)), is linear with respect to frequency, i.e. Efocused ( ) / Eunfocused ( ).
This is identical to a dierentiation in the time domain. Hence in the high-frequency limit the temporal shape of a pulse
at the focus of a lens is the time derivative of the unfocused pulse. In the low-frequency limit, the eld ratio approaches
a constant value, resulting in an unmodied pulse shape. The borderline between the two regions is at = 2cf=w02,
which under the experimental conditions applicable here (w0 = 3:4 mm, f = 120 mm) is approximately at 2 THz. The
bandwidth of the THz pulses emitted from biased, LA antennas typically extends up to 1.5 to 2 THz, so the shape of
the pulse is not expected to be modied dramatically. A pulse with higher bandwidth, e.g. emitted by a (3) -process,
will be stronger modied by focusing.
In Fig. (2) the beam propagation, based on Gaussian ABCD-formalism[15], is shown for the two situations outlined
here. The 1=e-diameter of the beam is plotted for dierent frequency components, starting at 0.1 THz (largest divergence)
in steps of 0.2 THz to 1.9 THz (smallest divergence). The initial spot size is 3.4 mm in accordance with our experimental
conditions. In part (a) of the gure the beam is propagating freely, in part (b) the beam is focused by a f = 120 mm
lens in a 2f conguration.
Experimental setup
In order to verify the numerical simulations presented in the previous section, we have performed experiments to measure
the spatio-temporal eld distribution in the near eld and in the far eld of a LA emitter. The near eld setup is
schematically illustrated in part (a) of Fig. (3), part (b) of the same gure shows the setup for determination of the far
eld distribution. The laser source used in the experiment is a regeneratively amplied Ti:sapphire system, delivering
1 mJ pulses of 80 fs duration and 800 nm wavelength at 1 kHz repetition rate. A part of the beam is used to generate
the THz radiation. A LA GaAs antenna with 1 cm electrode spacing is biased by 5-8 kV DC voltage, and excited by
a 50 J laser pulse (the pump pulse). The electrodes are attached to the emitter by a conducting mixture of epoxy
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(a )
6 0
S p o t s iz e [m m ]
4 0
2 0
0
-2 0
-4 0
-6 0
(b )
6 0
B = 1 2 0 m m
S p o t s iz e [m m ]
4 0
2 0
0
-2 0
-4 0
-6 0
0
5 0
1 0 0
1 5 0
2 0 0
P r o p a g a tio n d is ta n c e [m m ]
Figure 2: (a) Unfocused and (b) focused propagation of a Gaussian beam. The 1=e diameter of the beam is plotted for
frequencies from 0.1 THz to 1.9 THz
glue and silver paint. The transient surface conductivity set up by the pump pulse results in the emission of an intense
electromagnetic pulse with a duration of approximately 0.5 ps and a peak eld strength of several kV/cm.
The THz pulse is emitted into free space from the antenna. In the near eld measurements the pulse then propagates
a small distance (tens of millimeters) to the detector. In the far eld measurements the THz beam is guided to the
detector by either a plane gold mirror or an o-axis paraboloidal mirror in a 2f conguration. We use a mirror with
focal length f = 120 mm. In both situations the distance from emitter to detector is 240 mm.
The THz pulses are detected by free-space electro-optic sampling[16, 17]. Briey, the time-dependent electric eld
of the THz pulse is used to retard the phase of an initially linearly polarized femtosecond probe pulse, derived from
the pump pulse on a beamsplitter, which is propagating collinear with the THz pulse inside an electro-optic crystal, in
this case ZnTe. The duration of the probe pulse is shorter than the duration of the THz pulse, therefore the temporal
shape of the THz pulse can be measured by changing the relative delay between the THz pulse and the probe pulse
while monitoring the phase retardation of the probe beam. The phase retardation is measured as the dierence in light
intensity of the two polarization components of the probe beam by a balanced photodiode setup, as illustrated in Fig.
(3). If the balanced detector setup is optically biased to the quarter wave point, the instantaneous electric eld of the
THz pulse is proportional to the measured phase retardation. We send the probe pulse into the ZnTe detector crystal
from the back surface with respect to the direction of the incoming THz pulse. The advantage of this geometry is that
no pellicle beamsplitter is needed in front of the detector. Hence we can measure the shape of the THz pulse in the
extreme near eld of the emitter.
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o
4 5
(a )
p o la r iz e d
+ H V
P D
E O
A
l /4
s e n s o r
P D
E m itte r
(b )
N B S
B
P B S
+ H V
E m itte r
o
4 5
p o la r iz e d
P D
E O
A
l /4
s e n s o r
P D
G M
N B S
B
P B S
Figure 3: (a) Experimental setup for near eld measurements, (b) setup for far eld measurements. GM - guiding mirror
(plane or o-axis paraboloidal), NBS - nonpolarizing beam splitter, PBS - polarizing beam splitter, PD - photodiode.
Frequency-resolved spatial radiation patterns
In this section we describe a measurement of the spatial radiation pattern emitted from a biased, LA emitter. The emitter
area is 1 1 cm2 , and the emitter is biased by a 5 kV static voltage across the photoconductive gap. The laser beam
exciting the emitter is matched to the size of the gap so that the electrodes are not signicantly illuminated. The distance
from emitter to detector is 240 mm, and we measure the radiation pattern in two situations. First a plane mirror is used
to guide the THz beam to the emitter (unfocused propagation) and next a f = 120 mm o-axis paraboloidal mirror is
used as guiding optics in a 2f conguration. This conguration focuses the THz beam to a diraction-limited spot at
the detector. In Fig. (4) temporal traces of the THz pulse at various o-axis positions of the detector is shown for the
unfocused beam and for the focused beam. Experimental data for the unfocused beam are shown in part (a) of the gure,
2 5
(a )
2 0
(b )
O ff- a x is p o s itio n [m m ]
1 5
1 0
5
0
-5
-1 0
-1 5
-2 0
-3
-2
-1
0
1
2
3
4
-3
-2
-1
0
1
2
3
4
5
T im e d e la y [p s ]
T im e d e la y [p s ]
Figure 4: Experimental data in the time domain
data for the focused beam are shown in part (b) of the gure. The eect of focusing is clear. In the unfocused beam
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there is only little change in the shape and amplitude of the THz pulse over several millimeters near the propagation
axis, whereas the shape of the pulse in the focused beam is severely disturbed when moving away from the propagation
axis. In Fig. (5) we have performed a Fourier transformation of the individual traces in Fig. (4) in order to look at the
spatial distribution of the frequency components of the THz pulse. The Fourier transformed experimental data for the
2 5
(a )
2 0
(b )
O ff- a x is p o s itio n [m m ]
1 5
1 0
5
0
-5
-1 0
-1 5
-2 0
0 ,0
0 ,5
1 ,0
1 ,5
2 ,0
0 ,0
0 ,5
1 ,0
1 ,5
2 ,0
F re q u e n c y [T H z ]
F re q u e n c y [T H z ]
Figure 5: Fourier transform of the experimental data
unfocused beam are shown in part (a) of the gure, Fourier transformed data for the focused beam in part (b) of the
gure. Again the eect of focusing is obvious. In the unfocused beam the spectral distribution of the pulse is insensitive
to the exact position of the detector, and the 10% bandwidth limit (indicated by small vertical bars) is considerable over
the full range of detector positions. In the focused beam, on the other hand, there is only considerable bandwidth in
the region near the propagation axis, and we see an apparent enhancement of the bandwidth on the propagation axis.
Pulses in the focused beam are shorter than pulses in the unfocused beam, FWHM 0.6 ps vs. 0.8 ps. These observations
is explained by the focusing process, since in the present geometry, high frequency components are focused tighter than
low frequency components. Hence we expect to observe a larger bandwidth in a focused beam on the propagation axis.
The apparent reduction of spectral content at large o-axis position is an indication of the well-known fact that the spot
size of a focused beam is smaller than the spot size of the unfocused beam.
If we make cross-sections at specic frequencies of the data in Fig. (5) we obtain frequency dependent radiation
patterns of the beam. In Fig. (6) we show radiation patterns of the unfocused beam in part (a) of the gure and
radiation patterns of the focused beam in part (b) of the gure. The radiation pattern is shown at frequencies from 0.15
THz to 1.2 THz. Together with the experimental results we show ts, assuming a Gaussian prole of the beam at each
frequency component. We use a simple Gaussian beam prole since in the far eld, the integration of the diraction
integrals (1) and (3) in the far eld basically results in a Gaussian beam prole. In order to verify that the THz beam
propagates like a Gaussian beam we have plotted the frequency-dependence of the experimental spot size w( ), and the
spot size of an ideal Gaussian beam, obtained from Eqs. (9) and (10), in Fig. (7). The calculated curves in Fig. (7)
require as the only free tting parameter the initial spot size w0 at the emitter. The ts in Fig. (7) were obtained with
w0 = 3:4 mm, in perfect agreement with the experimental conditions. The unfocused beam size is too large by a factor
of 1.2 to 1.5 in the range from 0.5 to 1.0 THz. We believe that this is caused by diraction on a post mounted in the
vicinity of the THz beam. The diraction signal can be seen in Fig. (4) as a small oscillation starting at the +10 mm
trace at t = 3:5 ps, moving to shorter times at larger o-axis positions. In the case of the focused beam the spot size is
reduced, therefore the THz beam is unaected by the post, resulting in a better t.
If the normalized momentum (integrated electric eld) at the center of the THz beam is compared for the focused
and unfocused beam we nd that the unfocused pulse has a normalized momentum of 0.38 ps, compared to a normalized
momentum of 0.22 ps for the focused pulse. This, together with the modied pulse duration, is the direct manifestation
of the increased contents of high frequency components of the pulse in the focused beam. The decrease of the normalized
pulse momentum upon focusing is caused by an enhanced, negative feature following the main peak. This feature is the
rst indication of the onset of bipolarity of the pulse. We do not observe a complete transition from a unipolar pulse in
the unfocused beam to a symmetric bipolar pulse in the focused beam. In order to observe this experimentally, either a
pulse with larger bandwidth or a focusing element with shorter focal length must be used.
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(a )
(b )
0 .1 5 T H z
F ie ld s tr e n g th [a .u .]
0 .3 T H z
1 .2 T H z
-2 0
-1 0
0
1 .2 T H z
1 0
2 0
-2 0
O ff- a x is p o s itio n [m m ]
-1 0
0
1 0
2 0
O ff- a x is p o s itio n [m m ]
Figure 6: Experimental and theoretically predicted radiation patterns
5 0
F o c u s e d
U n fo c u s
G a u s s ia
G a u s s ia
B e a m
r a d iu s [m m ]
4 0
p r
e d
n b
n b
o p
p r
e a
e a
a g a tio n
o p a g a tio n
m
m
3 0
2 0
1 0
0
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
1 ,0
1 ,2
1 ,4
F re q u e n c y [T H z ]
Figure 7: Comparison between experimental beam and a Gaussian beam prole
Based on an absolute calibration of the electro-optic response we know that the peak eld strength in the unfocused
R ~
case can be adjusted in the range from 0 to 2 kV/cm. Hence the absolute momentum of the THz pulse (e A~ = e Edt
)
falls in the range from 10;3 to 10;2 atomic units. Precise knowledge of the absolute momentum is a prerequisite for the
analysis of the energy of photoelectrons liberated by THz radiation in THz eld ionization experiments.
Spatially resolved temporal pulse proles
We have measured the temporal shape of the THz pulse in the near eld and in the far eld of the THz emitter. The
pulse shape measured in the near eld contains information about the eld distribution on the emitter surface, as will be
shown. In Fig. (8) we show the THz pulse shape recorded at z = 16 mm from the emitter, at dierent o-axis positions.
The pulse shape consists in general of three major contributions. The rst feature (in local time at the observation point)
is the sharp rise from zero eld to the peak value of the eld. This is when the eld components from the central part
of the emitter reaches the observation point. Later the contributions from the eld-enhanced regions near the electrodes
reaches the observation point, resulting in two weaker maxima in the pulse shape. On the propagation axis these two
contributions arrive simultaneously, but they can in principle be separated at o-axis positions. In the experimental data
(Fig: (8) we only resolve the rst peak and a second shoulder, delayed corresponding to the o-axis position with respect
to the rst peak.
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1 ,0
0 m m
1 m m
E le c tr ic fie ld [a .u .]
0 ,5
2 m m
0 ,0
3 m m
4 m m
-0 ,5
5 m m
-1 ,0
6
8
1 0
1 2
1 4
1 6
1 8
T im e d e la y [p s ]
Figure 8: O-axis THz pulse shapes measured at z = 16 mm from the emitter
In Fig. (9) we show numerical integrations of the scalar and vectorial broadband diraction integrals (Eqs. (2) and
(4). On the same gure we show one set of experimental data to illustrate the agreement between the simulation and
1 ,0
S c a la r th e o r y
V e c to r ia l th e o r y
E x p e r im e n t
0 m m
E le c tr ic fie ld [a .u .]
1 m m
0 ,5
2 m m
3 m m
0 ,0
4 m m
5 m m
-0 ,5
-1 ,0
5 4
5 6
5 8
6 0
6 2
6 4
T im e d e la y [p s ]
Figure 9: Numerical determination of o-axis THz pulse shape at z = 16 mm. One experimental data set is shown to
illustrate the agreement with experiment
the experiments. In the simulation data the three main features of the pulse are visible. To obtain the simulated data
in Fig. (9) we used n = 6 and Ee =Ec = 2:5 in Eq. (8).
In Fig. (10) we show a measurement of the THz pulse shape as the detector is moved from the near eld towards
the far eld of the emitter. The pulse shape is shown for z = 11 mm, z = 20 mm, and z = 81 mm. The transition into
the far eld pulse shape is not complete in this series of pulses, but the general trend is clear. As the detector is moved
further away from the emitter the pulse shape develops from the typical multi-peaked shape observed in the near eld
to the single-peaked THz pulse shape normally observed in the far eld. Together with the experimental data we show
results of the numerical integration of the scalar diraction integral, Eq. (2).
Numerical simulations of THz pulse propagation
We have performed numerical integration of the vectorial broadband diraction integral (4) in order to visualize the
temporal evolution of the radiation pattern from a LA emitter. We have modeled the both eld components of the
radiation pattern from a 10 10 mm2 emitter. If the emitter is excited by a plane wavefront only two of the three
Tucson, Dec. 7-11 1998
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A. Gurtler et al.
SOQUE LASERS '98 Proceedings
[a .u .]
1 ,5
E le c tr ic fie ld
2 ,0
1 ,0
E x p e r im e n ta l r e s u lts
S c a la r d iff r a c t io n th e o r y
z = 1 1 m m
z = 2 0 m m
0 ,5
z = 8 1 m m
0 ,0
8
1 0
1 2
1 4
1 6
1 8
2 0
T im e d e la y [p s ]
Figure 10: On-axis THz pulse shape as function of distance to the emitter
THz eld components have non-vanishing values, that is the component parallel to the emitter bias eld (x) and the
component parallel to the propagation axis (z ). In Fig. (11) the x-component (perpendicular to the propagation axis)
of the electric eld distribution is shown for (a) t = 333 ps (wavefront at z = 99:8 mm) and (c) t = 32 ps (wavefront at
z = 9:6 mm) after excitation of the emitter. The z component (parallel to the propagation axis) of the eld distribution at
the same times is shown as parts (b) and (d). In these plots the transition from the near eld to the far eld can be seen.
As a result of the large emitting area retardation eects are important in the near eld. This leads to a highly structured
pulse of several ps duration. The structure of the pulse shape reects the spatial distribution of the electric eld on the
emitter surface. The eect of eld enhancement near the electrodes can be seen as sharp, triangular boundaries. In the
far eld these eects are no longer present. Now retardation of the electrical eld is no longer important, and the THz
eld has evolved into a single, short pulse. This reects the fact that in the far eld, the temporal shape of the THz pulse
proportional to the time derivative of the electric eld on the surface of the emitter. The density plot of the z -component
at t = 32 ps after excitation (d) has been scaled by a factor of 2.5, and the density plot showing the z -component at
t = 333 ps (b) has been scaled by a factor of 17, i.e. the z component of the electric eld at larger distances is very small.
On the propagation axis (x = 0 mm) the z -component vanishes, and increases in magnitude with increasing distance
from the propagation axis. We see a sign change at x = 0 mm (positive eld values are shaded in white, negative values
are shaded in black), indicative of the curved wavefront of the THz beam.
Conclusions
We have described the propagation of ultrashort THz pulses emitted from a biased, large-aperture THz antenna, and
found, by measuring the frequency-dependent radiation pattern of an unfocused and a focused THz beam, that the THz
beam essentially behaves like a fundamental Gaussian beam with a beam waist given by the spot size of the exciting
laser pulse at the THz emitter. This fact is also true for other types of LA emitters, since the propagation characteristics
is independent of the microscopic processes generating the THz beam. We have demonstrated that it is possible to
model the full electrical eld vector of the THz beam by a single diraction integral, and that we can obtain important
information about the eld distribution on the emitter surface by studying the THz pulse shape in the near eld. We
nd that focusing of the THz beam leads to an enhancement of the peak THz eld strength, but at the same time the
normalized pulse momentum, the important quantity in THz eld ionization experiments, is reduced.
This work was supported by the Deutsche Forschungsgemeinshaft, project number SFB 276, TP C14.
Appendix A: Derivation of the vectorial broadband diraction integral
In this appendix the derivation of the vectorial polychromatic diraction integral is outlined. The derivation is based on
an inverse Fourier transformation of the monochromatic vectorial diraction integral found in text books. This integral
can be written as
Z
0
E(x; ) = 21 r (n E(x0 )) exp(ir20r ) df 0 :
(12)
A
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SOQUE LASERS '98 Proceedings
(b)
Distance to emitter [mm]
a)100
99
98
97
96
Distance to emitter [mm]
(c) 10
(d)
9
8
7
6
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
Off-axis position [mm]
Figure 11: Density plots of the spatiotemporal distribution of the x and z components of the THz eld 333 ps after
excitation (a and b), and the x and z components of the THz eld 32 ps after excitation (c and d)
The inverse Fourier transformation of the electric eld is dened as
E(x; t) =
Z1
;1
E(x; ; ) exp(;i2t)d
(13)
To derive the time-dependent spatial distribution of the electric eld E(x; t) the frequency-domain expression of the eld
in Eq. (12) is inserted in the inverse Fourier transformation, Eq. (13).
Z1
Z
1
0 ; ; )) exp(i2r0 =c) df 0 ] exp(;i2t)d
E(x; t) =
[ r
(
n
E
(
x
r0
;1 Z2
AZ
1 1
0 ; ; ) exp(i2r0 =c) df 0 ] exp(;i2t)d
= r
[n E
(
x
2
r0
Z;1
Z A1
1 [n = r 2r
E(x0 ; ; ) exp(;i2 (t ; r0 =c))d ]df 0 :
0
A
;1
(14)
(15)
(16)
By comparison with Eq. (13) we see that the term in brackets in Eq. (16) is the Fourier transformation of E(x0 ; t ; r0 =c).
This leads to the nal result,
Z
1 (n E(x0 ; t ; r0 =c)) df 0 :
(17)
E(x; t) = r 2r
0
A
Hence, the electric eld vector can be found at an arbitrary point in space if the time-dependent surface eld of the THz
emitter is known.
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SOQUE LASERS '98 Proceedings
Appendix B: Correspondence between vectorial and scalar diraction theory
In this appendix we demonstrate that in the far eld the results from scalar and vectorial polychromatic diraction
theory merge. The vectorial diraction integral in the time domain is
Z
1 (n E(x0 ; t ; r0 =c)) df 0 :
E(x; t) = r 2r
(18)
0
A
The corresponding scalar diraction integral is
Z
n; r0 ) @ u(x0 ; t ; r0 =c)df 0 :
u(x; t) = cos(
(19)
A 2cr0 @t
In the far eld we can make the following approximations: r0 ' z , cos(x; r0 ) ' 1. In our situation the electric eld on
the emitter surface is applied in the x^ direction, hence the vector operations in Eq. (18) can be expressed as
@Ex
x
r (n E) = ; @E
@z x^ + @x z^ :
Hence Eq. (18) can be written as
1 ;x^ @ + z^ @
E(x; t) ' 2z
@z @x
Z
A
Ex (x0 ; t ; z=c)df 0 :
(20)
(21)
The time derivative in Eq. (19) is evaluated at the retarded time t0 = t ; r0 =c ' t ; z=c. Hence if we apply @=@t0 =
@z=@t = ;c@=@z we can rewrite Eq. (19) to
Z
@
1
u(x; t) ' ; 2z @z u(x0 ; t ; z=c)df 0 :
(22)
A
By comparison, this is identical to the x^-component of Eq. (21). This demonstrates that in the far eld, near the
propagation axis, the scalar and the vectorial diraction integrals are identical. The scalar eld function u is in this
situation identical to the x^-component of the electric eld vector E. Furthermore, near the propagation axis in the far
eld the z^-component of the electric eld vanishes, indicating that the THz beam can be treated as a plane wave.
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