PHYSICAL REVIEW X 11, 021067 (2021) - Ultrafast Control of Material Optical Properties via the Infrared Resonant Raman Effect
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PHYSICAL REVIEW X 11, 021067 (2021)
Ultrafast Control of Material Optical Properties via the Infrared Resonant Raman Effect
*
Guru Khalsa and Nicole A. Benedek
Department of Materials Science and Engineering, Cornell University, Ithaca, New York, 14853, USA
Jeffrey Moses
School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA
(Received 24 November 2020; revised 6 May 2021; accepted 10 May 2021; published 30 June 2021)
The Raman effect, inelastic scattering of light by lattice vibrations (phonons), produces an optical
response closely tied to a material’s crystal structure. Here we show that resonant optical excitation of IR
and Raman phonons gives rise to a Raman-scattering effect that can induce giant shifts to the refractive
index and induce new optical constants that are forbidden in the equilibrium crystal structure. We complete
the description of light-matter interactions mediated by coupled IR and Raman phonons in crystalline
insulators—currently the focus of numerous experiments aiming to dynamically control material properties
—by including a forgotten pathway through the nonlinear lattice polarizability. Our work expands the
toolset for control and development of new optical technologies by revealing that the absorption of light
within the terahertz gap can enable control of optical properties of materials over a broad frequency range.
DOI: 10.1103/PhysRevX.11.021067 Subject Areas: Condensed Matter Physics,
Materials Science, Optics
I. INTRODUCTION an optics community perspective, also as a route to uncover
new ways to control optical properties through the inter-
The Raman effect arises from the inelastic scattering
action of laser light with material structure. The recent
of light by phonons in crystals or vibrations in molecules.
development of bright midinfrared and THz sources
The simplest theories describing the effect assume that the
capable of resonantly exciting collective modes in crystals
optical polarizability—that is, the general frequency-
has created new opportunities for exploring such novel
dependent polarizability of a crystalline material—is pre-
optical phenomena [15–17].
dominantly electronic in origin [1,2], and that it is a
In this work, we show that resonant optical excitation of
function only of the displacements of so-called Raman-
IR phonons strongly contributes to the optical polarizability
active phonon modes. However, Herzberg [3] and Raman
via a Raman-scattering mechanism mediated by the dis-
[4] noted over 70 years ago that when infrared (IR)-active
placements of IR phonons and that such excitations can
vibrational modes are driven to large amplitudes, nonlinear
be exploited to significantly modify a material’s optical
coupling between infrared- and Raman-active modes alters
properties, including inducing new optical constants that
the optical polarizability. The relevant terms are complex
are forbidden in the equilibrium crystal structure. This
and were difficult or impossible to verify and explore using
mechanism, which we refer to as infrared-resonant Raman
the methods available at the time [5–14].
scattering (IRRS), differs from the conventional Raman
The Raman effect has widespread scientific importance
effect, in which typically only changes in the electronic
because it produces a basic optical response that is closely
polarizability due to displacements of Raman phonons are
tied to a material’s crystal structure. Raman spectroscopy
considered. In contrast, IRRS involves the inelastic scatter-
exploits this tie and is an essential tool for understanding
ing of light and subsequent changes in the optical polar-
crystal symmetry, structure, and defects. For this reason,
izability due to both changes in the electronic polarizability
the effect predicted by Herzberg and Raman noted above—
and ionic polarizability due to Raman and IR phonon
changes to optical polarizability induced as a result of
displacements. Additionally, IRRS is distinct from the ionic
strongly driven IR vibrational modes—invites exploration
Raman-scattering mechanism, which indirectly produces
as a potential tool for materials probing or control and, from an optical response through the coupling of phonons by the
anharmonic lattice potential, rather than directly through
*
guru.khalsa@cornell.edu the nonlinear lattice polarizability. IRRS has a broad and
distinct frequency response, which we find is orders of
Published by the American Physical Society under the terms of magnitude larger than ionic Raman scattering in an
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to archetypal perovskite due to its direct nature.
the author(s) and the published article’s title, journal citation, Our work is significant in two general respects. First,
and DOI. selective optical pumping of IR phonons by frequency and
2160-3308=21=11(2)=021067(16) 021067-1 Published by the American Physical Society
KHALSA, BENEDEK, and MOSES PHYS. REV. X 11, 021067 (2021)
field polarization direction can lead to control of the tensor control of specific Raman modes. (3) We employ a
components of the electric susceptibility, enabling control of perturbation method approach to derive a third-order
effective linear and nonlinear optical responses of the polarization and a Raman-scattering susceptibility that
material. This control of the linear and nonlinear optical capture the IRRS effect. These quantities may be used
response can enable enhancement or suppression of funda- to explore the intensity-dependent modification of the
mental optical properties such as absorption, birefringence, linear susceptibility tensor for a second laser as a result
and electro-optic coefficient. Such control may aid the of resonant or near-resonant IR pumping by a first laser.
development of ultrafast optical technologies including (4) We employ first-principles computational techniques
optical switches, amplitude and phase modulators, and for the perovskite SrTiO3 to investigate the physical
quantum optical logic gates. Second, we predict that the mechanism of IRRS and opportunities for ultrafast material
IRRS mechanism contributes to unidirectional displacement optical property control.
of Raman phonons and therefore changes of crystalline
symmetry, giving rise to opportunities for ultrafast control of II. SYMMETRY CONSIDERATIONS
functional properties of crystalline materials tied to sym-
metry. Thus IRRS provides a pathway to ultrafast optical A. The lattice potential
control of material symmetry and properties that is comple- We start by expanding the lattice energy for the simpler
mentary to the so-called nonlinear phononics effect of case of a centrosymmetric crystal, in which IR and Raman
phononic rectification, which has been the focus of much modes have different symmetry (in noncentrosymmetric
recent work for its ability to tune superconducting properties crystals, some phonons are both IR and Raman active; this
[18–21] and magnetic order [22–25] in perovskites by lack of distinction between IR and Raman phonons leads
intense laser pumping of an IR phonon. to additional terms in the nonlinear polarizability and
To illustrate IRRS, we develop a semiclassical perturba- anharmonic potential, which we do not consider here).
tion method approach in order to derive two-laser fre- The lattice energy U for the process we consider is then
quency contributions to the dielectric function from defined as
nonlinear contributions to the optical polarizability due
to IR phonon displacements and IR-Raman phonon cou- 1 1
U lattice ¼ K IR Q2IR þ K R Q2R
pling. We further employ first-principles computational 2 2
techniques (density-functional theory) to investigate the 2
− BQIR QR − ΔP⃗ lattice · EðtÞ;
⃗ ð1Þ
effect in SrTiO3 , an archetypal insulating perovskite [26]
that has received recent attention in nonlinear phononics where K σ ¼ Mσ ω2σ ðσ ¼ IR; RÞ is the effective spring con-
studies [27,28]. We find that the IRRS effect can be stant of the IR and Raman phonons, respectively. Qσ is the
measured using a two-laser optical experiment where real-space eigendisplacement of the phonon mode found
one laser is tuned to resonantly excite an IR-active phonon. by solving Mω2σ Qσ ¼ KQσ , where M is the mass matrix, K
In addition to resonant enhancement of Stokes and anti- the force constant matrix at zero crystal momentum, and ωσ
Stokes Raman peaks through IRRS, broad optical suscep- is the corresponding phonon frequency. In addition to the
tibility changes are predicted that extend far above and harmonic energy associated with the IR and Raman
below the IR phonon resonance. Furthermore, these phonons, we include an anharmonic potential term that
changes are polarization direction dependent and can result has been the focus of recent works on nonlinear phononics
in modification of the number of optical axes and their [16,29,30]. The last term contains the polarization change
directions. Additionally, we find that the mechanism in the crystal ΔP ⃗ due to optical excitation (E).
⃗
responsible for IRRS induces unidirectional displacement ⃗
There are both lattice and electronic contributions to ΔP.
of Raman-active phonons. IRRS therefore presents an
The lattice contribution to the polarization typically takes
alternative pathway to the nonlinear phononics mechanism
for quasistatic optical control of crystalline structure and the form Z̃ QIR , where Z̃ is the mode-effective charge
properties. induced by the displacement of the IR phonon [31].
In the following pages, we develop a theory of IRRS as Expanding ΔP ⃗ lattice to second order in phonon amplitude
follows: (1) We define a potential for describing a cen- for a centrosymmetric crystal, we find
trosymmetric crystalline lattice driven to large phonon
mode displacements by IR light including the lowest-order ⃗ lattice ¼ Z̃ QIR þ bQR QIR :
ΔP ð2Þ
nonlinear term in the lattice polarizability, which gives
rise to IRRS. (2) We summarize the dependence of the QIR induces a dipole to which an electric field can be
linear electric susceptibility tensor on Raman phonon ⃗
coupled directly. To preserve the dipole character of ΔP,
mode displacements and related symmetry considerations. QR must transform as either a monopole or a quadrupole.
The purpose of this section is to understand how basic Since the product of a dipole (QIR ) and a monopole or
optical properties and symmetries can be modified through quadrupole (QR ) includes a dipole component, it follows
021067-2
ULTRAFAST CONTROL OF MATERIAL OPTICAL PROPERTIES … PHYS. REV. X 11, 021067 (2021)
optical response of a crystalline material in the mid- or far
infrared when IR phonons are strongly excited.
B. The optical susceptibility
To lowest order in a perturbative expansion of the
polarizability in the electric field amplitude, the response
of a crystalline material to light is captured by its fre-
quency- and wave-vector-dependent linear susceptibility
χ ð1Þ ðω; kÞ, which connects the electric field of light to
ð1Þ
the induced polarization Pα ¼ χ αβ Eβ , where α and β are
FIG. 1. Contributions to ΔP ⃗ in centrosymmetric crystals. (a) In Cartesian directions. For frequencies within the electronic
an electric field, relative displacements of cations (blue) and gap of a centrosymmetric material, the susceptibility is
anions (green) contribute to the lattice (ionic) polarizability. The dominated by the frequency-dependent electronic polar-
ð1Þ
lattice polarizability is responsible for the dominant features of izability. χ αβ ðωÞ is a second-rank polar tensor, which is a
the low-frequency dielectric response in the mid- and far infrared.
function of the amplitudes of particular lattice vibrations. In
(b) Displacement of Raman phonons (shown, for example, as a ð1Þ
distortion of the octahedral oxygen environment) alters the general, χ αβ ðωÞ can be decomposed into spherically
strength, and possibly direction, of the lattice polarizability. irreducible components (we use the notation of Ref. [34]),
(c) A cloud of electrons bound to a nucleus displaces in response
to an electric field. This electronic polarizability is responsible for
ð1Þ 1 ðDÞ ðQÞ
the high-frequency dielectric response. (d) Displacement of χ αβ ðωÞ ¼ χ ðMÞ δαβ þ ϵαβγ χ γ þ χ αβ ; ð3Þ
Raman phonons alters the electronic polarizability. This is the 3
conventional Raman effect. A single atom is shown, although
collective motion of Raman phonons is necessary for this where δ is the Kronecker delta, ϵ is a Levi-Civita symbol,
effect. ðDÞ
and χ ðMÞ are the scalar (monopole), χ γ the vector (dipole),
ðQÞ
and χ αβ the deviator (quadrupole) contributions to the
susceptibility:
that bQR QIR is allowed in centrosymmetric crystals. Both
ð1Þ
terms of Eq. (2) are illustrated in Figs. 1(a) and 1(b). The χ ðMÞ ¼ χ αα ðωÞ; ð4Þ
schematic shows that the first term Z̃ QIR is produced by
relative displacements of nuclei through the IR phonon
ðDÞ 1 ð1Þ
[31]. Displacements of Raman phonons can change how χ γ0 ¼ ϵαβγ0 χ αβ ðωÞ; ð5Þ
the lattice polarizes through QIR , and this is captured 2
through the nonlinear contribution bQR QIR . The math-
ematical interpretation of this statement is that b can be ðQÞ 1 ð1Þ ð1Þ 1
χ αβ ¼ ½χ αβ ðωÞ þ χ βα ðωÞ − χ ðMÞ δαβ : ð6Þ
understood as the change in the mode-effective charge (Z̃ ) 2 3
with respect to the Raman phonon. We note that the
nonlinear lattice polarization bQR QIR is, in general, a Note, it is important to remember that while these irre-
wave-vector-dependent quantity so that bQIR QR → ducible components will always have the symmetry of
bð⃗q1 − q⃗ 2 ÞQ1 ð⃗q1 ÞQ2 ð−⃗q2 Þ, where we assume discrete the stated multipoles, the actual multipoles contributing to
translational symmetry in a crystal [32]. Since light cannot each can and usually are of higher order in the reduced
impart significant momentum on IR phonons, q⃗ 1 ≈ 0. ⃗ It (compared with free space) symmetry of a material’s crystal
follows that q⃗ 2 ¼ 0. ⃗ field [35].
Previous experimental and theoretical works on the In this work, we are considering systems that have
nonlinear phononics mechanism have focused on the ground states that are centrosymmetric and time-reversal
anharmonic lattice potential (BQ2IR QR ) and not nonlinear invariant (that is, nonmagnetic and/or in zero-applied
changes to the polarizability (bQR QIR ) [16,29,30]. In one magnetic field). In such cases, Onsager’s reciprocity
ð1Þ ð1Þ
recent exception, a nonlinear contribution to the polar- relation dictates that χ αβ ðωÞ ¼ χ βα ðωÞ. It follows that
ðDÞ
izability ΔP ⃗ NL ∝ Q3IR was included in order to describe the the antisymmetric tensor χ γ0 vanishes by symmetry. In
ð1Þ
optical changes in the reststrahlen band of SiC when IR that case, the susceptibility is totally symmetric χ αβ ðωÞ ¼
phonons were excited to large amplitude [33]. As we show ðQÞ
below, including the lowest-order nonlinear contribution to χ ðMÞ 13 δαβ þ χ αβ , and the 1 þ 5 independent components of
ð1Þ
the polarizability [Eq. (2)] is critical to understanding the χ αβ ðωÞ are written explicitly as
021067-3KHALSA, BENEDEK, and MOSES PHYS. REV. X 11, 021067 (2021)
0 1
1 ðMÞ ðQÞ ðQÞ ðQÞ Γ ¼ Γþ þ þ þ
1 ðA1g Þ þ Γ2 ðB1g Þ þ 2Γ4 ðB2g Þ þ 3Γ5 ðEg Þ
B 3χ þ χ xx χ xy χ xz
C
B C þ 4Γ−3 ðA2u Þ þ 6Γ−5 ðEu Þ þ 2Γþ
3 ðA2g Þ
B ðQÞ 1 ðMÞ ðQÞ ðQÞ C
B χ xy 3χ þ χ yy χ yz C:
B C þ Γ−1 ðA1u Þ þ Γ−2 ðB1u Þ;
@ A
ðQÞ ðQÞ 1 ðMÞ ðQÞ
χ xz χ yz 3χ þ χ zz
where the first two lines describe the symmetries of the
ð7Þ Raman-active phonons, the third line shows the symmetries
of the IR-active phonons (which include acoustic modes),
The crystallographic symmetry of a particular material and the fourth and fifth lines are, respectively, even- and
imposes additional constraints that dictate which elements odd-symmetry silent phonons. The Γþ 1 ðA1g Þ phonon is the
ð1Þ only monopolar Raman phonon. Displacing this phonon
are zero or nonzero and how many components of χ αβ ðωÞ
preserves the crystal symmetry and the number of optical
are independent. ð1Þ
constants. However, the relative size of each term in χ 0
may change, as we explain above. The remaining Raman-
C. The Raman tensor
active phonons induce quadrupoles, and therefore, new
Now considering the Raman effect, we express the optical constants. As an example, consider the two Γþ 4 ðB2g Þ
polarizability as a function of the amplitude Qi of the Raman phonons, which transform like xy. Displacing either
lattice displacements associated with a particular phonon one of these quadrupolar phonons introduces new optical
mode, constants χ xy ¼ χ yx , which induces, for example, a crystal
polarization field along the y direction when the crystal is
ð1Þ ∂χ ð1Þ illuminated with light polarized along the x direction—a
Pα ≈ χ 0;αβ ðωÞ þ Qi E; ð8Þ
∂Qi Qi →0 property that is forbidden in the equilibrium structure.
ð1Þ
where χ 0;αβ ðωÞ is the susceptibility of the undistorted III. THEORETICAL MODEL
crystal, and the second term describes the Raman effect. Having discussed symmetry principles that govern the
Note that Qi ð∂χ ð1Þ =∂Qi ÞjQi →0 is constrained by the same infrared-resonant Raman effect in crystals, we now present
ð1Þ a theoretical model that can be used to describe it. While the
symmetry as χ 0;αβ ðωÞ. This constraint results in two distinct
infrared-resonant Raman effect is expected to be present in
situations: (1) Qi transforms as a monopole, and therefore,
small-band-gap materials and metals, we focus our dis-
ð∂χ ð1Þ =∂Qi ÞjQi →0 has the same nonzero elements as cussion and modeling on wide-band-gap insulators in order
ð1Þ
χ 0;αβ ðωÞ. In this case, the modulation of the nonlinear to distill the physical implications of the infrared-resonant
susceptibility can affect only the relative size of each term Raman effect. This approach avoids obfuscation of the
ð1Þ infrared-resonant Raman effect from other considerations
in χ 0 , but not induce new components. (2) Qi transforms
in metals (e.g., free-charge screening) and semiconductors
as a quadrupole, and therefore, ð∂χ ð1Þ =∂Qi ÞjQi →0 has new (e.g., Zener breakdown [37], avalanche ionization [38]).
ð1Þ
nonzero elements when compared to χ 0;αβ ðωÞ. In this case,
new optical constants are induced by displacing Qi . A. Equations of motion
In both of these cases, Qi is called a Raman-active In order to derive the IRRS susceptibility, we first derive
phonon, and hence, we refer to the process described the equations of motion from Eqs. (1) and (2) in order to
above as the infrared-resonant Raman effect. In terms of identify Raman-scattering pathways. To capture the high-
Eq. (3), displacing a monopolar Raman phonon alters frequency response, we add the harmonic energy of an
ðMÞ ðQÞ
nonzero elements of χ αβ and χ αβ that already exist in effective electronic coordinate Qe (a phenomenological
the equilibrium structure. A quadrupolar Raman phonon parameter that describes the high-frequency dielectric
lowers the symmetry of the crystal when displaced. As a response) and its coupling to the electric field and define
result, the new induced optical constants—constrained by a total potential U ¼ U e þ Ulattice , where
ðQÞ
symmetry to be zero at equilibrium—are described by χ αβ . 1
We use SrTiO3 as a test material in this study. At ⃗ e · E:
Ue ¼ K e Q2e − ΔP ⃗ ð9Þ
2
temperatures below 105 K, SrTiO3 is a tetragonal crystal
[36] with space-group symmetry I4=mcm (#140—D4h K e is an effective spring constant, and the electronic
point group). SrTiO3 therefore has only two distinct non- polarizability includes coupling to the Raman phonon
ð1Þ ð1Þ ð1Þ
zero optical constants, χ 0;xx ¼ χ 0;yy ≠ χ 0;zz . The primitive through
unit cell contains ten atoms and therefore has 30 phonons at
⃗ e ¼ ζQe þ βQR Qe :
ΔP ð10Þ
the Γ point that transform as
021067-4ULTRAFAST CONTROL OF MATERIAL OPTICAL PROPERTIES … PHYS. REV. X 11, 021067 (2021)
Here, ζ is an effective charge that describes the high- the anharmonic lattice potential, and coupling through the
frequency optical response of the crystal. The coefficient β nonlinear polarizability.
describes the conventional electronically mediated Raman Numerous scattering pathways present themselves
effect. Figures 1(c) and 1(d) show schematic representa- through Eq. (11) as illustrated in Fig. 2.
tions of linear and nonlinear electronic polarizability terms (i) Conventional (electronic) Raman scattering. The
in Eq. (10). Incidentally, comparing Eqs. (2) and (10), we conventional Raman scattering mechanism (top
can see that since both QIR and Qe transform as vectors, panel of Fig. 2) is described by the β coefficient.
they both couple to Raman phonons in the same way. It The effective electronic coordinate Qe can respond
follows that all of the symmetry arguments presented above to any frequency of light at or below its fundamental
are true for both the conventional electronically mediated frequency ωe. The combined motion of Qe with the
Raman effect and the infrared-resonant Raman effect. electric field induces a driving force for the Raman
Considering Eqs. (1), (2), (9), and (10), and taking coordinate QR through the parameter β [first term on
derivatives of U with respect to Qe , QIR , and QR , we find the right-hand side of Eq. (11c)].
the following equations of motion: (ii) Infrared resonant Raman scattering. The IRRS
mechanism (middle panel of Fig. 2) is analogous
Le Qe ¼ ζE þ βQR E; ð11aÞ to the conventional Raman-scattering pathway, ex-
cept it substitutes the displacement of the electron
LIR QIR ¼ Z̃ E þ bQR E þ 2BQIR QR ; ð11bÞ cloud characterized by Qe, with the displacement of
an IR phonon QIR . Displacement of an IR phonon in
LR QR ¼ βQe E þ bQIR E þ BQ2IR : ð11cÞ the presence of an electric field induces displace-
ments of Raman phonons through the nonlinear
We define the linear operator Lσ ¼ Mσ ½ðd2 =dt2 Þ þ polarizability described by the coefficient b. The
2γ σ ðd=dtÞ þ ω2σ for notational convenience. We include largest IR phonon response is when a component of
a damping parameter γ σ and note that Mσ is a reduced mass the applied field is tuned to the frequency ωIR. Once
coordinate. The right-hand side of Eq. (11) describes the resonantly excited, the combined motion of QIR and
driving terms through the electric field, coupling through the electric field drive QR through b.
FIG. 2. Comparison between the conventional (top), infrared-resonant (middle), and ionic Raman-scattering (bottom) mechanisms.
Blue, green, and orange panels represent the effect of the three electric field components, while the induced polarization is shown in the
red panels. Driving terms in Eq. (11) are shown in ovals with induced motion represented in rectangles with the allowed frequency
response written below. The connection to linear and nonlinear polarization, and the anharmonic potential are shown in the text in each
panel with resonance conditions for each effect called out in the polarization panels. Dashed arrows leading into the light-blue region in
the middle of the figure show that cross-coupling is allowed between all three effects. The conventional and infrared-resonant Raman
responses have potential for third-order resonance in the polarization [see Eqs. (13) and (A7)–(A10)], while ionic Raman scattering has
potential for a fifth-order resonance [see Eqs. (A7)–(A10)].
021067-5KHALSA, BENEDEK, and MOSES PHYS. REV. X 11, 021067 (2021)
(iii) Ionic Raman scattering. An additional Raman- To simplify the presentation, we focus on a scenario
scattering mechanism (bottom panel of Fig. 2) is where one laser is used to drive an IR phonon on, or
provided by the anharmonic lattice potential. After near, resonance, while a second laser samples the optical
resonant excitation of QIR , there is a driving force on response at another frequency taken to vary over a large
QR induced by the parameter B proportional to Q2IR . range. This scenario can be viewed as intensity-dependent
This scattering pathway is an example of nonlinear dielectric changes due to the excitation of an IR phonon.
ionic source terms in the dielectric response arising The electric field is taken to be
from the anharmonic lattice potential [39] and has
been described in the literature as ionic Raman 1
EðtÞ ¼ ðE1 e−iω1 t þ E−1 eiω1 t þ E2 e−iω2 t þ E−2 eiω2 t Þ
scattering [10–12,40]. It should be noted that this 2
pathway is distinct from the other two in that the ð12Þ
applied field directly induces only a displacement to
an IR phonon; i.e., the presence of an electric field is with the field strength defined with the constraint E−n ¼ En
not required to couple IR phonon motion to Raman to enforce real fields. Notice that with this convention,
phonon motion. It should also be noted that this we can take ω−n ¼ −ωn and definePthe general multi-
pathway prevents fully resonant excitation of the component electric field as EðtÞ ¼ 12 n¼1;2;… En e−iωn t
Raman phonon except for the singular case where
by summing over both negative and positive components of
the frequency of the applied electric field is half the
the field. The third-order susceptibility for the process
frequency of the Raman phonon. In contrast, in
conventional Raman scattering and in IRRS, reso- under investigation χ ð3Þ ðω2 ;ω1 ;−ω1 ;ω2 Þ ≡ χ ð3Þ ð2;1;−1;2Þ
nant Raman excitation may be achieved through the describes the macroscopic nonlinear polarizability at ω2
combination of any two laser frequencies such due to the mixing of light at ω1 (taken to be near the IR
that ω1 ω2 ≈ ωR . resonance) and ω2 . To simplify χ ð3Þ further, we assume
that all laser frequencies are far below the electronic
B. Perturbative approach to the resonance ωe , but include general expressions for the
nonlinear susceptibility polarizability in the Appendix [see Eq. (A7)]. The non-
We derive expressions for the optical susceptibility using linear polarization of the crystal is then defined by Pð3Þ ¼
a perturbative approach in the parameters β, b, and B and χ ð3Þ ð2; 1; −1; 2ÞjE1 j2 E2 e−iω2 t þ c:c: ≡ δχ ð1Þ ðI 1 ÞE2 e−iω2 t þ
solve Eq. (11) in the presence of a multicomponent electric c:c: We see that the polarization is measured at the
field, expressing Qσ in terms of components of E. These frequency ω2 and can be interpreted as an intensity-
solutions are used to remove Qe , QIR , and QR from Eqs. (2) dependent change to the linear susceptibility
and (10) and express the polarizability in terms of only the δχ ð1Þ ðω2 ; I 1 Þ ∝ I 1 , where the intensity I 1 ¼ jE1 j2. With
electric field so that the linear and nonlinear contributions this definition, the effective linear susceptibility at ω2 takes
ð1Þ
to the polarizability can be collected. By symmetry, we can the form χ eff ðω2 ; I 1 Þ ¼ χ ð1Þ ðω2 Þ þ δχ ð1Þ ðω2 ; I 1 Þ, which
see that the dipole active coordinates Qe and QIR will be reflects that the linear optical properties for E2 are altered
replaced by a parallel electric field E, while each Raman by the presence of a strong E1 .
phonon will be replaced by two factors of the electric field. Even with these assumptions, the perturbative approach
From Eqs. (2) and (10), we see that Z̃ QIR and ζQe will leaves us with cumbersome expressions. Focusing on the
contribute to first order in E, while bQR QIR and βQR Qe component of the third-order susceptibility due solely to the
will contribute terms to third order in E. In this way, we find nonlinear contribution to the polarizability with coefficient
that the polarizability includes both linear and third-order b (corresponding to the IRRS effect introduced here; see
susceptibilities P⃗ ¼ χ ð1Þ E
⃗ þ χ ð3Þ E
⃗ E
⃗ E.
⃗ middle pathway in Fig. 2), we define and find
ð3Þ ðbZ̃Þ2
χ b;b ð2; 1; −1; 2Þ ¼ 1 þ G−1 þ G2 þ G−2 ÞG2 G ðω1 − ω1 ¼ 0Þ
× ½2ðGIR IR IR IR IR R
ϵ0 V
þ ðGIR
2 G2 þ G1 G2 þ G2 G−1 þ G1 G−1 ÞG ð−ω1 þ ω2 Þ
IR IR IR IR IR IR IR R
2 G2 þ G1 G2 þ G2 G−1 þ G1 G−1 ÞG ðω1 þ ω2 Þ:
þ ðGIR IR IR IR IR IR IR IR R ð13Þ
In this expression, Gσn ¼ Gσ ðωn Þ is the inverse of the linear operator Lσ in the frequency domain [see Eq. (11)] evaluated
at ωn . That is,
021067-6ULTRAFAST CONTROL OF MATERIAL OPTICAL PROPERTIES … PHYS. REV. X 11, 021067 (2021)
1=MIR ð3Þ
seen in the magnitude of χ b;b at the difference and sum
n ¼ G ðωn Þ ¼
GIR ð14aÞ
IR
;
ðω2IR − ω2n Þ − 2iγ IR ωn frequencies of 8 and 32 THz, respectively. While the
magnitude remains constant below 8 THz, it falls off as
1=MR f −2 for high frequency (above 32 THz). At the difference
GRmn ¼ GR ðωmn Þ ¼ ; ð14bÞ
ðω2R − ω2mn Þ
− 2iγ R ωmn frequency, the phase of χ ð3Þ approaches −π=2, suggesting
ð3Þ
χ b;b is purely imaginary and that there will be emission of
where ωmn ¼ ωm þ ωn . Equation (13) is organized by the
radiation at 8 THz. At the sum frequency, the phase
frequency response of the Raman phonon. The right-hand
approaches þπ=2, suggesting absorption of radiation at
side of the first line shows the contribution from static
32 THz. This is consistent with the classical theory of
displacement of the Raman phonon through rectification of
Stokes and anti-Stokes Raman scattering. We therefore find
the electric field components (ωmn ¼ 0, remembering that
that the nonlinear contribution to the polarization shown in
ωnðmÞ is summed over positive and negative frequencies).
Eq. (2) resonantly enhances the Stokes and anti-Stokes
The remaining lines are due to dynamical responses of the response known from the conventional Raman effect
Raman phonon. The second line gives the contribution through excitation of an IR phonon.
from the motion of the Raman phonon at −ω1 þ ω2 . In between the difference and sum frequencies, the phase
Since the function GR is peaked at the Raman frequency, of the response is zero, indicating that the real part of the
the second line of Eq. (13) is resonant when ð3Þ
ð−ω1 þ ω2 Þ ¼ ωR , while the third line is resonant when dielectric constant will increase through χ b;b . Below the
ðω1 þ ω2 Þ ¼ ωR . difference frequency and above the sum frequency,
ð3Þ
Each term in Eq. (13) includes two contributions the phase of χ b;b becomes ∓ π, suggesting a decrease in
from GIR and one from GR . This corresponds to a double the real part of the dielectric constant in these frequency
resonance due to the IR phonon response and a single ranges. These effects are also consistent with conventional
resonance from the Raman response. An overall triply Raman scattering.
resonant response is therefore possible in the two-laser
scenario we are describing. As an example, consider a first IV. FIRST-PRINCIPLES CALCULATIONS
laser resonant with the IR phonon (ω1 ¼ ωIR ) and a second
laser tuned to a frequency ω2 away from the IR resonance The model described in the previous section requires
so that −ω1 þ ω2 ¼ ωR , as is expected with a Stokes various inputs, which we obtain from first-principles
(−ωR ) or anti-Stokes Raman (þωR ) response. We note that calculations. Density-functional-theory calculations are
this triply resonant response is analogous to conventional performed using projector-augmented-wave potentials
resonant stimulated Raman scattering, wherein the first and the local density approximation (LDA), as imple-
laser is tuned to an electronic resonance (ω1 ¼ ωe ). mented in VASP. We use a 600-eV plane-wave cutoff pffiffiffiand
However, these responses are found in disparate ranges a ffiffi6ffi × 6 × 4 Monkhorst-Pack k-point grid for a 2a ×
p
of the optical spectrum. 2a × 2a supercell, where a is the optimized lattice
Figure 3 shows the magnitude and phase of constant of SrTiO3 in the cubic Pm3̄m phase. The
χ ð3Þ ð2; 1; −1; 2Þ schematically when coupling between a calculated lattice constants a ¼ b ¼ 5.44 Å and c ¼
single IR phonon at f IR ¼ ωIR =2π ¼ 20 THz, and a single 7.75 Å underestimate the experimental [41] lattice con-
Raman phonon at f R ¼ ωR =2π ¼ 12 THz is allowed stants of a ¼ b ¼ 5.51 Å and c ¼ 7.81 Å as expected in
through the nonlinear dipole moment. The curves indicate LDA. Phonons and Born effective charges are calculated
ð3Þ
the contribution from χ b;b , i.e., from IRRS. Two peaks are using density-functional-perturbation theory within VASP
and used to calculate dynamical-mode-effective charges.
Further details are given in Ref. [42].
For the anharmonic potential terms B, we compute
the total energy U lattice on a mesh of IR and Raman
phonon amplitudes up to 20 pm in 2-pm steps and fit
to a symmetry-constrained polynomial. B is then inter-
preted as the derivative of the total energy fit with respect
to the real-space eigendisplacements of each phonon
B ¼ ð∂ 3 U=∂Q2IR ∂QR Þ. Similarly, the coefficient control-
FIG. 3. Magnitude (black) of the third-order IRRS susceptibil-
ð3Þ
ity χ b;b resulting from bQIR QR for an IR phonon at 20 THz with ling the strength of the nonlinear contribution to the lattice
effective charge Z̃ ¼ 1e coupled to a Raman phonon at 12 THz polarization b is evaluated by taking derivatives of a
with a coupling strength b ¼ 1e=Å. The sum and difference symmetry-constrained-polynomial fit to the polarization
frequencies are 32 and 8 THz, respectively. The phase (red) is b ¼ ð∂ΔP=∂Q ⃗ IR ∂QR Þ calculated using the modern theory
referenced to the right vertical axis. of polarization [43].
021067-7KHALSA, BENEDEK, and MOSES PHYS. REV. X 11, 021067 (2021)
The damping parameters γ σ are temperature-dependent
quantities. Because they require evaluation of, minimally,
all third-order force constants, and often all electron-
phonon coupling pathways and defect scattering pathways,
their calculation from first principles is prohibitively
expensive. Instead of calculating damping terms from first
principles, we explore a range of values 0.1–1.0 THz
consistent with typical damping parameters for phonons.
In what follows, we fix damping parameters to 0.1 THz
corresponding to 10-ps lifetimes and comment on any
strong dependence of the optical response on this value
when necessary.
V. RESULTS
We employ the theoretical model of Sec. III together
with the first-principles calculations of Sec. IV to inves-
tigate the changes to optical material properties that can be FIG. 4. Simulated magnitude (black) and phase (red) of the
ð3Þ
third-order susceptibility due to IRRS χ b;b in SrTiO3 when
induced in SrTiO3 through IRRS. Table I summarizes the −
the highest-frequency Γ5 ðEu Þ IR phonon is resonantly excited
first-principles parameters for SrTiO3. As we are most ð3Þ
interested in the changes to optical constants and optical (f IR ¼ 16.40 THz). (a) χ xxxx ð2; 1; −1; 2Þ exhibits coupling to the
symmetry that arise from resonant or near-resonant IR Γþ þ
1 ðA1g Þ and Γ2 ðB1g Þ Raman phonons. The difference and sum
pumping, we explore cubic susceptibility terms, in the frequencies that identify peaks are 12.18 and 20.62 THz for the
Γþ1 ðA1g Þ phonon, and 1.19 and 31.61 THz for the Γ2 ðB1g Þ
þ
form of the intensity-dependent change to the linear ð3Þ
susceptibility, for an applied field E⃗ 2 due to the applica- phonon. (b) χ yxxy ð2; 1; −1; 2Þ exhibits coupling to the two
Γþ
4 ðB2g Þ phonons. The difference frequencies are 3.38 and
tion of an infrared-resonant field E ⃗ 1 , δχ ð1Þ
ij ½ω2 ; I 1 ðω1 Þ ¼ 11.88 THz, while the sum frequencies are 29.42 and
ð3Þ
χ ijji ð2; 1; −1; 2ÞjE1;j j2 , as defined in Sec. III B. The 20.92 THz. Also shown (dashed), the corresponding magnitude
of the third-order susceptibility due to the anharmonic lattice
symmetry of χ ð3Þ for the infrared-resonant Raman ð3Þ
response depends on the symmetry of the displaced potential (ionic Raman scattering) χ B;B .
Raman phonons as we discuss in Sec. II. The frequency
dependence of the tensor components of χ ð3Þ , as is seen phonons of SrTiO3 . Through these IR-Raman couplings
below, has the expected form of a Raman scattering mediated by IRRS, intensity-dependent changes to the
susceptibility as seen in Fig. 3, with Stokes and anti- effective linear susceptibility for a second field E ⃗ 2
Stokes peaks corresponding to differences and sums of the ⃗ ð1Þ
occur for E2 fields polarized along x, δχ xx ðω2 ; I 1 Þ ¼
IR and Raman phonons of the material, respectively. The ð3Þ ð1Þ
response, however, is complicated by the presence of χ xxxx ð2; 1; −1; 2ÞjE1;x j2 and along y, δχ yy ðω2 ; I 1 Þ ¼
ð3Þ
multiple IR-Raman phonon couplings for each activated χ yxxy ð2; 1; −1; 2ÞjE1;x j2 .
IR phonon and the different χ ð3Þ symmetries correspond- We first consider the case where both E ⃗ 1 and E
⃗ 2 are
ing to each Raman phonon. As a result, the intensity- polarized along the x axis. The Raman phonons that
dependent dielectric response through the excitation of the contribute to the optical response in this case [via
IR phonons [see Eq. (13) and discussion] and the resulting ð3Þ
χ xxxx ð2; 1; −1; 2Þ] are the Γþ 1 ðA1g Þ mode, which is fully
changes to the optical symmetry are intimately tied to the þ
frequency range spanned in the experiment and the symmetric, and the Γ2 ðB1g Þ, which transforms as x2 -y2
polarizations of the applied fields. [Fig. 4(a)]. The resonance peaks are then the sums (anti-
Stokes peaks) and differences (Stokes peaks) of the IR
resonance and the Γþ þ
1 ðA1g Þ and Γ2 ðB1g Þ Raman phonons.
A. Optical symmetry breaking We see peaks at 12.18 and 20.62 THz through coupling
Figure 4 shows the simulated third-order susceptibility to Γþ 1 ðA1g Þ, and 1.19 and 31.61 THz through coupling
þ
⃗ 1 is polarized along the
χ ð3Þ ð2; 1; −1; 2Þ for SrTiO3 when E to Γ2 ðB1g Þ Raman phonons. The phase of the nonlinear
x direction (taken to be parallel to the crystallographic susceptibility is also complicated, showing multiple
a axis) and resonant with the Γ−5 ðEu Þ IR phonon at absorption and emission pathways. However, there are
f IR ¼ 16.40 THz. The nonlinear susceptibility is notably broad spectral regions where the IRRS susceptibility
more complex than that of the simplified response in Fig. 3. remains purely real [that is, when argðχ ð3Þ Þ ¼ f0; πg]
This is due to the coupling to multiple Raman-active in ranges of frequencies not near resonances.
021067-8ULTRAFAST CONTROL OF MATERIAL OPTICAL PROPERTIES … PHYS. REV. X 11, 021067 (2021)
The intensity-dependent changes to the effective linear The splitting and shift of the optical axes due to the
susceptibility induced by the Γþ 1 ðA1g Þ Raman phonon IRRS response in SrTiO3 through excitation of the Γ−5 ðEu Þ
preserve the uniaxial optical symmetry. That is, IR-active phonon at 16.40 THz and the Γþ 4 ðB2g Þ Raman
ð1Þ ð1Þ ð1Þ
δχ xx ðω2 ; I 1 Þ ¼ δχ yy ðω2 ; I 1 Þ ≠ δχ zz ðω2 ; I 1 Þ, a result of phonon at 13.02 THz—both of which primarily involve
ð3Þ ð3Þ ð3Þ motion of oxygen atoms—is shown schematically in Fig. 5.
χ xxxx ð2; 1; −1; 2ÞjE1;x j2 ¼ χ yyyy ð2; 1; −1; 2ÞjE1;y j2 ≠ χ zzzz ð2;
Here the Cartesian coordinates fx; y; zg are taken to be
1; −1; 2ÞjE1;z j2 . However, the response of the Γþ 2 ðB1g Þ parallel to the crystallographic axes fa; b; cg, and the
Raman phonon alters the linear susceptibility so that amplitudes of distortions are exaggerated for viewing.
ð1Þ ð1Þ
δχ xx ðω2 ; I 1 Þ ≠ δχ yy ðω2 ; I 1 Þ, which is the property of a For light incident along the z axis and polarized in the
ð3Þ
biaxial optical symmetry, since χ xxxx ð2; 1; −1; 2ÞjE1;x j2 ≠ x-y plane, only Γ−5 ðEu Þ IR-active phonons can be excited
ð3Þ directly. For E ⃗ 1 kx;
b the induced polarization P ⃗ through
χ yyyy ð2; 1; −1; 2ÞjE1;y j2 . Since both Raman modes are −
driven by the IR mode simultaneously through IRRS excitation of the Γ5 ðEu Þ IR phonon is along the x axis
and thus their effects on the optical response cannot be [top panel of Figs. 5(b) and 5(c)]. Considering only the
disentangled, the net result of resonant pumping of the linear susceptibility, SrTiO3 remains uniaxial [top panel of
Γ−5 ðEu Þ IR phonon is a shift from uniaxial to biaxial Fig. 5(d)]. Through IRRS measured by a second field E ⃗ 2 kŷ,
þ
symmetry, with the angle that splits the two optical axes motion of the Γ4 ðB2g Þ is also induced [middle panel
controlled by I 1. of Figs. 5(b) and 5(c)] but on its own does not induce a
If we instead consider the case where E ⃗ 2 is polarized dipole and therefore cannot affect the polarization direc-
−
along the y axis [and the Γ5 ðEu Þ IR phonon at f IR ¼ tion. The combined response of these IR and Raman
⃗ 1 polarized along the x phonons [bottom panel of Figs. 5(b) and 5(c)] causes
16.40 THz is still pumped by an E
the frequency-dependent canting of the polarization
axis], we now observe a cross-polarization effect. The two
which is seen in the induced off-diagonal terms in the
Γþ4 ðB2g Þ Raman phonons at 13.02 and 4.52 THz now effective linear susceptibility, as we describe above. The
ð3Þ
facilitate the response through χ yxxy ð2; 1; −1; 2Þ [Fig. 4(b)]. biaxial optical axes induced by the IRRS are shown in the
Again, complicated amplitude peaks and phase responses bottom panel of Fig. 5(d), where the off-diagonal compo-
are seen with difference frequencies (Stokes peaks) nents of the effective linear susceptibility have now forced
and sum frequencies (anti-Stokes peaks) of 3.38 and the direction of the in-plane principle axes (shown as
11.88 THz, and 29.42 and 20.92 THz, respectively. fx0 ; y0 ; z0 g).
Interpreting this response from the perspective of the We note that the canting of the polarization in SrTiO3 is
intensity-dependent linear susceptibility, we find that not directly due to the displacement of ions through the
ð1Þ ð1Þ motion of the IR and Raman phonons. Rather, the com-
δχ xy ðω2 ; I 1 Þ ¼ δχ yx ðω2 ; I 1 Þ are nonzero. That is, an opti-
cal constant not allowed by the symmetry of the equilib- bined motion of the IR and Raman phonons lowers the
rium structure appears and becomes intensity dependent. symmetry of the crystal; this induces changes to the
This implies an intensity-dependent shift to the principal electronic structure of the occupied states, which conse-
optical axes. quently cants the polarization. Although motion of the ions
Figure 4 also shows that the contribution to does not contribute directly to the canting of the polariza-
χ ð3Þ ð2; 1; −1; 2Þ coming purely from the anharmonic lattice tion in SrTiO3 , it is unclear how general this is in arbitrary
potential (blue dashed line in Fig. 4) is many orders of crystalline systems.
magnitude smaller than the contribution due to IRRS in
SrTiO3 . This could have been anticipated by the definition B. Giant refractive index shifts
of the induced polarization [Eqs. (2) and (10)] and the When E⃗ 1 is instead resonant with the lowest-frequency
equations of motion Eq. (11) along with the perturbation Γ−5 ðEu ÞIR-active phonon at 2.67 THz, χ ð3Þ can be remark-
approach shown schematically in the bottom panel of Fig. 2
ably large—approaching unity at resonance peaks when
and developed in Sec. III B. In the absence of b and β, the jE1 j approaches 1 MV=cm. That is, the shift in the effective
only possible changes to the polarization are through linear susceptibility for the infrared-resonant Raman
changes in QIR . While the anharmonic lattice potential response in SrTiO3 can be comparable to the typical linear
induces a first-order change in the Raman coordinate QR , it
dielectric response of many optical materials (χ ð1Þ ≈ 1–10).
induces only a second-order change in QIR ; thus, we should ð3Þ
expect the pure anharmonic lattice contribution to the third- Figure 6 shows χ zxxx ð2; 1; −1; 2Þ for this case. Here, the
þ
order susceptibility to be small. An expression for the Γ5 ðEg Þ Raman phonons, which have fyz; zxg symmetry,
changes to χ ð3Þ induced by the anharmonic lattice potential are activated in the optical process, so the optical changes
and all possible cross-coupling pathways for χ ð3Þ up to are again along a perpendicular axis (that is, E⃗ 2 is polarized
second order in perturbation theory are given in the along the z axis). The 1.25- and 4.35-THz Γþ 5 ðEg Þ phonons
Appendix [see Eqs. (A9) and (A10)]. dominate the IRRS process, while the 13.09-THz Γþ 5 ðEg Þ
021067-9KHALSA, BENEDEK, and MOSES PHYS. REV. X 11, 021067 (2021)
FIG. 5. Activation of IR and Raman phonons, and symmetry breaking in SrTiO3 through IRRS. (a) Equilibrium crystal structure of
SrTiO3 from two perspectives. (b) Schematic representation of the IR, Raman, and combined IR and Raman phonons. The Γ−5 IR
phonon at 16.40 THz and the Γþ 4 Raman phonon at 13.02 THz are shown here. (c) Distorted structure when IR, Raman, and the
combined IR and Raman phonons are included. The induced polarization is canted by the activation of the Γ−5 IR phonon polarized in the
x-y plane and the Γþ 4 Raman phonon that transforms like xy. (d) Away from equilibrium, the uniaxial nature of SrTiO3 is altered—
splitting and tilting the optical axes. Here, x0 ; y0 ; z0 are the principle axes of the distorted crystal with the optical axes canted from z0 in the
z0 -x0 plane.
is not significantly altering the optical response because ð1Þ ð1Þ
δχ yz ðω2 ; I 1 Þ ≠ 0 and δχ zx ðω2 ; I 1 Þ ≠ 0. Focusing on the
of its weak coupling to the 2.57-THz IR phonon. response above 10 THz, we again note a phase of π=2 and a
Difference-frequency (Stokes) and sum-frequency (anti- falloff with frequency proportional to f −2 showing that the
Stokes) responses are clearly seen in the amplitude and large intensity-dependent changes to the real parts of the
phase of χ ð3Þ at 1.68 and 1.42 THz, and 7.02 and 3.92 THz,
respectively. Interpreting the nonlinear optical changes
as intensity-dependent susceptibility changes suggests TABLE I. Coupling coefficients for SrTiO3 used in Figs. 4 and
6. The reduced mass and effective charges for the high- and low-
frequency Γ−5 ðEu Þ IR phonons are M ¼ 16.01μ and Z̃ ¼ 6.92e,
and M ¼ 20.12μ and Z̃ ¼ 14.97e, respectively.
Γ−5 ðEu Þ IR phonon at 16.40 THz
Symmetry fðTHzÞ Mðmu Þ bðe=ÅÞ
Γþ
1 ðA1g Þ: 1 4.22 16.00 0.33
Γþ 2
2 ðB1g Þ: x − y
2 15.21 16.00 −0.35
þ −0.68
Γ4 ðB2g Þ: xy 13.02 16.02
4.52 86.95 −0.28
FIG. 6. Simulated magnitude (black) and phase (red) of the
ð3Þ
third-order susceptibility due to IRRS χ b;b in SrTiO3 when the
−
lowest-frequency Γ5 ðEu Þ IR phonon is resonantly excited Γ−5 ðEu Þ IR phonon at 2.67 THz
ð3Þ
(fIR ¼ 2.67 THz). χ zyxx ð2; 1; −1; 2Þ is now facilitated by the Symmetry fðTHzÞ Mðmu Þ bðe=ÅÞ
þ
Γ5 ðEg Þ Raman phonons. The difference and sum frequencies are
Γþ
5 ðEg Þ: fyz; zxg 13.09 16.03 −0.03
now 1.68 and 1.42 THz, and 7.02 and 3.92 THz, respectively. The
4.35 80.05 −1.33
coupling to the 13.09-THz Raman phonon is weak and does not
1.25 16.25 0.26
alter the response significantly.
021067-10ULTRAFAST CONTROL OF MATERIAL OPTICAL PROPERTIES … PHYS. REV. X 11, 021067 (2021)
ð1Þ ð1Þ constants on Raman phonon displacements. Since some
effective χ yz and χ zx extend far above the region where
optical phonons are present. Raman phonons in noncentrosymmetric media have the
same symmetry as IR-active phonons, evaluation of IRRS-
induced shifts of χ ð2Þ is considerably more involved.
VI. DISCUSSION
However, such media present a complex response to IR-
The findings of Secs. III and V suggest a broad relevance driven light that might be used to simultaneously tailor
to the fields of optics and materials physics. linear, electro-optic, and higher-order constants through the
In optics, IRRS offers a previously unidentified route to application of IR-resonant light fields. For example, in
strong control of optical material properties. In the analysis AlN, the Γ1 ðA1 Þ optical phonon is simultaneously IR and
of SrTiO3 in Sec. V, large, broad-frequency changes to the Raman active, and invariant under all symmetry operations.
dielectric response are observed in the form of new tensor Extending the discussion of Sec. II C, every optical con-
elements—in one case reflecting an intensity-dependent stant, of every order, will be proportional to the displace-
shift from uniaxial to biaxial symmetry—as well as the ment of this phonon while also directly excitable.
induction of strong absorption and emission peaks, and We anticipate that the potential utility provided by IRRS
large refractive-index shifts. Extending these ideas to through the effects seen in Sec. V for optical technologies
include circularly polarized fields, we might expect will rely on three considerations: the applicability of having
IRRS to give strong control of chiral optical response as strong absorption of a control pulse over a short length
well. Such large effects are possible via a triply resonant scale, the possible utility of achieving relative shifts in
Raman-scattering susceptibility through the coupling of IR propagation constants with field polarization, and the
and Raman phonons mediated by nonlinear contributions degree to which the use of multiple laser frequencies is
to the polarizability, as shown in Eq. (2). practical. Through the application of propagating or stand-
Whereas it is akin to the conventional electronically ing-wave infrared light fields, one might imagine signifi-
mediated resonant Raman response, which couples Raman cant applications on the scale of integrated optical devices,
phonons to virtual electronic dipole transitions through the e.g., in which side pumping of a waveguide is used to
nonlinear polarizability, IRRS is relevant over a vastly induce effects such as light-induced wave retardation,
different frequency range relevant to the emergent fields of polarization ellipse rotation, Bragg reflection, etc.
midinfrared and THz optics. The theoretical approach we We note that the optical effects described in this article
use is quite general and can be used to analyze IRRS in a offer a route to control of optical symmetry akin to that
wide variety of optical materials, including other complex provided by the Pockels effect (linear electro-optic effect),
oxides and III-V semiconductors. The main features of a but by a very different physical mechanism. The Pockels
material allowing strong IRRS response are large nonlinear effect induces an optical-field-dependent shift to the linear
polarizability b ¼ ð∂ΔP=∂Q ⃗ IR ∂QR Þ and effective charge
electric susceptibility via the action of a strong dc or low-
frequency electric field through a nonzero quadratic electric
Z̃. SrTiO3 has modest values of these parameters, and while ð1Þ ð2Þ
the evaluation of other materials is beyond the scope of this susceptibility, δχ ij ¼ 2χ ijk Ek ðω ≈ 0Þ. The low-frequency
article, we might anticipate optical materials with signifi- field shifts the electronic distribution from its equilibrium
ð3Þ coordination, thus, breaking the original symmetry of the
cantly larger χ b;b.
The parameter b can be viewed as the change in the electronic potential with respect to new electronic displace-
ments induced by a second electric field. In contrast, IRRS
mode-effective charge Z̃ due to the Raman phonon [see
induces a change to the lattice from its equilibrium
discussion below Eq. (2)]. In SrTiO3 we find that the
structure, thus, also breaking the original symmetry of
parameter b is dominated by changes to so-called anoma-
the electronic potential, as the electronic displacements
lous charge contributions to the mode-effective charge—
induced by a second electric field are now with respect to
changes to the electronic structure of occupied electronic
the electronic distribution of the changed lattice.
states. This finding suggests that materials with large
Experimental verification of IRRS in mid-IR experi-
anomalous charge should be the focus of the initial search
ments is most straightforward through observation of an
for materials with sizable infrared-resonant Raman
IR-resonant enhancement of Stokes or anti-Stokes Raman
response. Perovskites are well known to exhibit large peaks. Inspecting Fig. 4, it is clear that the spectral features
anomalous charges and are therefore a useful theoretical for IRRS and ionic Raman scattering in SrTiO3 are
and experimental test ground for IRRS. different, allowing for the experimental distinction between
The principle of IR resonant modification of χ ð1Þ via the two effects. Additionally, our first-principles evaluation
IRRS could be readily extended to explore infrared-driven of the coupling terms in SrTiO3 suggests that the resonant
control of higher-order susceptibilities in both centrosym- enhancement is overwhelmed by IRRS, with many orders
metric and noncentrosymmetric crystals where large of magnitude smaller contribution from ionic Raman
ð2Þ ð3Þ
changes to χ eff , χ eff , etc., may be anticipated due to the scattering, that is, through the anharmonic lattice potential.
dependence of the symmetry and size of these optical The degree to which IRRS overwhelms ionic Raman
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