PHYSICAL REVIEW B 102, 094313 (2020)

Selective imaging of the terahertz electric field of the phonon-polariton in LiNbO3
Keita Matsumoto
1

1,2,*

and Takuya Satoh1,†

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan2
Department of Physics, Kyushu University, Fukuoka 819-0385, Japan(Received 24 June 2020; revised 8 September 2020; accepted 10 September 2020; published 28 September 2020)
Coherent phonon-polaritons have attracted a considerable amount of interest owing to their relevance tononlinear optics and terahertz (THz)-wave emissions. Therefore, it is important to analyze the THz electricfields of phonon-polaritons. However, in the majority of previous measurements, only a single component of
the THz electric field was detected. In this paper, we demonstrate that pump-probe electro-optical imaging
measurements using the Stokes parameters of probe polarization enable the phase-resolved selective detection of
THz electric-field components that are associated with the phonon-polariton. We experimentally distinguish themode profiles of ordinary and extraordinary phonon-polaritons, and clarify the excitation mechanism as opticalrectification. These results are explained by numerical calculations of Maxwell equations for the THz electricfield. The technique of selectively observing the THz electric field components may be useful for designingefficient THz-wave emitters.
DOI: 10.1103/PhysRevB.102.094313

I. INTRODUCTION

The phonon-polariton, which is a coupling mode of a terahertz (THz) electric field with a phonon in a medium [1–4],
has been studied extensively and observed in noncentrosymmetric materials owing to its relevance to nonlinear optics andTHz-wave emissions. The coherent phonon-polariton can be
excited by an ultrashort optical pulse via impulsive stimulated
Raman scattering (ISRS) [5–8] or optical rectification [9–14],
and the phase-matching condition results in Cherenkov radiation. In particular, this Cherenkov-type phonon-polariton
has been used for efficient THz-wave emission and can beclearly observed in LiNbO3 [7,8,15–19], which is a typicalferroelectric material.
LiNbO3 offers a wide range of optical applications, suchas THz-wave emitters [20–22] and quasi-phase-matched devices [23–26], owing to its significant nonlinear optical effects
and ferroelectricity. LiNbO3 exhibits high optical-to-THz
conversion efficiency, particularly when using the tiltedpump-pulse-front scheme [13,22,27–31] or Si-prism coupling
method [32–36]. Thus, it is necessary to survey the responseto the ultrashort pulse laser and to analyze the THz electricfield. In the majority of previous pump-probe experiments
on LiNbO3 , only one component of the THz electric field
associated with the phonon-polariton was detected because alinearly polarized probe pulse was used.
A recent experiment demonstrated that several dynamic
magnetization components were selectively resolved withpump-probe measurements by detecting the change in the
Stokes parameters of a circularly polarized probe pulse [37].
The Stokes parameters (S0 , S1 , S2 , S3 ) describe the complete*†

k-matsumoto@phys.kyushu-u.ac.jp
satoh@phys.titech.ac.jp2469-9950/2020/102(9)/094313(6)polarization states of light, where S0 denotes the light intensity, S1 and S2 denote the light polarization direction, and S3
represents the helicity of the circular polarization. Therefore,
in addition to the magnetization, the detection of the change
in the Stokes parameters S1 and S2 of the incident circularly
polarized probe pulse can disclose the phase and amplitude of
the THz electric field components selectively.
In this study, we performed a detailed analysis of the THz
electric field associated with the phonon-polariton in LiNbO3
using the phase-resolved pump-probe imaging technique. The
numerical calculations of the THz electric field along the x
and y axes showed good agreement with the experimental
detections of the Stokes parameters S1 and S2 , respectively,
indicating the capability of the selective detection of theTHz electric field components. Moreover, we analyzed thedispersion relation and mode profiles of the excited phononpolariton in terms of the Cherenkov radiation through opticalrectification.
II. EXPERIMENTAL METHODS

Figure 1(a) illustrates the experimental setup of the pumpprobe electro-optical (EO) imaging measurement. The pumpand probe pulses were generated by a Ti:sapphire regenerative
amplifier with a repetition rate of 1 kHz and a pulse durationof τ = 60 fs. The optical parametric amplifier converted thelinear polarized pump pulse to a pulse with the same pulse
duration and 1300-nm central wavelength; the converted pulse
was focused in a circular shape to with half width at half
maximum of 35 μm. The angle of incidence of the pumppulse was 6◦ . The probe pulse was circularly polarized witha central wavelength of 800 nm, incident perpendicularly on
the sample surface without focusing, and was detected by
a complementary metal oxide semiconductor camera [38].
For the pump and probe pulses, the pulse fluences were094313-1©2020 American Physical Society

KEITA MATSUMOTO AND TAKUYA SATOH

PHYSICAL REVIEW B 102, 094313 (2020)

FIG. 1. (a) Experimental configuration of pump-probe imaging.
GTP: Glan–Taylor prism; λ/2: half-wave plate; λ/4: quarterwave plate; BPF: band-pass filter, which passes a wavelength of
800 nm; CMOS: complementary metal oxide semiconductor camera.
(b) Crystal orientation of LiNbO3 (z cut), pump polarization, anddefinition of cylindrical coordinates.
−2−2

160 mJ cm and 0.3 mJ cm , respectively. All the measurements were performed at room temperature.
The pump polarization was oriented along the y axis, as
illustrated in Fig. 1(b), to minimize the influence of oblique
incidence [see Fig. 1(a)]. We measured the transmitted intensities for four settings of the analyzer, namely, 0◦ , 45◦ , 90◦ ,and 135◦ from the x axis. From these data, we obtained S1 by
subtracting 90◦ from 0◦ , and S2 by subtracting 135◦ from 45◦ ,which are referred to as S1 probe and S2 probe, respectively.
Our sample was a uniaxial LiNbO3 (z-cut) single crystal
with a point group of 3m, grown by the Czochralski method
and with a thickness of 500 μm. The permittivity for the
extraordinary wave polarized along the optic (z) axis was
εe = 26, whereas the permittivity for the ordinary wave polarized along the x and y axes was εo = 41.5 in the sub-THz
regime [17]. Thus, without pump excitation, no birefringenceoccurred in the x−y plane, enabling the transmission of acircularly polarized probe pulse.
III. NUMERICAL CALCULATIONS

A pump pulse traveling in the z direction with a speed of
c/ng generates THz electric polarization PTHz through opticalrectification, where c is the speed of light in vacuum and ng =2.27 is the group refractive index at 1300 nm [39]. By using
the optical rectification tensor [dyyy = −8.8 × 10−8 , dzxx =−12.4 × 10−8 cm/statV (dyyy = −37 and dzxx = −52 pm/V
in SI units)] for LiNbO3 [36], the THz electric polarization ina cylindrical coordinate (r, φ, z), as indicated in Fig. 1(b) isexpressed as follows:PTHz = PrTHz , PφTHz , PzTHz · G(r)F (t )= (dyyy sin φ, dyyy cos φ, dzxx )I · G(r)F (t ),(1)where I is the pump-pulse intensity, G(r) = exp[−r 2 /(2l⊥2 )],l⊥ = 25.5 μm was set for our calculation, and F (t ) =(ε−1 )i j = δi j /εo +ri jk EkTHz =kexp[−(t − zng /c)2 /τ 2 ]. We note that excitation of thephonon-polariton via ISRS was not likely to occur because thepump polarization contained only the ordinary wave [7,8,40],
as discussed in the next section.
We numerically calculated the THz electric field ETHzof the phonon-polariton to explain the experimental results.
In the calculation, we referred to a theoretical framework[33,35,36,41] that describes the one-dimensional propagation of the THz electric field in LiNbO3 (x cut) byusing the Maxwell equations. We extended this theory to
two-dimensional propagation in LiNbO3 (z cut) with thecylindrical coordinate system:0ω2 εo 04π ω2 THz
THz
P̃ ,∇ × ∇ × Ẽ
= 2 0 εo 0 ẼTHz +
c
c2
0
0 ε
e

(2)
where ∇ = (∂/∂r, 0, −iωng /c) and ˜ represents quantities in
the Fourier space (ω). The influence of the phonon is incorporated into εo and εe , which we assumed to be nondispersive,
and the validity thereof is discussed in the following section.
In this calculation, we neglected the propagation along
the angular direction φ because the pump spot was focused
circularly. Thus, according to Eq. (2), the governing wave
equations can be expressed as follows:ω2∂ 2 ẼrTHz+
εeff ẼrTHz
∂r 2
c2
  2

∂ G(r)
ω2
4π
ω ∂G(r)−Pz F̃ (ω)=+εe 2 G(r) Pr +i ng
2
εo
∂r
c
c
∂r
∂ 2 ẼφTHz
∂r 2

+

ω2 εo
εeff ẼφTHz
c2 εe

4π ω2
Pφ G(r)F̃ (ω)
c2∂ 2 ẼzTHz
ω2+εeff ẼzTHz
∂r 2
c2

4π iωng ∂G(r)ωεoPr + i=εeff G(r)Pz F̃ (ω),εo c∂rcng εe=−(3)where εeff = εe (1 − ng2 /εo ) is the effective permittivity. Finally, we used the inverse Fourier transform of Eq. (3) andapplied a coordinate transformation from the cylindrical toCartesian coordinates, so as to obtain the spatial dynamics ofETHz = (ExTHz , EyTHz , EzTHz ).
IV. RESULTS AND DISCUSSION

Once the THz electric field ETHz is generated by the pumppulse, birefringence is induced in the x−y plane, which modulates the polarization of the circularly polarized probe pulse
via the EO effect [34]. As the EO effect yields the permittivity
change, the inverse matrix of the in-plane permittivity (ε−1 )i j
is expressed as follows:
1/εo + rxzx EzTHz − ryyy EyTHz−ryyy ExTHz
094313-2

−ryyy ExTHz,
1/εo + rxzx EzTHz + ryyy EyTHz(4)

SELECTIVE IMAGING OF THE TERAHERTZ ELECTRIC …

PHYSICAL REVIEW B 102, 094313 (2020)

(a)

(b)
1
500

250

250

y [ m]

0

0

-250

0

0

-250
-500

-500

-250

0
250
x [ m]

-1

500

(c)

-500

x [ m]

250

500

250

250
y [ m]

S2 [arb.units]

500

0

-1

1

500

-250

0

(d)

1

0

-250

Ey [arb. units]-500

y [ m]

Ex [arb. units]

500
S1 [arb. units]y [ m]

1

0

0

-250
-500

-500

-500

-250

0
250
x [ m]

-1

500

-500

(e)

-250

0

x [ m]

250

500

-1

1
500
Ez [arb. units]y [ m]

250

0

0
-250
-500

-500

-250

0

x [ m]

250

500

-1

FIG. 2. (a) Experimental result of S1 -probe configuration and (b) corresponding numerical calculation of ExTHz ; (c) experimental result of
S2 -probe configuration and (d) corresponding calculation of EyTHz at 10 ps after pump excitation. The black arrows in (b) and (d) indicate thedirections of the THz electric field. (e) Calculated waveform of EzTHz .

where δi j is the Kronecker delta, ri jk is the EO tensor [17,42]
for a 3m crystal, {i, j} = {x, y}, and k = {x, y, z}. Because theEO modulation is sufficiently small, we can approximately
determine the permittivity change δεi j from the permittivity
εi j = εo δi j + δεi j , as follows:−rxzx EzTHz + ryyy EyTHz ryyy ExTHz
2.
δεi j ≈ εoryyy ExTHz−rxzx EzTHz − ryyy EyTHz(5)Previous research [37,43] demonstrated the relation between the permittivity change and Stokes parameters: S1 ∝δεxy + δεyx and S2 ∝ δεxx − δεyy . Therefore, in our setup,S1 ∝ ExTHz and S2 ∝ EyTHz , which implied selective detectionof the THz electric field components by using the Stokes
parameters.
Figures 2(a) and 2(b) depict the comparison of the experimental results by the S1 -probe configuration with thecalculated waveform of ExTHz at 10 ps following pump

094313-3

KEITA MATSUMOTO AND TAKUYA SATOH

PHYSICAL REVIEW B 102, 094313 (2020)

FIG. 3. Experimentally observed spatiotemporal map along the x axis for (a) S1 - and (b) S2 -probe configurations. The dashed lines on eachpanel express the propagation speed.

excitation, and the similarity of these results confirmed thatS1 ∝ ExTHz . At x = y = 0, the pump pulse was shone on thesample at t = 0 ps. Thereafter, the THz electric polarizationand electric field were generated through optical rectification,such that the THz electric field of the phonon-polariton was
propagated in the r direction with two distinct speeds of ≈50
and 60 μm/ps. Similarly, Figs. 2(c) and 2(d) depict the comparison results of the experiment by S2 -probe configurationand the numerical calculation of EyTHz at 10 ps following pumpexcitation. The propagating speeds of both wave forms wereidentical to those in Fig. 2(a), and the relation of S2 ∝ EyTHz
was confirmed. The agreements between the experiments andnumerical calculations prove the selective detection of theTHz electric-field components. Furthermore, the calculated
wave form of EzTHz is illustrated in Fig. 2(e) (see also themovie in the Supplemental Material [44]). The wave formof EzTHz was nonzero only for the outer wave form, which
implied that the THz electric field of the inner wave form did
not contain an extraordinary wave.
The black arrows mapped onto Figs. 2(b) and 2(d) indicate
the direction of the THz electric field. The THz electric field of
the inner wave form was polarized along φ and perpendicular
to the propagation direction, whereas the outer wave-form
polarization was along r and z. Therefore, these propagating
modes could be attributed to the ordinary and extraordinary
phonon-polaritons [40,45,46] for the inner and outer wave
forms, respectively.
Figures 3(a) and 3(b) illustrate the spatiotemporal mapsof the wave form propagating along the x axis, which wereobtained experimentally in the S1 - and S2 -probe configurations, respectively. The spatial profiles along the y axis wereaveraged; thus, Fig. 3(a) reflects the extraordinary phononpolariton propagating at a speed of ≈60 μm/ps, whereas
Fig. 3(b) indicates the ordinary phonon-polariton propagating
at a speed of ≈50 μm/ps. These speeds were determined by√
Eq. (3): ErTHz and EzTHz had a wave number of ω εeff /c, with√a corresponding speed of c/ εeff ≈ 60 μm/ps; EφTHz had a√wave√number of ω εo εeff /εe /c, with a corresponding speedof c/ εeff εe /εo ≈ 50 μm/ps.

The theoretical dispersion curves of these phononpolariton modes are depicted in Fig. 4(a). The blue curve isthe ordinary one expressed by
ωo2 εo − ω2 εo∞k 2 c2=,ω2ωo2 − ω2

(6)

where k is a wave number perpendicular to the wave frontof the phonon-polariton, ωo /(2π ) = 4.6 THz is the frequencyof the E phonon at 152 cm−1 , polarized perpendicular tothe optic axis [47], and εo∞ = 19.5 is the permittivity of theordinary wave at the limit of ω → ∞ [17].
The red curve indicates the extraordinary phononpolariton, represented byk 2 c2=ω2ωe2 εe −ω2 εe∞ωe2 −ω2ωe2 εe −ω2 εe∞ωe2 −ω2ωo2 εo −ω2 εo∞ωo2 −ω22 ε −ω2 ε ∞ωcos2 θe + o ωo2 −ω2 oosin2 θe,(7)

where θe is the propagation angle from the z axis, ωe /(2π ) =7.4 THz is the frequency of the E phonon at 248 cm−1 ,polarized parallel to the optic axis [47], and εe∞ = 10
is the permittivity of the extraordinary wave at the limitof ω → ∞ [17].
As the spatial distribution of the pump pulse isexp[−r 2 /(2l⊥2 )] according to Eq. (3), the wave number of theexcited phonon-polariton follows ∝ exp(−k 2 l⊥2 /2) [48–51].
Thus, the upper limit of the excited wave number is 2π /l⊥ ≈0.25 rad/μm. At such a small wave number, the dispersions√
are approximately linear with slopes of vo = ω/k = c/ εo =√ 2c cos θo /ng and ve = ω/k = c/ ng + εeff = c cos θe /ng forthe ordinary and extraordinary photon-polaritons, respectively, where θo and θe are the Cherenkov angles, as
schematically depicted in Fig. 4(b).
The group velocity of the pump pulse c/ng is faster thanthe phase velocity of the generated THz electric field, leading to Cherenkov radiation with angles of θo and θe for theordinary and extraordinary phonon-polaritons, respectively.

094313-4

SELECTIVE IMAGING OF THE TERAHERTZ ELECTRIC …

PHYSICAL REVIEW B 102, 094313 (2020)

Finally, we discuss the possible excitation of the phononpolariton through ISRS. The Raman tensors of the E phononin the x−y plane are [40,43]




f
0
0
−f
, (10)
Ê (−x) =and Ê (y) =
−f
0
0 −f
where f is a component of the Raman tensor. The driving
force of the E phonon is obtained by operating the pumppolarization vector on the tensors as

 
f
0
0
= −I · F (t )c,FE (x) = I · F (t )(0 1)
0 −f
1

 
0
−f
0
FE (y) = I · F (t )(0 1)
= 0.
(11)
−f
0
1
Thus, the permittivity modulation δεiISRS
isjδεiISRS∝ FE (x) Ê (x) + FE (y) Ê (y) = I · F (t )
j

 2
−f
0


0
.
f2
(12)

Equation (12) indicates the signal of S2 probe because
ISRS
ISRS
δεxx
− δεyyis finite, while no signal can be expected for
ISRS
ISRS
+ δεyxis zero. However, as
the S1 probe because δεxy
depicted in Figs. 2 and 3, the signals in the S1 probe werediscernible, leading us to conclude that the excitation was
mainly caused by optical rectification rather than ISRS.
V. CONCLUSIONS

FIG. 4. (a) Dispersion relation of phonon-polariton. The inset
presents a magnification of the shaded region, indicating the different
propagation speeds of the ordinary and extraordinary phononpolaritons. The horizontal axis is a wave number perpendicular to thewave front of the phonon-polariton. (b) Schematic of ordinary andextraordinary phonon-polariton modes propagating at Cherenkov angles of θo and θe . The propagation directions of the wave fronts areillustrated in the right panels.

These angles can be obtained from equations using the variables in the panels on the right side of Fig. 4(b), as follows:



√ 
2
ε
−
n
ε
ε
o
eff
g
o
= tan−1θo = tan−1= 64.5◦ ,
εe ng
ng2
(8)
 
√ 

ng2
εeff
εe
−1
−1
= tan
1−
θe = tan= 69.3◦ .
ng
ng2
εo
(9)

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We demonstrated a selective detection technique of theelectric-field components of a THz wave propagating as aphonon-polariton in LiNbO3 (z cut) by observing the Stokes
parameters S1 and S2 , which can provide more information
regarding the phonon-polariton than previous studies. The
experimental results were confirmed by the numerical calculations of ExTHz and EyTHz using the Maxwell equations. This
technique enables phonon-polaritons with two different mode
profiles of the THz electric field to be distinguished: the ordinary phonon-polariton that is polarized perpendicular to boththe propagation direction and optic axis, and the extraordinary
phonon-polariton. Based on the experimental results, we canconclude that the excitation mechanism is optical rectification
rather than ISRS. The detection technique may provide a toolfor understanding the fundamental nature of the propagationof THz waves in anisotropic materials.
ACKNOWLEDGMENTS

This study was supported by the Japan Society for
the Promotion of Science (JSPS) KAKENHI (Grants
No. JP19H01828, No. JP19H05618, No. JP19J21797, No.
JP19K21854, and No. JP26103004) and the JSPS Coreto-Core Program (A. Advanced Research Networks). K.M.
would like to thank the Research Fellowship for Young Scientists by the JSPS.

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