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Terahertz generation through optical rectification in reflection
Mathias Hedegaard Kristensen,1Emilie Herault,2Dongwei Zhai,2,3 Esben Skovsen,1and Jean-Louis Coutaz2
1)Department of Materials and Production, Section for Physics and Mechanics, Aalborg University, Skjernvej 4A, DK-9220,
Aalborg East, Denmark
2)IMEP-LAHC, UMR CNRS 5130, Université Savoie Mont Blanc, 73 376 Le Bourget du Lac Cedex,
France
3)College of Physics, Qingdao University, Qingdao 266071, China
(*Electronic mail: mhkr@icloud.com)
(Dated: 5 May 2023)
In this paper, we study terahertz generation through optical rectification in reflection at normal incidence in a dielectric
nonlinear crystal. We first analyse, with a nonlinear optical model, the sample parameters (thickness, absorption at both
laser and terahertz wavelengths, etc.) for which a terahertz optical rectification reflection scheme is preferable to the
common transmission scheme. Then we report our experimental observations of a reflected terahertz signal generated
at the surface of a ZnTe crystal. The reflected terahertz signal shares all the characteristics of a signal generated in
transmission, but is not limited by absorption losses in the crystal, thereby providing a broader bandwidth. At high
pump laser power, the signal exhibits saturation, which is caused by the decrease of the nonlinear susceptibility due to
photocarriers generated by two-photon absorption. This reflection scheme could be of great importance for terahertz
microscopy of opaque materials like, e.g., humid samples or samples exhibiting strong absorption bands, or to study
samples for which the transmitted signal cannot be recorded.
I. INTRODUCTION
Optical rectification (OR) is an elegant way to produce tera-
hertz (THz) pulses by irradiating a dielectric nonlinear crystal
with femtosecond laser pulses. Because of the almost instan-
taneous polarization of the crystal atoms or molecules, broad-
band THz pulses may be generated1. Moreover, very intense
THz peak power can be delivered2. Actually, OR corresponds
to the difference of frequencies between all the spectral com-
ponents of the ultra-short laser pulse. In fact, difference-
frequency generation was the first opto-electronic technique
used to generate far infrared beams3–5. Here, difference-
frequency generation is named OR because the generated
THz frequencies are much smaller than the exciting laser fre-
quency. Usually, generation of THz waves through OR in a
crystal is performed in transmission, because the generated
THz field magnitude increases linearly with the crystal thick-
ness if phase-matching between the incident laser beam and
the generated THz beam is realized. Recently, Sotome et al.6
employed OR in transmission to obtain THz images of fer-
roelectric samples, where the laser beam was scanned over
the sample, and each irradiated point of the sample gener-
ated a THz signal whose magnitude was related to both the
crystallinity and nonlinearity of the sample. More recently,
some of us used the same technique to get THz images of
a caster sugar grain with a sub-wavelength resolution7. We
named this technique Optical Rectification Terahertz Imaging
(ORTI). Then, by improving our setup, we published an ORTI
image of the domains of a periodically-poled KTP crystal with
a lateral resolution of λ/200.8A next progress towards ORTI
of actual samples will be to record images in reflection. It will
allow one to characterize opaque or bulky samples, or samples
whose rear face is rough.
Generation of THz pulses through OR in reflection has only
been studied to a lesser extent. In 2005, Reid et al.9have re-
ported OR THz generation from a semi-conductor recorded
in reflection at 45◦incidence. They pumped an InAs sam-
ple below its bandgap at 800 nm and by a proper polarimet-
ric study of both THz generation and second-harmonic gen-
eration (SHG), they were able to discriminate the respective
contributions of the bound and free photo-excited electrons.
Moreover, they demonstrated the different contributions of
both bulk and surface regions of the sample, this difference
arising thanks to the surface electric-field. The effect is strong
in narrow bandgap semiconductors like InAs, but insignifi-
cant in larger bandgap materials, like GaAs. Later in 2007,
Zinov’ev et al.10,11 and Bakunov et al.12,13 developed theoret-
ical models of THz generation through OR including a field
generated in the backward direction (reflection). Zinov’ev et
al. presented a thorough description of all the THz pulses
generated when an optical pulse propagates through a slab of
nonlinear material. Their theoretical calculations clarify that
the THz radiation is generated at the surfaces due to the in-
stantaneous creation and acceleration of polarization charge
at the front surface, and subsequent deceleration and extinc-
tion at the back surface. They supported their theory by ex-
periments measured in transmission. Bakunov et al. extended
the usual Fresnel formulas for transmission and reflection of
free-propagating electromagnetic pulses to forced pulses gen-
erated in a nonlinear crystal and showed that the free and
forced waves obey different boundary conditions at the crystal
surfaces.12 In the second paper13, they expanded their model
to include the focusing of the pump beam and calculate the
Cherenkov angular spreading of the generated THz waves.
Later on, Hargreaves et al.14 published a detailed modeling on
THz OR generation versus the crystal orientation in view of
clearly discriminating OR and photo-induced current transient
contributions. Finally, A. Schneider15 performed a complete
analysis of the THz pulses generated in a nonlinear slab con-
sidering dispersion, absorption of both optical and terahertz
waves, and multiple reflections. Furthermore, OR THz gen-
eration in reflection has been performed when dealing with
2
opaque materials, like metals16.
In most of these publications, the research was focused on
the theoretical description of the OR THz generation. When
experimental results were reported, they were performed ei-
ther in transmission11,17 or under oblique incidence.9,16 From
a practical point of view when dealing with applications like
ORTI, THz OR generation in reflection under normal inci-
dence is preferable. The goal of this paper is not to propose
a highly efficient scheme for high power THz generation, but
to show that OR in reflection can be used to study the THz
response of samples. Moreover, generation in reflection is the
only available OR technique when samples are absorbing and
too thick, or if their rear face does not allow a good transmis-
sion of the THz beam (rough surface, surface covered with
opaque films like metallic ones, etc.). Especially, we will 1)
evaluate for which samples OR in reflection supplies stronger
signals than OR in transmission, and thus should be preferred;
2) experimentally demonstrate THz OR generation in reflec-
tion under normal incidence. This study is performed with
ZnTe as dielectric nonlinear crystal.
II. MODELING
Let us recall some basic expressions of OR generated THz
fields. We will deal only with the case of normal inci-
dence. The laboratory reference frame is (xyz)and the laser
beam propagates normally to the crystal surface along the z-
direction. We suppose that the irradiating laser beam is a plane
wave with two spectral components at ωand ω+Ω(ωand
Ωare respectively the optical and THz angular frequencies).
The plane wave is a good approximation when the laser beam
is not strongly focused onto the sample, i.e. the laser Rayleigh
length is larger than the crystal thickness d. We choose the
following notation for the electrical field of this plane wave in
air:
Eo,ω(z,t) =
Eo,ω(z)e−jωt=
Eo,ωejko,ωze−jωt.(1)
ko,ω=ω
cuzis the incident wave vector (cis the velocity of
light in vacuum, uzis the unit vector along direction z). We
neglect the anisotropy of the crystal. Inside the crystal, the
laser field is
Eω(z) =
Eωej˜
kωz=˜
tω
Eo,ωej˜
kωz.(2)
with wavevector
˜
kω=ω
c˜nωuzand transmission coefficient ˜
tω
at the crystal surface (we employ a tilde to indicate complex
values except for complex fields). Here, ˜nω=nω+jκωis the
complex refractive index of the crystal at the laser frequency.
The THz wave at frequency Ωis generated through OR, i.e. a
second order nonlinear effect. The related nonlinear polariza-
tion is
PNL
Ω(z) = εo↔
˜
χ(2):
Eω+Ω(z)·
E∗
ω(z).(3)
εois the permittivity of vacuum, ↔
˜
χ(2)is the nonlinear OR ten-
sor, and the asterisk denotes the complex conjugate. The non-
linear Helmholtz propagation equation for the THz field
EΩ
is:
∇2
EΩ(z)+ Ω2
c2˜
εΩ
EΩ(z) = −Ω2
c2
↔
˜
χ(2):
Eω+Ω(z)·
E∗
ω(z).(4)
In order to derive the THz fields reflected and transmitted out-
side the crystal, we neglect the rebounds of the laser and THz
pulses inside the crystal. We then substitute the laser field in
Eq. (4) with the expression from Eq. (2), and thus, the non-
linear term writes:
↔
˜
χ(2):
Eω+Ω(z)·
E∗
ω(z) = ↔
˜
χ(2):
Eω+Ω·
Eωej(˜
kω+Ω−˜
k∗
ω)z.(5)
The wave vector difference
˜
kω+Ω−˜
k∗
ω=ω+Ω
cnω+Ω−ω
cnω
+jω
c1+Ω
ωκω+Ω+κω(6)
simplifies as
∆˜
k=˜
kω+Ω−˜
k∗
ω≈Ω
cnG,ω+jαω≡Ω
c˜nG,ω,(7)
assuming Ω≪ωand using the group index nG,ω=nω+
ω∂nω
∂ ω . In Eq. (7), αω=2ω
cκωis the coefficient of absorp-
tion at the laser wavelength. It follows that
˜nG,ω≡nG,ω+jc
2ωαG,ω⇒αG,ω=2ω
Ωαω.(8)
Solving the boundary equations at the surface (z=0) within
this hypothesis, the reflected THz field is given by
ER,Ω(z) = −
↔
˜
χ(2):
Eω+Ω·
E∗
ω
(˜nΩ+1)( ˜nG,ω+˜nΩ)e−jkR,Ωz.(9)
˜nΩis the refractive index at the THz frequency, and kR,Ω=
Ω/c. When dealing with ultrashort laser pulses, expression
(9) must be integrated over the whole laser pulse spectrum
(see for example eqs. (7) and (20) in the paper by Schneider
et al.17). This is compulsory when one desires to determine
the upper spectral limit of the generated THz signal, or to fit
the experimental spectra. Let us notice that in Eq. (9)
˜nG,ω+˜nΩ=nG,ω+nΩ+jc
2Ω(2αω+αΩ),(10)
which is deduced from Eq. (7). As expected, the reflected
field does not depend on the crystal thickness and exists even
if the crystal thickness tends towards zero: The generation in
reflection is a pure surface effect. The reflected field magni-
tude depends on the crystal nonlinearity and only slowly on
the crystal refractive index. Therefore the reflected THz sig-
nal exhibits all the spectral features related to the linear and
nonlinear properties of the crystal at both laser and THz fre-
quencies. Especially, phonon resonances at THz frequencies
should be clearly observed in the reflected THz spectrum. Fig.
1(a) shows the magnitude normalized to the nonlinear source,
i.e.
ER,Ω/↔
˜
χ(2):
Eω+Ω·
E∗
ω
=
(˜nΩ+1) ( ˜nG,ω+˜nΩ)
−1
,
3
FIG. 1. Maps of the THz field magnitudes generated in reflection (a) and transmission (b) normalized to the nonlinear source using λ=
2πc/ω=800 nm, f=1 THz, nG,ω=2.5, nΩ=3, and d=1.52 mm, versus the optical and THz absorption coefficients. The contour map of
the ratio (Eq. (13)) is plotted in the case of phase-matching (nG,ω=nΩ=3) for d=0.1 mm (c), d=1.52 mm (d) and d=10 mm (e). The
red lines indicate the unity ratio.
for λ=2πc/ω=800 nm, f=1 THz, nG,ω=2.5 and nΩ=3
versus the absorption coefficients αG,ωand αΩ. We see that
the reflected field magnitude is almost constant up to αω≈
106cm−1and αΩ≈103cm−1(the yellow plateau), where-
upon it decreases strongly and becomes zero. We can con-
clude that for most of materials, even if they exhibit a rather
large absorption at both laser and THz wavelengths, the re-
flected THz field does not depend so much on the loss. It
vanishes only for materials that are practically opaque.
Let us now treat the transmitted THz field. At a distance z
inside the crystal, the field is the sum of the propagating free
and forced waves:
EΩ(z) =
↔
˜
χ(2):
Eω+Ω·
E∗
ω
˜n2
G,ω−˜n2
Ωej∆˜
kz −˜nG,ω+1
˜nΩ+1ej˜
kΩz(11)
with ˜
kΩ=Ω
c˜nΩ. The field transmitted at the exit face of the
crystal (z>d) must be multiplied by the THz transmission
coefficient ˜
tΩ:
ET,Ω(z) =
EΩ(d)˜
tΩejkR,Ω(z−d).(12)
The magnitude of the transmitted field normalized to the non-
linear source,
ET,Ω/↔
˜
χ(2):
Eω+Ω·
E∗
ω
=
2 ˜nΩ
1+˜nΩ
2ej∆˜
kd −˜nG,ω+1
˜nΩ+1ej˜
kΩd
(˜nΩ+1)˜n2
G,ω−˜n2
Ω
,
is plotted in Fig. 1(b) using similar parameter values as for
the reflected field and crystal thickness d=1.52 mm (this is
the thickness of the sample studied in the experimental part.
It corresponds to a typical value for a rather efficient and
broadband THz OR generation in transmission). The trans-
mitted THz signal decreases strongly when the visible and/or
THz absorption increase. Typically, it is almost null when
αΩ>103cm−1or αω>107cm−1. Finally, let us compare
the magnitudes of the reflected and transmitted THz fields:
ER,Ω
ET,Ω
=
(˜nΩ+1) ( ˜nG,ω−˜nΩ)
4 ˜nΩej∆˜
kd −˜nG,ω+1
˜nΩ+1ej˜
kΩd
.(13)
The ratio is independent on the crystal nonlinearity, and thus,
on the polarization of the laser and THz beams. For weak vis-
ible and THz absorption, the transmitted THz signal is much
stronger than the reflected signal due to cumulative generation
4
throughout the crystal. At higher absorption, laser and/or THz
beams no longer propagate inside the crystal. This appears
clearly on the contour maps plotted in Fig. 1(c) for d=0.1
mm, Fig. 1(d) for d=1.52 mm, and Fig. 1(e) for d=1 cm.
The red line represents the unity ratio. For the d=1.52 mm
sample, the red line limit occurs at αΩ≈103cm−1for crystals
with an absorption αωat the laser frequency less than ∼103
cm−1. For large absorption values at the laser frequency, the
reflected signal is of course stronger than the transmitted one
at an absorption weaker than for thinner samples. In conclu-
sion, for most of the common materials with standard thick-
nesses, THz generation by OR in transmission is more effi-
cient than in reflection, and hence, the reflection technique is
practically useful only if the sample or the experimental ge-
ometry does not allow to measure the transmitted signal. This
is the case for samples whose exit face is rough or covered by a
metallic film. For absorption values beyond this red contour,
i.e. for materials like water18 or carbon-fiber composites19
(αΩ∼1000 cm−1), the reflected signal is stronger than the
transmitted signal. However, we must keep in mind that its
magnitude is very weak, even if stronger than the transmit-
ted one. The difference between the reflected and transmitted
signals is emphasized when plotting the field magnitudes ver-
sus the difference nG,ω−nΩ. Fig. 2(a) shows the transmitted
and reflected THz fields versus nG,ω−nΩfor nG,ω=3 and
f=1 THz. We assume that the crystal is d=1.52 mm thick
and transparent at the laser wavelength (λ=800 nm). The
transmitted field is plotted for different values of the THz ab-
sorption. Because the reflected field depends weakly on the
THz absorption, we plot it only for αΩ=0 cm−1. The phase-
matching oscillations in transmission are clearly seen when
the THz absorption is null, while the much weaker reflection
curve does not show any phase-matching feature. Typically,
at 1 THz with nΩ=3.2, nG,ω=3.16 and d=1.52 mm, one
obtains ER/ET=1/90. With increasing THz absorption, the
transmitted THz signal decreases and its oscillations are at-
tenuated. However, when the sample is almost opaque to
THz waves (αΩ=1000 cm−1, like water18 or carbon-fiber
composites19), the phase-matching maximum is erased and
the transmitted field is comparable to the reflected field. Fig.
2(b) presents similar curves, but calculated for a given THz
absorption (αΩ=20 cm−1) and different crystal thicknesses.
Because of the phase-matching phenomenon, the transmitted
curves show oscillations whose pseudo-periodicity is shorter
with thinner crystals. Here the effect of THz absorption is
compensated by increasing the crystal thickness when phase-
matching is realized. Thus, as before (Fig. 2(a)), the trans-
mitted field is stronger than the reflected one by 1-2 orders of
magnitude when phase-matching is realised.
Hence, it appears that, even for transparent crystals and
without achieving phase-matching, the signal generated in re-
flection is much smaller than the one in transmission. There-
fore, OR performed in reflection is not a technique that pro-
duces high power THz pulses. It should be used when study-
ing materials that are opaque or scatter in the THz range, or
whose rear face is rough, or covered by nontransparent or
diffracting layers. Also, it could be of interest when a reflec-
tion scheme is easier to implement than a transmission one,
for example in microscopy. In all other cases, a transmission
arrangement is more efficient.
Let us now address the error on the transmitted THz field
made when the reflected THz is omitted. Instead of expression
(11), one gets:
Eapprox
Ω(d) =
↔
˜
χ(2):
Eω+Ω·
E∗
ω
˜n2
G,ω−˜n2
Ωej∆˜
kd −ej˜
kΩd.(14)
The relative error writes:
Eapprox
Ω(d)−
EΩ(d)
EΩ(d)=˜nG,ω−˜nΩ
(1+˜nΩ)ej(∆˜
k−˜
kΩ)d−1−˜nG,ω
.
(15)
Typically, this error is almost constant with ˜nΩ−˜nG,ω, but it
depends strongly on the crystal thickness d. For a transpar-
ent crystal, and at 1 THz, the error is much less than 1 %
for dlarger than 1 mm, and thus, it could be neglected. But
it increases to ∼7% for d=100 µm and up to ∼30% for
d=10 µm. In such very thin crystals, used to generate very
broadband THz signals,20 generation in transmission is weak
because the crystal thickness is small. Thus, the amplitudes
of the reflected and transmitted THz signals become of the
same order of magnitude. Therefore, the reflected field can no
longer be omitted.
The generated THz fields depend on the nonlinear source
term ↔
˜
χ(2):
Eω+Ω·
E∗
ω. This nonlinear term has to be calculated
in the crystal frame (XY Z ), in which the nonlinear tensor ↔
˜
χ(2)
is known. To switch from the laboratory to the crystal frame, a
first rotation by θaround yis performed followed by a second
rotation by φaround z.21 The whole rotation matrix Ris:
R(φ,θ) =
cosφcosθ−sinθcos φsin θ
sinφcosθcosθsin φsin θ
−sinθ0 cosθ
.(16)
The nonlinear polarization
PNL
Ω,XY Z in Eq. (4) is calculated in
the crystal frame using
Eω,XY Z =R(φ,θ)
Eω,xyz
and then multiplied by the inverse rotation matrix R−1(φ,θ)
to obtain the expression in the laboratory frame:
PNL
Ω,xyz =R−1(φ,θ)εo
×↔
˜
χ(2):R(φ,θ)
Eω+Ω,xyz·R(φ,θ)
E∗
ω,xyz.
(17)
Expression (17) must be calculated for each crystallographic
class and each orientation of the crystal. In the case of cubic
crystals (432, ¯
43m, 23) addressed here, the nonlinear suscep-
tibility tensor is:
↔
˜
χ(2)=
000 ˜
χ(2)
14 0 0
0 0 0 0 ˜
χ(2)
14 0
0 0 0 0 0 ˜
χ(2)
14
.(18)
5
FIG. 2. Calculated transmitted (continuous lines) and reflected (dashed line) THz field magnitudes versus nG,ω−nΩ. All the curves are
normalized to the reflected signal at nG,ω−nΩ=0. (a) The sample (d=1.52 mm) is assumed to be transparent at the laser wavelength and
nG,ω=3. (b) The THz absorption is assumed to be αΩ=20 cm−1and the curves are calculated for different crystal thicknesses, namely
d=0.1, 0.5, 1 and 10 mm.
For the most common crystal cuts ⟨110⟩and ⟨111⟩, the rota-
tion angles are respectively
θ=π
2,φ=π
4and θ=arccos1
√3,φ=π
4,
which leads to the following dependence of the THz field on
the laser polarization angle ψ:21
⟨110⟩ →
ER,Ω,
ET,Ω∝1
4
cos2ψ−1
−2sin2ψ
0
(19)
⟨111⟩ →
ER,Ω,
ET,Ω∝1
√6
cos2ψ
−sin2ψ
−1/√2
(20)
Here, we model the THz OR generation in the framework
of plane waves interaction. However, if strongly focusing the
exciting laser beam in view of performing ORTI microscopy,
the issue of focusing the pump laser beam is of utmost im-
portance. Such an issue is very difficult to address, as both
laser and THz beams encounter diffraction, and the nonlin-
ear process in the crystal is made complicated because each
spatial plane wave component of the incident laser beam ex-
hibits a E-field polarized in a slightly different direction. A
complete modeling taking into account the finite size of the
exciting laser beam has already been treated by Bakunov et
al.13: When the laser beam is not strongly focused, the gen-
erated THz beam is very similar to a one-dimensional beam
(plane wave-like of limited radial size). On the other hand,
when the laser spot size at the crystal entrance is smaller than
the involved THz wavelengths, a Cherenkov cone is generated
inside the crystal and both the transmitted and reflected THz
beam are highly diverging outside the crystal: The excited
area of the crystal behaves almost as a THz point-source. In
the hypothesis of a crystal thickness that is smaller than the
laser beam Rayleigh length, Schneider et al.17 came to the
same conclusion and gave a simple analytical expression of
the generated THz beam, assuming the realization of phase-
matching and no absorption at both laser and THz frequen-
cies. THz generation by optical rectification in transmission,
with a sub-wavelength Gaussian-shaped spot source, has been
modeled and measured by H. Lin and coworkers22. When the
laser spot size is much smaller than the THz wavelength, the
THz beam is diffracted in nearly all directions from the crys-
tal independently of the frequency, and it obeys the obliquity
factor law. The main result is a decrease of the signal col-
lected by an aperture-limited receiving system. Oppositely,
for large laser spot size, the generated THz beam is almost a
paraxial Gaussian-like beam. For intermediate laser spot size,
the diffraction effect is more pronounced for the lower fre-
quency range, making the corresponding signal detected with
a weaker efficiency than the high frequency range. Here, a
similar behavior is expected, as only the excited spot at the
sample surface radiates the THz field. In the case of ORTI
microscopy, as-large-as-possible aperture optics must be em-
ployed in front of the receiver in order to collect the maximum
of all the THz reflected light spread in a 2πsolid angle. Let
us point out that, even in the case of strong focusing, all the
spectral features of the THz field will be saved.
III. EXPERIMENT
The experimental setup used for THz generation through
OR in reflection (Fig. 3) resembled a THz time-domain
spectroscopy setup. A beam of 100-fs linearly polarized
laser pulses at 80 MHz repetition rate was delivered by a
Ti:sapphire oscillator (Spectra-Physics Tsunami, 786 nm cen-
ter wavelength, and 12.5 nJ pulse energy). A beam splitter
6
femtosecond
pulse
optical delay
ZnTe
⟨110⟩
beam
dump
BS
chopper
ND
HWP
lens
lens
PCA
OAP
FIG. 3. Illustration of the experimental setup. ND, variable neutral
density filter. BS, beam splitter; HWP, half-wave plate; PCA, photo-
conductive antenna; OAP, off-axis parabolic mirror.
(BS) divided the pulses into a pump and a probe branch. A
half-wave plate (HWP) and a mechanical chopper in the pump
beam controlled the pump polarization and triggered the lock-
in detection, respectively. The pump beam was focused by a
150-mm focal lens on a 1.52-mm thick ⟨110⟩-cut ZnTe crys-
tal at normal incidence through a hole in an off-axis parabolic
mirror (OAP). At the crystal, the spotsize diameter was 37
µm and the maximum average laser power was approximately
400 mW without the chopper. A neutral density filter (ND)
was used to control the laser power at the crystal. The re-
flected THz radiation generated through OR was collected by
the same parabolic mirror and focused onto a photoconductive
antenna (PCA) by a second OAP. The bow-tie PCA detec-
tor (BATOP GmbH) was oriented such that it was sensitive to
vertically polarized THz radiation. Finally, the measurements
were done under ambient conditions.
IV. RESULTS AND DISCUSSION
Fig. 4(a) shows two THz pulses separated by ∆t≈32.8
ps (main-peak-to-main-peak). The first THz waveform R was
generated through OR in reflection, while the later waveform
T was generated in transmission and then reflected at the exit
surface of the crystal. The time delay ∆tbetween the two
THz waveforms is thus equal to ∆t=2dnG,Ω/c≈2dnΩ/c.
With the crystal thickness d=1.52 mm and nΩ=3.20 mea-
sured around 1 THz, we get ∆t=32.4 ps. Let us examine
the magnitudes of the R and T pulses. Here, the T pulse is
not measured in transmission but is reflected by the exit face
of the crystal towards the detector. The great advantage of
this scheme over the common transmission one is that both
the R and T pulses are excited by the same laser pulse and are
measured by the same detector and the same receiving elec-
tronics. Therefore, any error due to a difference in sensitivity
of a double separated detection (one in reflection and one in
transmission) is avoided. Moreover, we took a great care in
the alignment of the sample in the THz beam, in such a way
the T pulse is directed exactly in the same direction as the
reflected one. Laser beams reflected at the entrance and exit
faces of the crystal were adjusted to superimpose at the detec-
tion system, within a precision better than the laser beam size,
i.e. less than 1 mm at a 30-cm distance from the crystal. The
angular precision is thus better than 0.003 rad = 11 arc min.
Let us notice that, generally, crystals are supplied with a par-
allelism of a few arc min. The ratio of the R and T pulse can
be calculated using Eq. (13). However, the transmitted THz
field must be multiplied by ˜rΩe−αΩd/2to include the back re-
flection at the second crystal surface and transmission through
the crystal. With αΩ≈0, we get
ER,ΩET,Ω
calc ≈1/47. In
the recorded trace, we unexpectedly find that the THz peak-
to-peak magnitude of waveform R is 7 times stronger com-
pared to waveform T. A possible explanation of the weak T
pulse is THz absorption by free carriers generated through a
two-photon absorption (TPA) process. Bose et al.23 measured
the photogenerated carrier lifetime of ZnTe to be τZnTe =25
ns. The pulse period of the Ti:sapphire oscillator is 12.5 ns.
Thus, TPA could lead to a steady state free carrier population
that may decrease the THz generation through the crystal, and
in turn, absorb the THz radiation reflected at the exit surface.
Using the TPA coefficient β≈5 cm/GW 24,τZnTe ≈25 ns,
and the Drude model to determine the THz absorption by the
mean density of photogenerated carriers, we obtain αΩ≈80
cm−1. Contrary, if we extract the absorption coefficient from
the measurement using the modified expression (13), we get
αΩ≈77 cm−1in good agreement with the estimate. As ex-
pected, the measured absorption coefficient is slightly lower
compared to our estimate, since we do not consider the change
of THz transmission/reflection due to the increase of THz ab-
sorption.
The THz spectra of waveform R (black) and waveform T
(blue) are plotted in Fig. 4(b). The R spectrum is rather
smooth (excluding water vapour absorption lines), while the
T spectrum exhibits the known sinc function shape with a
first zero around 1.5 THz ( fcut−off =c/πd(nΩ−nG,ω)≈1.57
THz) and two plateaus at 0.9 and 2 THz. The T spectrum
is evidently narrower than the R spectrum due to non-perfect
phase-matching and exhibits a poor signal-to-noise ratio be-
cause its intensity is degraded by linear and TPA absorption
when propagating backwards in the crystal. Previously, we
hypothesized that the T pulse corresponds to a THz pulse gen-
erated in transmission and reflected at the exit face of the crys-
tal. Within this hypothesis, the T spectrum is equal to the R
spectrum multiplied by the calculated ratio
ET,ΩER,Ω
and
by ˜rΩe−αΩd/2to include the reflection and absorption through
the sample. The TPA-induced THz absorption is calculated
using a Drude model with the free carrier density as the only
adjustable variable. We assume that the crystal is transparent
in the visible range. This allows us to get rid of the spectral
response of our setup. The calculated spectrum is plotted as
a red curve in Fig. 4(b). We see that it corresponds nicely to
the measured T spectrum. The best fit is obtained for a car-
rier density of 1.19 ×1015 cm−3. The photogenerated carrier
density can also be estimated from the laser power used in
the measurement. Using the TPA coefficient β≈5 cm/GW
and τZnTe ≈25 ns, we get a carrier density of 6.49 ×1015
cm−3. This is in fairly good agreement with the above fit as
we neglect the TPA effect in the nonlinear propagation equa-
tion, which would lead to a higher value of the carrier density
calculated from the T spectrum fit. The recorded R spectrum
7
FIG. 4. (a) Waveforms of THz pulses R and T generated through OR in reflection and in transmission, respectively. (b) THz spectra of
waveform R (black) and T (blue) and the calculated T spectra (red). The inset shows the R and T spectra normalized to their peak value, which
points out the narrower T spectrum as compared to the R spectrum (see comments in the text).
spreads up to 2.7 THz, which is the upper limit (-20 dB) of the
Batop bow-tie detector bandwidth. As the reflected pulse is
weakly affected by absorption in the crystal, one may expect
that its actual spectrum is directly proportional to the laser
beam spectral width and could reach some tens of THz with
sub-100 fs laser pulses. However, because of the weak mag-
nitude of the reflected signal, detection of this latter requires
very sensitive receivers that usually exhibit a long response
time, like bolometers. This could be overcome by performing
an interferometric measurement that will supply the autocor-
relation trace of the generated THz pulses.25
The efficiency of the THz generation through OR in reflec-
tion vs. pump laser fluence is plotted (circles) in Fig. 5(a) to-
gether with a linear fit (dotted) and a TPA-induced saturation
fit (red). At weak pump laser fluence, the efficiency is linear,
while TPA-induced saturation begins at 200 µJ/cm2in good
agreement with published results on OR in transmission26.
This saturation effect is explained by the absorption of the
THz signal induced by the TPA photogenerated carriers in the
sample. A rigorous analysis requires solving coupled propa-
gation equations with the TPA effect. This is a rather tricky
task, which is outside the scope of this paper, as the carrier
population dynamics is usually treated in the time domain,
while propagation equations are solved in the frequency do-
main (see for example the pump-and-probe THz studies per-
formed by P. Kužel and his coworkers27,28). However, a sim-
ple evaluation of the order of magnitude of the influence of
TPA can be performed as follows. The photocarrier popula-
tion modifies both the refractive indices ˜nΩand ˜nG,ω∝˜nω
at THz and laser frequencies, respectively. Thus, it changes
the magnitude of the reflected THz signal, whose expression
is given by Eq. (9). The variation of the real part of ˜nG,ω
is due to the Kerr effect, while its imaginary part is modified
by TPA. In ZnTe, the value of the TPA absorption coefficient
is β=5 cm/GW and the Kerr coefficient is n2≈5×10−18
m2/W 29. In the present experiment, the maximum laser in-
tensity is Iω(max)≈5.5 GW/cm2, therefore the photoinduced
variation of ˜nG,ωis ∆nω≈2.5×10−4and ∆κω≈2×10−4.
We conclude that the variation of ˜nωis too small to explain
the saturation of the reflected THz field. Let us now address
the variation of ˜nΩ. To take into account the influence of the
TPA photocarrier population, we use the Drude model:
˜
εΩ=ε∞−ω2
p
ω(ω+jΓ)= (nΩ+jκΩ)2,ω2
p=NTPA e2
meffεo
.(21)
Here, NTPA is the TPA photocarrier density, meff the effective
mass of the free electrons, Γthe damping angular frequency,
and ethe charge of electron. As already explained, we do not
take into account the dynamics of NTPA, but we simply take
an averaged value in time and over the sample thickness. The
absorbed laser intensity due to the TPA effect is
∆Iω=Iω1−e−βIωd.(22)
The number of absorbed photons per laser pulse is
Npulse =∆Iω
τlaser
ℏωS(23)
with laser pulse duration τlaser, the reduced Planck constant ℏ,
and laser spot size S. The TPA-induced photocarrier density
is given by
NTPA =ηNpulse
Sd .(24)
The coefficient η(not calculated here) renders for the dy-
namics of the photocarrier population (carrier lifetime, dif-
fusion inside the sample. . . ). We performed the calculation
using ZnTe parameters determined by Constable and Lewis30 :
meff =0.151 ×me,ε∞=7.3 and Γ=0.3 THz. The TPA in-
duces both an increase of absorption and a decrease of the
refractive index at THz frequencies. However, the effect of
8
FIG. 5. (a) Measured R pulse amplitude versus laser peak fluence (circles). The dotted straight line is a linear fit while the continuos red curve
is calculated taking account TPA as explained in the manuscript. (b) Polarimetric measurements of the R pulse (circles) fitted with the model
(red).
the refractive index decrease is stronger in Eq. (9), leading
to a small increase of the reflected signal versus the laser in-
tensity. Thus, this can neither explain the observed saturation
effect. The only possible reason left for this saturation is the
dependence of the nonlinear susceptibility on the carrier den-
sity. In a classical model for nonlinear susceptibility31,32 , the
magnitude of the nonlinear susceptibility ↔
˜
χ(2)is proportional
to the magnitudes of the linear susceptibilities at ωand Ω,i.e.
↔
˜
χ(2)
∝
↔
˜
χ(1)
ω2↔
˜
χ(1)
Ω
=
(˜
εω−1)2(˜
εΩ−1)
(25)
=
(nω+jκω)2−12(nΩ+jκΩ)2−1
.
We calculated the term
↔
˜
χ(2)
Iωof Eq. (9) using the above-
described Drude model and the ZnTe parameters by Constable
and Lewis30. The normalized calculated curve is plotted as a
continuous red curve in Fig. 5. We can see the good agree-
ment with our experimental data. The only adjustable param-
eter is the coefficient η, whose best fitting value is η=0.259.
This means that just about a quarter of the photoexcited carri-
ers interact with the THz pulse. The reason could be that they
both recombine and diffuse inside the sample (Dember effect,
both longitudinal and transversal33,34) in between two succes-
sive laser pulses. Of course, this crude model and the related
explanations must be validated by a complete rigorous analy-
sis. Nevertheless, the variation of the nonlinear susceptibility
appears to be a prevailing phenomenon when dealing with OR
THz generation in a crystal exhibiting a strong TPA effect.
Finally, we perform a polarimetric study of the R pulse.
The angle of the linear polarized pump beam ψ=2ψHWP is
scanned 360◦by adjusting the half-wave plate angle ψHWP in
5◦increments. The bow-tie detector is not strictly sensitive to
a single polarization due to its antenna geometry. Therefore,
we must fit a weighted expression
(1−γ)ER,Ω,x+γER,Ω,y(26)
of the field components given in Eq. (19) to the data. Addi-
tionally, the fit takes into account an angular shift δψ due to
disorientation of the crystal axes compared to the laboratory
frame. Inspecting Fig. 5(b), we see an excellent agreement
of the recorded THz peak magnitude (circles) and our fitted
model (red, δ ψ =66.5◦,γ=0.34). This value of γcorre-
sponds to a 10% sensitivity of the antenna to the cross po-
larization, which is within the specifications given by Batop
GmbH. Thus, the reflected THz signal contains information
of the crystalline orientation of the sample.
V. CONCLUSION
In conclusion, we experimentally demonstrated the gener-
ation of a reflected THz signal at normal incidence through
OR in a ZnTe crystal. The reflected signal originates in the
boundary conditions for the nonlinear fields at the crystal sur-
face. All the characteristics of THz OR generation in the crys-
tal (polarization symmetry, spectral features...) are retrieved
in the reflected signal. Its bandwidth is wider than in trans-
mission, because it is not limited by absorption losses in the
crystal. At high laser power excitation, the reflected THz sig-
nal from ZnTe saturates: It seems that its origin is the effect of
TPA, which reduces the magnitude of the second order non-
linear susceptibility. However, for crystals of common mm-
thickness that are transparent or exhibit moderate absorption
in both the THz and visible domains, the THz reflection mag-
nitude is much smaller than the one in transmission. When
dealing with crystals that are opaque in one or both of these
spectral domains, or whose rear surface is rough or covered
by opaque films like metals, the reflected THz signal is of
great interest since the transmitted THz signal is weak or even
9
zero. This could be applied to THz microscopy of opaque ma-
terials, like humid biological samples, e.g. when performing
sub-wavelength OR THz imaging7,8.
ACKNOWLEDGMENTS
We acknowledge fruitful discussions with Prof. F. Lau-
rell and Prof. V. Pasiskevicius, both at the Dept. of Applied
Physics, Royal Institute of Technology, Stockholm (Sweden).
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