ARTICLES

PUBLISHED ONLINE: 11 AUGUST 2013 | DOI: 10.1038/NPHOTON.2013.213

Terahertz polarization pulse shaping with arbitrary

field control
Masaaki Sato1,2†, Takuya Higuchi2,3,4†, Natsuki Kanda2,3,5,6, Kuniaki Konishi2,6, Kosuke Yoshioka2,4,
Takayuki Suzuki1,2,7, Kazuhiko Misawa1,2,7 and Makoto Kuwata-Gonokami2,3,4,6*

Polarization shaping of terahertz pulses enables us to manipulate the temporal evolution of the amplitude and direction of
electric-field vectors in a prescribed manner. Such arbitrary control of terahertz waves has great potential in expanding the
scope of terahertz spectroscopy, the manipulation of terahertz nonlinear phenomena and coherent control. This is
analogous to the use of pulse-shaping techniques for optical frequencies that involve light’s polarization states as a
controllable degree of freedom. Here, we propose and demonstrate a method for generating a prescribed terahertz
polarization-shaped waveform by the optical rectification of a laser pulse whose instantaneous polarization state and
intensity are controlled by an optical pulse shaper. We have developed a deterministic procedure to derive input
parameters for the pulse shaper that are adequate to generate the desired terahertz polarization-shaped waveform, with
the benefit of simple polarization selection rules for the rectification process of light waves propagating along the three-
fold axis of a nonlinear optical crystal.

M
anipulating the direction and timing of electromagnetic electric-field vectors can be manipulated in a prescribed manner.
fields has played a pivotal role in handling the coherent For this purpose, we converted a tailored laser pulse (where its
interactions between light fields and matter, for example, instantaneous intensity and polarization state were controlled by
in increasing the multiphoton ionization yield1–3, in polarization an optical pulse shaper) to terahertz waves using a nonlinear
gating for higher harmonic generation4,5, in coherent multidimen- optical crystal. Such conversion of a polarization-shaped pulse by
sional spectroscopy6–8 and in photoassisted asymmetric synthesis frequency conversion has been demonstrated in various frequency
of chiral molecules9–11. Recent developments in terahertz spec- ranges, including ultraviolet35,36 and mid-infrared37 frequencies.
troscopy and coherent control12–16 require the extension of these However, extension of these techniques into the terahertz regime
optical techniques to terahertz spectral ranges. In other words, it has not yet been achieved. Here, wavelength conversion from the
is essential to be able to generate arbitrary terahertz polarization- optical to terahertz frequency range was performed along the
shaped waveforms, where the amplitude and direction of the electric three-fold axis of the crystal to take advantage of the simple polar-
field evolve in a prescribed manner with time. Various methods to ization selection rules for optical rectification determined by crystal-
control the scalar properties of terahertz pulses (that is, shaping a line symmetry38. We developed a deterministic algorithm to derive a
terahertz waveform with a fixed polarization state) have been set of parameters for the optical pulse shaper, resulting in the gen-
demonstrated, for example, by using a poled nonlinear optical eration of a targeted terahertz polarization-shaped waveform. We
crystal17, by varying the intensity profile of the laser18–20 and by experimentally demonstrate that our method indeed provides the
synthesizing terahertz waves with different frequencies generated targeted terahertz polarization-shaped waveforms.
by spatially distributed sources21. However, electromagnetic fields
are vector quantities, and the use of their directions as an active Results
tool can substantially broaden the applications of terahertz technol- We begin with the relationship between the incident laser pulse and
ogies22–24. Researchers have demonstrated the control of terahertz the emitted terahertz wave. When a femtosecond laser pulse is intro-
polarization states using wire-grid polarizers and Fresnel-rhomb duced into a nonlinear optical crystal without an inversion centre, a
THz
wave plates25,26, or by combining terahertz waves generated by mul- rectified dielectric polarization P THz(t) (P̃ (V) being its Fourier
tiple laser pulses with different polarization states26–29. However, the form) is induced via the second-order nonlinear optical response39:
arbitrariness and flexibility of the generated terahertz polarization- v0 +Dv
shaped waveforms have been quite limited, and inferior to those ∗
x(2)
THz
P̃i (V) = ijk (V; −v, v + V)Ẽj (v)Ẽ k (v + V)dv (1)
at optical frequencies, such as those achieved through Fourier v0 −Dv
synthesis using spatial light modulators (SLMs)30–34. Only the
polarization states can be manipulated22,25–29 and the achievable where v0 and Dv are the central frequency and bandwidth of the
terahertz waveforms are dictated by the set-up23. laser pulse, respectively, x(2)
ijk is the second-order susceptibility
We report the first demonstration of the generation of polariz- tensor and Ẽ(v) is the Fourier form of the laser electric field.
ation-shaped terahertz pulses, where the temporal evolution of Terahertz waves with electric field E THz(t) are radiated by the

1
Department of Applied Physics, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan, 2 CREST, Japan
Science and Technology Agency, Sanbancho Bldg, 5 Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan, 3 Department of Applied Physics, The University of
Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, 4 Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan,
5
RIKEN Advanced Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan, 6 Photon Science Center, The University of Tokyo, 7-3-1 Hongo, Bunkyo-
ku, Tokyo 113-8656, Japan, 7 Interdisciplinary Research Unit in Photon-nano Science, Tokyo University of Agriculture and Technology 2-24-16 Naka-cho,
Koganei, Tokyo 184-8588, Japan; † These authors contributed equally to this work. * e-mail: gonokami@phys.s.u-tokyo.ac.jp

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ARTICLES NATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.213

dipole radiation process from P THz(t). We used GaP as the non- In the following, we use instantaneous Stokes parameters in
linear optical material because it has a large nonlinear optical coef- addition to the instantaneous intensity and polarization angles,
ficient, and the phase-matching condition is relatively easily satisfied especially in the figures.
for optical rectification in the collinear geometry40. In the current To shape the incident laser pulse, a Fourier-synthesis-based
study, the crystal is sufficiently thin and the pump laser frequency optical pulse shaper and a quarter-wave plate (QWP) were used,
is well-detuned from any resonances, and the generated terahertz as shown in Fig. 1d (see Methods for details). A horizontally polar-
wave frequency is well below the phonon resonances. Accordingly, ized pulse is introduced into the SLM, and phases u+458(v) of the
the effect of the material’s absorption and dispersion can be neg- diagonally polarized components of each frequency are controlled
lected. We can therefore assume that x(2) ijk does not depend on the independently. After passing through a QWP whose fast axis is
optical frequency. With this assumption, we can also say that the horizontal, these phase parameters are converted to the spectral
rectified polarization is impulsively excited, with the amplitude pro- phase uopt (v) ; (1/2)(u458 (v) + u−458 (v)) and azimuthal angle
portional to the instantaneous intensity Iins(t) of the laser pulse. fopt (v) ; u458(v) 2 u2458(v). Note that the spectral ellipticity
As well as the instantaneous laser intensity, equation (1) also hopt (v) is always zero. As a result, the output is given by
indicates that P THz(t) depends on the instantaneous polarization     
state of the laser pulse. In this study, we chose a geometry Ex (t) dv −ivt iuopt (v)
cos fopt (v)
wherein a femtosecond laser pulse propagates along a three-fold = e Ẽinc (v)e (4)
Ey (t) 2p sin fopt (v)
axis [111] of GaP to make use of the simple polarization selection
rules for its optical rectification. This polarization dependence is
obtained by considering the non-zero elements of x(2) ijk , which are where Ẽinc (v) is the Fourier form of the electric field of the incident
determined by crystalline symmetry via Neumann’s principle29,38,41. Fourier-transform-limited pulse. (See Supplementary Information
When the coordinates considered are x[112], y[110] and z[111] for the detailed derivation of equation (4).)
in GaP, only x(2) (2) (2) (2)
xxx = −xxyy = −xyxy = −xyyx are non-zero, and the As a first example, we demonstrate the control of the terahertz
remaining elements in the x–y plane vanish. As a result, E THz(t) is polarization state by generating few-cycle circularly polarized tera-
described as an impulsive response to Iins(t) of the laser pulse, and hertz pulses and by controlling their helicities, as shown in
E THz(t) also depends on the instantaneous ellipticity hins(t) and Fig. 1b,c. To generate a circularly polarized terahertz wave, we
polarization azimuthal angle fins(t) of the laser following the polar- made the terahertz electric-field vector rotate with a prescribed
ization selection rule: angular velocity. This makes the x and y components of the elec-
 THz      tric-field vector have the same spectral amplitude, while the phase
Ex (t) 1   cos −2fins (t) of the x component is shifted by p/2 relative to that of y (ref. 25).
= a(t − t)Iins (t) cos 2hins (t)   dt To achieve this, we introduced the following spectral parameters
EyTHz (t) −1 sin −2fins (t)
into the pulse shaper:
(2) b
uopt (v) = (v − v0 )2 (5)
2
where a(t) is the impulse response function determined by the
strength of the second-order nonlinear response, absorption of fopt (v) = g(v − v0 ) (6)
the terahertz waves by the crystal and phase-matching conditions42.
According to equation (2), one can generate a desired terahertz which are shown in Fig. 1e (v0 is the laser central frequency).
polarization-shaped waveform by designing the instantaneous The role of the spectral phase in equation (5) is to stretch the
intensity and polarization state of the incident laser pulse. light pulse in the time domain. A linear chirp is introduced so
Namely, for a given ExTHz (t) and EyTHz (t), deconvolution of these that each frequency component has a frequency-dependent delay
functions by a(t) gives the desired Iins(t) and fins(t), when hins(t) of t(v) = (∂/∂v)uopt (v) = b(v − v0 ) (ref. 44). Because we set |b|
is maintained at 0 (that is, cos(2hins(t)) ¼ 1) to maximize the con- to be sufficiently larger than the inverse square of the bandwidth
version efficiency. In particular, the intensity and phase of ã(V), of the laser pulse, a one-to-one relation can be introduced
the Fourier form of a(t), are nearly flat in the region between between time and frequency (the validity of this assumption is dis-
1 and 3 THz (experimental data are shown in Supplementary cussed in Supplementary Section S1). In other words, the instan-
Section S1) in our experiment. Therefore, a laser pulse with an taneous frequency vins(t) is well-defined as the inverse function of
instantaneous azimuthal angle of fins(t) impulsively generates an t ¼ t (v) (Fig. 1e2)44.
electric-field vector within this frequency region, with angle According to this one-to-one relation between time and fre-
22fins(t). Using this relation, we can control the amplitude and quency, the change in the polarization state in the frequency
direction of the terahertz electric-field vector by modulating Iins(t) domain (Fig. 1e4) is imprinted into the time domain (Fig. 1e5).
and fins(t) of the incident laser pulse, respectively. Note that the The modulation in equation (6) results in the rotation of the instan-
terahertz waves were linearly polarized without shaping the incident taneous azimuthal angle, fins(t) ¼ fopt (vins(t)) ¼ gb 21t and
laser, and the azimuthal angle of the terahertz wave was constant, as hins(t) ¼ 0, as depicted in Fig. 1e5. The result is that the direction
determined by the relation fTHz(t) ¼ 22fins(t) (Fig. 1a). (See of the terahertz electric-field vector rotates as a function of time:
Methods for measurement of the terahertz electric-field vector.) f THz(t) ¼ 22fins(t) ¼ 22gb 21t. This means that the terahertz
The instantaneous intensity and polarization states can also be wave is circularly polarized, and the angular velocity of the elec-
described in terms of instantaneous Stokes parameters43. In particu- tric-field vector is 22gb 21. Therefore, by changing the sign of
lar, S1(t) ; Iins,x(t) 2 Iins,y(t) and S2(t) ; Iins,458(t) 2 Iins,2458(t) are 22gb 21 the helicity of the terahertz wave can be controlled, as
defined as the difference between the instantaneous intensities of shown in Fig. 1b,c.
two cross-linearly polarized components. They satisfy S1(t) ¼ As seen in this example, this pulse shaper controls the optical
Iins(t) cos2hins(t) cos2fins(t) and S2(t) ¼ Iins(t) cos2hins(t) sin2fins parameters in the frequency domain, resulting in a change in inten-
(t). Using these relations, equation (2) can be simplified as sity and polarization profile of the laser with time. This enabled us to
  control the terahertz polarization-shaped waveforms in the time
1  
ExTHz (t) S1 (t) domain. However, one question remains: how can we find the spec-
= a(t − t) dt (3) tral parameters that result in the laser pulse shaping that will achieve
EyTHz (t) −1 −S2 (t) the desired temporal evolutions of laser parameters Iins(t), fins(t)

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NATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.213 ARTICLES
a Ey b Ey c Ey

Ex Ex Ex

S2 S2 S2
ϕins S1
S1 S1
ϕTHz
2ϕins

z 1p z z
s×
c
z−c t z−c t z−c t

e 385
e1 e2 e3 e4
d

CM SLM ~ θ(ω) ϕ(ω)

I (ω)

Frequency (THz)
t = τ(ω)
375 ω = ωins(t)

Linearly polarized Grating

pulse
365
Polarization-shaped 1 0 −6 −3 0 3 6 0 75π −π 0 π
QWP pulse Intensity (a.u.) Time (ps) Phase (rad) Azimuthal
angle (rad)
π
S1(t) and S2(t) (a.u.) Intensity (a.u.)

and elliptical
angle (rad)

Azimuthal
e5
ηins(t) Iins(t)
0
ϕins(t)
0 −π

1
e6 S1(t)
0
S2(t)
−1
−6 −3 0 3 6
Time (ps)

Figure 1 | Generation of terahertz waves by shaped laser pulses with prescribed polarization states. a, Experimentally obtained trajectory of the electric-
field vector (purple curve) of a linearly polarized terahertz pulse generated by a linearly polarized laser pulse. The time-dependent intensity and azimuthal
angle of the incident laser pulse are schematically depicted as a red curve heading to the orange GaP crystal. These parameters of the pulse were
reconstructed from its instantaneous intensity and instantaneous polarization states, which were measured by the polarization-selective cross-correlation
method32. Experimentally obtained trajectories of S1(t) and S2(t), the first and second instantaneous Stokes parameters, are also shown below. b, A few-cycle
right-circularly polarized terahertz pulse generated by a polarization-shaped laser pulse. c, Reversing the direction of the twists in the polarization envelope of
the incident laser results in the reversal of the terahertz helicity, and a left-circularly polarized terahertz wave is obtained. All values are plotted in arbitrary
units. d, Schematic of the optical pulse shaper. The laser is focused to the liquid-crystal cells of the SLM, colour by colour, by a cylindrical mirror (CM). The
phase and azimuthal angle of each colour component are modulated by the SLM, and the light then passes through a QWP. e, Design of the laser
parameters to achieve the circularly polarized terahertz pulse in b: spectral intensity (e1), frequency-dependent delay (e2), spectral phase (e3) and spectral
azimuthal angle (e4). These parameters determine the instantaneous intensity and polarization state, as depicted in e5. The corresponding S1(t) and S2(t),
the first and second instantaneous Stokes parameters, are plotted in e6 as red and blue curves, respectively.

and hins(t) from a given Fourier-limited pulse with a power spec- Iins (t)dt = Ĩ(v)dv. This differential equation can be directly
1 1
trum of Ĩ(v) ; |Ẽinc (v)|2 ? Such a scheme is necessary to manip- solved by integrating both sides. Here −1 Iins (t)dt and 0 Ĩ(v)dv
ulate the terahertz waveforms as desired, for example, to achieve a are the same, because both describe the total energy of a laser pulse.
very complicated terahertz waveform, as depicted in Fig. 2a. The second step is to obtain the shaper parameters from the fre-
We developed a deterministic algorithm to solve this problem, quency-dependent delay and the targeted instantaneous azimuthal
which was accomplished with the following two steps (Fig. 2b). angle. Because the frequency-dependent delay is the derivative
The first step is to obtain a frequency-dependent delay of the spectral phase as a function of frequency,
v
t(v) ; (∂/∂v)uopt (v) that shapes the laser pulse with incident uopt (v) = 0 dv′ t(v′ ) gives the spectral phase (Fig. 2b6). In
Ĩ(v) (Fig. 2b4) to form the targeted Iins(t) (Fig. 2b2). We are addition, the one-to-one relation t ¼ t (v) between frequency and
looking for a t (v) that monotonically increases as a function of v time from the frequency-dependent delay (Fig. 2b5) allows us to
(Fig. 2b5). In this case, the pulse energy around a frequency v find the spectral azimuthal angle fopt (v) from the instantaneous
within an infinitesimal frequency range dv appears around azimuth fins(t) by fopt (v) ¼ fins(t (v)) (Fig. 2b7). Figure 2b8
t ¼ t (v) with an infinitesimal duration of dt. Accordingly, shows the resultant instantaneous Stokes parameters of the light
Iins (energy density as a function of time) should satisfy pulse that are calculated from its power spectrum (Fig. 2b4), spectral

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ARTICLES NATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.213

a 2 b 1

S1(t) and S2(t)

b1 S2(t)
1

(a.u.)
)
ps 0
e(
Tim 0 S1(t)
−1
−1
1
b2

Intensity
−2

(a.u.)
1.0 Iins(t )
0.5
Ey (a.u.)

0
0.0 π

angle (rad)
Azimuthal
−0.5 b3
−1.0 0
−1.0 ϕins(t )
0.5
−π
E 0.0
x (a
.u. 0.5 379
b6
) 1.0 b4 b5 b7
~
I (ω) ϕ(ω)
θ(ω)
c 2

1 Frequency (THz)
)
ps
e( 374 t = τ(ω)
Tim 0
ω = ωins(t)
−1

−2
1.0
0.5
Ey (a.u.)

0.0 369
1 0 −1.5 0 1.5 0 10π −π 0 π
−0.5
−1.0 Intensity (a.u.) Time (ps) Phase (rad) Azimuthal
−1.0 angle (rad)
S1(t) and S2(t) (a.u.)

−0.5 1
E 0.0 b8 S2(t)
x (a
.u. 0.5
) 1.0 0
S1(t)
−1
−1.5 0 1.5
Time (ps)

Figure 2 | Deterministic procedure to find a set of shaper parameters from a given targeted terahertz polarization-shaped waveform, and its experimental
verification. a, Example of targeted terahertz polarization-shaped waveforms. b, Deterministic procedure to find a set of spectral parameters that achieves the
generation of the prescribed terahertz waveform. From the given target waveform, the instantaneous intensity and polarization state of the laser pulse are
calculated, which are shown in the form of S1(t) and S2(t) (b1), Iins(t) (b2) and fins(t) (b3). Instantaneous ellipticity hins(t) is always set to zero. From a
given spectral intensity (b4, experimental value) and targeted Iins(t), t(v ) is derived (b5). t(v ) and fins(t) determine the spectral phase (b6) and spectral
azimuthal angle (b7). b8 shows the resultant S1(t) and S2(t) of the laser output, which reproduces the desired waveforms in b1. c, Experimentally obtained
terahertz polarization-shaped waveform generated by the laser pulse shaped with the laser parameters depicted in b.

phase (Fig. 2b6) and spectral azimuthal angle (Fig. 2b7). One can spectral intensity of a radiated terahertz wave by independently
see that the resultant laser pulse reproduces the designed temporal changing its bandwidth and centre frequency. We input the
evolution of the instantaneous intensity and the instantaneous shaper parameters in the form of equations (5) and (6), and
polarization state (Fig. 2b1). Note that this algorithm does not varied b and g for this purpose. The bandwidth of the terahertz
require any learning process by feedback loops, as is often applied waves is inversely proportional to the duration of the incident
in studies using computer-controlled pulse shapers. laser pulse, which is modified by tuning the laser chirp parameter
Using the laser pulse shaped by this set of parameters, we generated b. In addition, the angular velocity of rotation of the laser
the targeted terahertz polarization-shaped waveform. Figure 2c shows azimuth, gb 21, determines the centre frequency of the radiated
the experimentally obtained electric-field trajectory of the generated terahertz waves VTHz0 as 2gb 21. Figure 3a shows the instantaneous
terahertz polarization-shaped waveform. At the beginning of the Stokes parameters of the laser pulses with various durations while
waveform, the electric-field vector rotates anticlockwise with time, maintaining gb 21 constant. As the duration increases, the spectra
but by the end it rotates clockwise. In addition to the controlled direc- of the terahertz waves narrow (Fig. 3b). Therefore, by maintaining
tion of the electric-field vector, its amplitude is also found to be as the pulse duration and varying the angular velocity of the azimuthal
designed. The excellent agreement between the design (Fig. 2a) and rotation of the laser pulse (Fig. 3c), the centre frequency of the tera-
the resultant waveform (Fig. 2c) verifies the validity of our method hertz waves was changed (Fig. 3d). See Supplementary Section S3
for arbitrary terahertz polarization-shaped waveform generation. for the corresponding laser parameters.
Deterministic derivation of the shaper parameters enabled us to In Fig. 4, we also show a way to manipulate the terahertz spectral
manipulate the terahertz polarization-shaped waveforms at will. In phases by introducing an up chirp or down chirp to these waves.
the following, we demonstrate applications of our method to These chirps were achieved by changing the azimuthal angle
control the spectral intensity and phase of circularly polarized tera- fins(t) of an incident laser pulse to a quadratic function of t in
hertz pulses, which is important in terahertz spectroscopy and order that VTHz (t) = −2(d/dt)fins (t) (the instantaneous frequency
coherent control. We first demonstrate a method to control the of the terahertz field) changes linearly as a function of time (see

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NATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.213 ARTICLES
a 1 b 1 c 1 d 1
0 1.0 THz 0 3.0 THz
−1 0 −1 0
1 1 1 1
S1(t) and S2(t) (a.u.)

S1(t) and S2(t) (a.u.)

0 0 2.5 THz

Intensity (a.u.)
Intensity (a.u.)
−1 0 −1 0
1 1 1 1
0 0 2.0 THz
−1 0 −1 0
1 1 1 1
0 0 1.5 THz
−1 0 −1 0
1 1 1 1
0 0.3 THz 0 1.0 THz
−1 0 −1 0
−3 −2 −1 0 1 2 3 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −3 −2 −1 0 1 2 3 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Time (ps) Frequency (THz) Time (ps) Frequency (THz)

S1(t) first Stokes Left-circular S1(t) first Stokes Left-circular

S2(t) second Stokes Right-circular S2(t) second Stokes Right-circular

Figure 3 | Control of spectra of circularly polarized terahertz pulses. a, Experimentally obtained instantaneous Stokes parameters of shaped laser pulses
with various pulse durations. b, Intensity spectra of the terahertz waves generated by the laser pulses in a. The bandwidth is inversely proportional to the
pulse duration, and varies from 0.3 THz (bottom) to 1.0 THz (top). c, Instantaneous Stokes parameters of the laser pulses with various angular velocities of
the twists in the azimuthal angles. d, Intensity spectra of the terahertz waves generated by these laser pulses, the centre frequencies of which were adjusted
from 1.0 THz (bottom) to 3.0 THz (top), while keeping the bandwidth at 0.3 THz. In all panels, dots indicate experimental data. Light-coloured solid lines in
a and c indicate the designed values of the instantaneous Stokes parameters of the laser pulses, which were calculated from the laser spectrum and the
parameters introduced into the pulse shaper. The spectral intensities of the generated terahertz waves were calculated from these designed instantaneous
Stokes parameter values; measured values of a(V) are also depicted as light-coloured solid lines in b and d.

a c
Electric field (a.u.)

1 1 2π
IR(Ω)
Intensity (a.u.)

Up chirp θR(Ω)

Phase (rad)
π
0 0 0
Ex(t) IL(Ω)
−π
Ey(t)
−1 −1 −2π
−2 −1 0 1 2 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Time (ps) Frequency (THz)

b 1 d 1 2π
Electric field (a.u.)

IR(Ω)
Intensity (a.u.)

Down chirp

Phase (rad)
π
IL(Ω)
0 0 0
Ex(t) θR(Ω) −π
Ey(t)
−1 −1 −2π
−2 −1 0 1 2 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Time (ps) Frequency (THz)

Figure 4 | Control of spectral phase of circularly polarized terahertz pulses. a,b, Temporal waveforms of right-circularly polarized terahertz pulses with an
up chirp (a) and a down chirp (b). c,d, Circular-based intensity spectra and phase spectra of the up- (c) and down- (d) chirped terahertz waves are also
shown. IR(V) and IL(V) are intensities of the right- and left-circular components, respectively. Quadratic phases can clearly be seen in the phase spectra uR
(V) of the right-circular components plotted in c and d, where the chirp constants of the terahertz waves, bTHz , were þ0.28 ps2 rad21 and 20.28 ps2 rad21,
respectively. Note that bTHz is defined as the second-order coefficient of the spectral phase by u(2) (2)
THz (V) = (bTHz /2)(V − V0 ) , where uTHz (V) and V0 are the
2

second-order components of the phase spectrum and centre frequency of the terahertz wave, respectively.

Supplementary Section S4 for the design of the laser parameters). Discussion

This method is readily applicable to any higher-order dispersion, Our experimental findings demonstrate that the proposed method
because VTHz(t) can be prescribed in the time domain by the allows the generation of arbitrary terahertz polarization-shaped
time-dependent laser azimuthal angle. waveforms. One might be tempted to suspect that the numbers of
As a final demonstration, we controlled the intensity and polariz- controllable parameters may not be sufficient to handle all degrees
ation state of a terahertz pulse as a function of time (Fig. 5). We of freedom of terahertz electromagnetic fields. In general, to
divided a terahertz pulse into a pair of pulses, and independently con- achieve complete control of waveforms with two transverse dimen-
trolled the polarization states of each of the pair. This was achieved sions, one should control four degrees of freedom for each fre-
by adding a spectral phase in the form of an absolute function quency, for example, spectral amplitudes and spectral phases for
T|v 2 v0|/2 in addition to a linear chirp for the laser (Fig. 5c). This two polarization components. However, the optical pulse shaper
additional phase divides the input pulse into two pulses separated by controls only two sets of laser parameters. This is of no concern,
T in time, as shown in Fig. 5b. Control of the helicity of the paired because the laser spectral parameters and the terahertz spectral par-
terahertz pulses was done by manipulating the azimuthal rotation of ameters are not tied by a one-to-one relationship: optical rectifica-
the laser pulse with time (Fig. 5d,e). The two terahertz pulses had tion does not depend on the change in laser carrier frequency as a
either the same helicities (Fig. 5f) or opposite helicities (Fig. 5g). function of time within a laser pulse. This degree of freedom

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ARTICLES NATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.213

a 385 b c d e

~ θ(ω) ϕ(ω) ϕ(ω)

I (ω)
Frequency (THz)

375
t = τ(ω)
ω = ωins(t)

365
1 0 −2 −1 0 1 2 0 3π −π 0 π −π 0 π
Intensity (a.u.) Time (ps) Phase (rad) Azimuthal Azimuthal
angle (rad) angle (rad)

f Ey g Ey

Ex Ex

Right Right

2.5 2.5
ps Right Left
×c ps
×c

z–c t z–c t

Figure 5 | Control of intensity and direction of terahertz waves as functions of time. a, Intensity spectra of the incident laser pulse. b,c, Frequency-
dependent delay shows that the incident pulse is devided into two parts in the time domain (b), which is achieved by the spectral phase that consists of a
combination of a quadratic function (b/2)(v − v0 )2 and an absolute value function T|v 2 v0|/2 (c). d–g, Spectral azimuthal angles were introduced either
linearly (d) or with different slopes above and below v0 (e) as functions of frequency, resulting in the generation of a pair of circularly polarized terahertz
pulses, with experimentally obtained trajectories as shown in f and g, respectively.

cannot be independently controlled by the pulse shaper employed phase modulation is discrete in frequency, because the number of
here, but it is lost during the rectification process. On the other liquid-crystal cells is finite. This limits the duration of a terahertz
hand, Iins(t) and fins(t), which are essential for shaping terahertz pulse, because the spectral phase change within a pulse cannot be
waves, can be controlled by two shaper parameters uopt (v) and extremely large47. This condition determines the spectral resolution
fopt (v), as discussed above. Accordingly, this rather simple pulse in the frequency domain. Finally, the intensity of the generated tera-
shaper provides sufficient degrees of freedom to control terahertz hertz pulse is limited by the energy of the input laser pulse, the effi-
waveforms with two dimensions. Note that the set of spectral par- ciency of the terahertz generation process and the damage threshold
ameters of a laser pulse is not a unique solution: if one can of the optical elements. The maximum field amplitude is
control all four parameters, one may find a different solution that 300 V cm21 in the current experimental conditions. By optimizing
results in the same Iins(t) and fins(t) but with a different temporal the set-up, the electric-field amplitude is expected to exceed
evolution of carrier frequency. 10 kV cm21 (refs 48,49; see Supplementary Section S5 for detailed
An important requirement for arbitrary waveform generation is estimation), enabling observation of the nonlinear phenomena in
that the carrier-envelope phase of a pulse should be prescribed; the terahertz spectral range13–16.
otherwise, the waveform will vary pulse by pulse, even if its envelope
is controlled. This is particularly important when such waveforms Summary
are employed to observe nonlinear phenomena using few-cycle To summarize, we have proposed and demonstrated the generation
pulses. Arbitrary waveform generation in optical frequencies has of terahertz waves with polarization-shaped waveforms by the
been extensively studied by combining pulse-shaping and fre- optical rectification of a tailored incident laser for a targeted tera-
quency-comb techniques45; however, the generation of optical arbi- hertz waveform. In optical rectification, the instantaneous polariz-
trary polarization-shaped waveforms has not been reported. We ation states and intensities of laser fields determine the terahertz
were able to achieve the generation of arbitrary terahertz polariz- vector field by the selection rules determined by the symmetry of
ation-shaped waveforms with a rather simple set-up. This is possible the nonlinear optical crystal. By shaping the laser parameters
because of the fact that the carrier-envelope phase of a terahertz within a pulse, they are imprinted into the terahertz field. This
pulse generated by optical rectification is passively locked, because method achieved our objective of arbitrary terahertz wave gener-
the rectification is a difference-frequency process46. ation. It is worth noting that the various terahertz polarization-
The bandwidth, duration and peak intensity of the waveform shaped pulses demonstrated here were generated by a single set-
achievable by this method are restricted by the practical set-up. up. The rich variety of waveforms is achieved by simply changing
The bandwidth of the terahertz field is limited by the bandwidth the parameters of the computer-controlled SLM. Therefore, our
of the laser and/or response function a(t) in equation (2). In the method is readily applicable for adaptively optimizing terahertz
current set-up, the latter is more critical. In addition, the SLM waveforms for a certain terahertz process, a commonly used

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NATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.213 ARTICLES
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40. Tanabe, T., Suto, K., Nishizawa, J., Sato, K. & Kimura, T. Tunable terahertz Acknowledgements
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Author contributions
M.S. and T.H. contributed equally to the work. All authors contributed to the design, data
46. Baltuška, A., Fuji, T. & Kobayashi, T. Controlling the carrier-envelope phase of
analysis and preparation of the manuscript. M.S. and N.K. performed all experiments at the
ultrashort light pulses with optical parametric amplifiers. Phys. Rev. Lett.
University of Tokyo. M.S. and K.M. designed and developed the optical polarization shaper.
88, 133901 (2002). T.H. and M.K-G. provided the theory for designing pulse shapes and terahertz generation.
47. Weiner, A. M. Femtosecond pulse shaping using spatial light modulators. K.K. and T.S. gave technical support. K.Y. and M.K-G. provided experimental support.
Rev. Sci. Instrum. 71, 1929–1960 (2000). K.M. and M.K-G. planned and supervised the project.
48. Hoffmann, M. C., Yeh, K-L., Hebling, J. & Nelson, K. A. Efficient terahertz
generation by optical rectification at 1035 nm. Opt. Express 15, Additional information
11706–11713 (2007). Supplementary information is available in the online version of the paper. Reprints and
49. Stobrawa, G. et al. A new high-resolution femtosecond pulse shaper. Appl. Phys. permissions information is available online at www.nature.com/reprints. Correspondence and
B 72, 627–630 (2001). requests for materials should be addressed to M.K-G.
50. Kanda, N., Konishi, K. & Kuwata-Gonokami, M. Terahertz wave polarization
rotation with double layered metal grating of complementary chiral patterns. Competing financial interests
Opt. Express 15, 11117–11125 (2007). The authors declare no competing financial interests.

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