Theory of nonlinear polaritonics: $\chi^{(2)}$ scattering on a $\beta$-SiC surface

0.62 MB·

by  Christopher R. Gubbin

Theory of nonlinear polaritonics: χ(2) scattering on a β-SiC surface
Christopher R. Gubbin and Simone De Liberato
School of Physics and Astronomy, University of Southampton,
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Southampton, SO17 1BJ, United Kingdom
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Abstract
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a In this article we provide a practical prescription to harness the rigorous microscopic, quantum-
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4 leveldescriptions oflight-matter systemsprovided byHopfielddiagonalisation forquantum descrip-
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tion of nonlinear scattering. A general frame to describe the practically important second-order

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l optical nonlinearities which underpin sumand differencefrequency generation isdeveloped forarbi-
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trarilyinhomogeneousdielectricenvironments. Specificattentionisthenfocussedonplanarsystems
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withopticalnonlinearity mediatedby apolardielectric β-SiChalfspace. Inthissystemwecalculate
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. the rate of second harmonic generation and the result compared to recent experimental measure-
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ments. Furthermore the rate of difference frequency generation of subdiffraction surface phonon
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d polaritons on the β-SiC halfspace by two plane waves is calculated. The developed theory is easily
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integrated with commercial finite element solvers, opening the way for calculation of second-order
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nonlinear scattering coefficients in complex geometries which lack analytical linear solutions.

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I. INTRODUCTION
At the quantum level light-matter interactions are conveniently described in a micro-
scopic Hopfield picture, taking matter degrees of freedom as bosonic fields coupled to the
photons. The system Hamiltonian is then diagonalised utilising coupled light-matter quasi-
particles, termed polaritons [1]. This prescription can be followed for quantization of fields
in homogeneous, linear dielectrics in the absence of [2] and including [3] losses. Hopfield
approaches are particularly advantageous as frames to describe nonlinear processes, which
can be understood as scattering cross-sections for the polaritons [4–12].
Exploitingrecentadvancesinphotonics,demonstratedarchetypicallybytheplasmonsextant
at metal-dielectric interfaces formed by hybridisation of photons with coherent oscillations
of the electron gas, energy can be confined in deep subdiffraction volumes. This allows for
downscaling of photonic devices to the nanoscale [13, 14] and enables measurable nonlinear
scattering at diminished pump threshold. Plasmonic nonlinear devices oftenexploit intrinsic
metal nonlinearities, arising from free carriers or band to band transitions and first demon-
strated for harmonic generation mediated by Kretschmann-coupled surface plasmons of a
sub-wavelength silver film [15]. Metal-mediated plasmon generation by four-wave mixing
[16] and frequency difference generation [17] have also been demonstrated, however the cen-
trosymmetry of plasmonic metals precludes second order nonlinear effects in the bulk and
limits studies to symmetry-breaking interfaces [18]. Instead, theoretical studies in nonlinear
plasmonics focus toward third-order nonlinear processes such as self-phase modulation [19]
or other Kerr nonlinearities [20].
Interfaces between polar and non-polar dielectrics also permit deep subdiffraction localisa-
tion by hybridising photons with coherent oscillations of the polar crystal atomic lattice, in
surface phonon polaritons. Supported in the Reststrahlen region, between transverse and
longitudinal optic phonon frequencies which ordinarily occupy the midinfrared spectral re-
gion, these modes absent themselves from the Ohmic losses inherent in plasmonic systems
and render achievable quality factors an order of magnitude larger. Moreover, interplay
between extreme energy localisation in user-defined deep sub-wavelength resonators [21, 22]
and bulk second-order nonlinearity in polar crystals make phonon polaritons an excellent
testbed for midinfrared nonlinear optics [23, 24].
These inhomogeneous photonic systems require polaritonic descriptions which also account
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for real-space variation in the dielectric environment, a necessity recently demonstrated in
description of the healing length in polaritonic condensates [25]. Following initial efforts
[26, 27], a full polaritonic description was recently developed for non-magnetic, arbitrarily
inhomogeneous, linearly polarisable media [28], opening the path for quantum descriptions
of nonlinear processes in sub-wavelength devices. In this Paper we embark on development
of a quantum, microscopic description of second-order nonlinear scattering in inhomoge-
neous phonon-polariton systems. In Sec. II, after having sketched the main points of the
theory of Hopfield diagonalisation in linear inhomogeneous media previously reported [28],
we specialise our discussion to the β-SiC/vacuum interface which has been the subject of
contemporary study [23], deriving its relevant polaritonic modes. In Sec. III we develop the
theory of polaritonic χ(2) scattering, exploiting data regarding plane-wave scattering from
recent second harmonic generation experiments [23] to both validate our theory and deter-
mining the phenomenological couplings that parameterise it. Exploiting those coefficients,
that are in good agreement with data from ab initio simulations present in the literature
[29], we will then be able to investigate nonlinear processes involving subdiffraction phonon
polariton modes.
II. LINEAR REGIME
A. General theory
Followingapriortreatment[28], wedescribeanon-magnetic, inhomogeneouslight-matter
system, characterised by a single dipolar resonance with Hamiltonian
Dˆ2 µ Hˆ2 Pˆ2 ρω2 Xˆ2 κ
Hˆ = dr + 0 + + LO Xˆ Dˆ , (1)
0
2ǫ 2 2ρ 2 − ǫ ·
" 0 0 #
Z
where Dˆ (Hˆ) is the electric displacement (magnetic) field operator and Xˆ (Pˆ) is the material
displacement (momentum) field operator. The system is fully parameterised by spatially
dependant density ρ, longitudinal frequency ω , and macroscopic dipole moment κ. In the
L
spirit of the Hopfield prescription, the Hamiltonian in Eq. 1 can be diagonalised in terms of
linear superpositions of light and matter conjugate operators through the polariton operator
Kˆ = dr α Dˆ +β Hˆ +γ Pˆ +η Xˆ , (2)
n n n n n
· · · ·
Z h i
3
where the Greek symbols are the space-dependent Hopfield coefficients [28] and n indexes
the different positive-frequency normalmodes. Such operatorsobeythe Heisenberg equation
Kˆ ,Hˆ = ~ω Kˆ , (3)
n 0 n n
h i
which are equivalent to Maxwell equations over the Hopfield coefficients in an inhomoge-
neous dielectric. Together with the requirement that the polaritonic operators obey bosonic
commutation relations
Kˆ ,Kˆ† = δ , (4)
m n m,n
h i
this allows us to solve the system and to recover the properly normalised light and matter
fields as linear superpositions of polaritonic modes weighted by the wavefunctions α
n
Eˆ = ~ ω α¯ Kˆ +α Kˆ† , (5)
n n n n n
Xn h i
κω
Xˆ = ~ n α¯ Kˆ +α Kˆ† ,
ρ(ω2 ω2) n n n n
Xn T − n h i
where the transverse resonance of the matter is
κ2
ω2 = ω2 . (6)
T L
− ρǫ
0
B. Application to surface phonon polaritons
The theory sketched above can be naturally applied to the case of an halfspace filled
with a polar dielectric, sketched in Figure 1a, identifying ωL and ωT with longitudinal and
transverse optical phonon frequencies ωLO and ωTO. Developing Eq. 3 this would lead to
the solve Maxwell equations with the inhomogeneous dielectric function
ǫ(ω,z > 0) = ǫ (ω) = 1,
1
ω2 ω2
ǫ(ω,z < 0) = ǫ (ω) = ǫ LO − , (7)
2 ∞ω2 ω2
TO −
where we labelled the mediums 1 and 2 and include off-resonance effects through the phe-
nomenological high frequency dielectric constant ǫ . These effects could be formally in-
∞
cluded by considering an additional matter resonance with longitudinal frequency far above
the Reststrahlen band. In this geometry the eigenproblem in Eq. 3 has been solved [28],
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giving rise to seven classes of solutions. Two (TMv and TEv) describe photons incident
upon the surface coming from the vacuum and their respective reflected and transmitted
waves, with either Transverse Magnetic or Transverse Electric polarizations. Four (TMl,
TEl, TMu, TEu) describe excitations incident upon the surface from the dielectric side and
their respective reflected and transmitted waves, indexed by both the polarization and by
the polaritonic branch, either lower or upper, they belong to. An example of each group
is shown in Figure 1a. The last one (S) is instead a surface bound evanescent solution de-
scribing the branch of surface phonon polaritons. The mode index n will thus consist of the
solution class and the relevant wavevector: in-plane for the evanescent solutions and both
in- and out-of-plane in the medium of origin for the propagative ones. The solution of Eq. 3
leads to the Helmholtz equation, linking for each solution the in-plane wavevector k and
k
the, generally complex, out-of-plane components k in each halfspace i by
i,z
ǫ (ω)k2 = k2 +k2 , (8)
i 0 k i,z
where k = ω is the vacuum wavelength. As an example in the case of propagative, TM
0 c
polarised radiation incident from z > 0 (TMv) with in- and out-of-plane wavevectors k and
k
k , the solution of Eq. 3 is
1,z
αTMv r ,z > 0 = NTMv p e−ik1,zz +p rTMeik1,zz eikk·rk,
1,k k k 1− 1+ 12
αTMv(cid:0)r ,z < 0(cid:1) = NTMvt(cid:2)TMp e−ik2,zzeikk·rk, (cid:3) (9)
2,k k k 12 2−
(cid:0) (cid:1)
TMv
where N is a normalization constant to be determined by the modal orthonormality
k
TM TM
condition in Eq. 4, t (r ) are the Fresnel transmission (reflection) coefficients for a
12 12
TM polarised beam propagating from medium 1 to medium 2 and the unit vectors for TM
polarisation are given by
k e k e
p = k z ∓ i,z k, (10)
i±
k √ǫ
0 i
where the subscript i indicates the medium of propagation and the index the direction.
±
Here (e ,e ,e ) form a right-handed orthonormal basis and e is the in-plane unit vector
x y z k
parallel to k . The TE polarised solutions can be found by simply replacing the p with the
k i±
plane unit vectors perpendicular to k and the Fresnel coefficients with their TE polarised
k
analogues. For real incident wavevectors the normalisation constant can be calculated as
1
TMv
N = , (11)
k ~ǫ ωV
r 0
5
where V is the volume of quantization. The evanescent, surface bound modes (S) shown in
Figure 2 with in-plane wavevector k and dispersion obeying
k
ω ǫ (ω)
2
k = , (12)
k
csǫ2(ω)+1
have instead the form
αS r ,z > 0 = NS p eik1,zzeikk·rk, (13)
1,k k k 1+
k k
(cid:0) (cid:1) ǫ (ω)k
αS r ,z < 0 = NS 2 1,zp e−ik2,zzeikk·rk,
2,kk k kk k 2−
p 2,z
(cid:0) (cid:1)
with the normalisation
k ǫ (ω)
S 0 2
N =
kk kkv~ǫ ωA ǫ2(ω)−1 +ν(ω) 1−ǫ2(ω)
u 0 2|k1,z| 2|k2,z|
u
t h i
k ǫ (ω)
0 2
= , (14)
kks~ǫ0ωVeff (ω)
where A is the quantization area and ν is the ratio of the group and phase velocities in the
dielectric halfspace [28]. Wecan therefore write thequantized field in the dielectric halfspace
in the simple form
αS r ,z < 0 = eiκ2,zzeikk·rk e kk e , (15)
2,kk k ~ǫ0ωVeff (ω) (cid:18) k − κ2,z z(cid:19)
(cid:0) (cid:1)
with κ = ik real and V can be intperpreted as a dispersive effective mode volume for
2,z 2,z eff
the evanescent mode [30].
III. RESULTS AND DISCUSSION
A. Nonlinear Quantum Theory
1. General Theory
The quadratic treatment implicit in Eq. 1 amounts to the lowest order term in the Taylor
th
expansion of the full Hamiltonian. The most general N order neglected term in the case
of a local interaction will consist of the product of light or matter operators
3 N
H(N) = dr Φj1···jN Oˆjl, (16)
NL t1···tN tl
Z t1X···tN j1·X··jN=1 Yl=1
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where the spatial coordinates are indexed by js and the ts index the different light and
matter operators, in our case Oˆ Dˆ,Hˆ,Xˆ,Pˆ . Using the expression of the field operators
t
∈
in Eq. 5 and their equivalents for thhe conjugateifields, we can rewrite Eq. 16 in the form of
th
a N order polaritonic scattering term
N
H(N) = Ξn1···nN K , (17)
NL nl
n1X···nN Yl=1
where the scattering tensor Ξ can be foundby carrying out integrals over the relevant spatial
variables of the usual nonlinear tensor Φ.
In the following we will aim to describe χ(2) (N = 3) effects in inhomogeneous β-SiC.
As the interaction is mediated by electric field coupling to charges position, there are only
three nonzero field combinations in Eq. 16: the purely mechanical nonlinearity XˆXˆXˆ, the
electrical one XˆXˆEˆ, and the Raman scattering XˆEˆEˆ. Due to the crystal lattice symmetry,
in each case the nonlinear tensor is characterised by the single parameter coupling three
orthogonal field components [31], leaving us with three independent coupling coefficients to
be fixed.
Inserting Eq. 5 inside of Eq. 16 and introducing the second order susceptibility
χ(2)(ω ,ω ,ω ) = G [D(ω )+D(ω )+D(ω )] (18)
1 2 3 1 1 2 3
+G [D(ω )D(ω )+D(ω )D(ω )+D(ω )D(ω )]
2 1 2 1 3 2 3
+G D(ω )D(ω )D(ω ),
3 1 2 3
with
ω2
TO
D(ω) = , (19)
ω2 ω2
TO
−
the interaction Hamiltonian can be put in the usual nonlinear optics form
H(2) = dr ǫ χ(2)(ω ,ω ,ω )Eˆjn1Eˆjn2Eˆjn3, (20)
NL 0 n1 n2 n3 n1 n2 n3
Z n,jn
X
where from Eq. 5
Eˆj = ~ω α¯jKˆ +αjKˆ† , (21)
n n n n n n
h i
is the Cartesian component j of the electric field in the polariton mode n, and the sum is
over all the triplets of polariton modes with different Cartesian coordinates. Introducing the
Born effective charge per primitive cell Z [32], primitive cell reduced mass M, and primitive
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cell volume v, we can relate the three macroscopic dimensionless coupling parameters in
Eq. 18 to the microscopic parameters usually derived in lattice dynamics simulations as
α Z
TO
G = , (22)
1 2v Mω2
(cid:18) TO(cid:19)
µ(2) Z 2
G = , (23)
2 2v Mω2
(cid:18) TO(cid:19)
φ(3) Z 3
G = , (24)
3 2v Mω2
(cid:18) TO(cid:19)
where α is the Raman polarisability per primitive cell, µ(2) is the second order dipole
TO
moment per primitive cell, and φ(3) the third order lattice potential [31, 33]. Notice that we
have ignored the contribution due to the high-frequency static second order susceptibility
(2)
χ , that would be negligible in the resonant regime studied.
∞
2. Second Harmonic Generation at a β-SiC/vacuum interface
We will now apply the theory developed to the case experimentally investigated by Paar-
mann et al [23]: second harmonic generation at a β-SiC/vacuum interface. We have thus to
calculate the scattering tensor Ξ by performing the integral inEq. 20 using the TM polarised
plane-wave eigenmodes Eq. 9or their TEpolarised analoguesinto thenonlinear Hamiltonian
and perform the space integral. As explained in Sec. II B, each modal index n stands for
the solution class and the relevant two- or three-dimensional wavevector in the medium of
origin. For simplicity we consider the case illustrated in Figure 1a where the harmonic pump
photons are azimuthally coplanar (all the k are aligned). Remembering that the nonlinear
k
third order tensor in zincblende crystals is characterised by a single value, coupling three
orthogonally polarised fields [29], we can calculate the scattering coefficient of two TEv har-
monic pumps (TE polarised incident from z > 0, indexed by 1 and 2) generating a TMu
second harmonic (TM polarised upper polariton mode, indexed by 3)
~3ω ω ω
Ξn1,n2,n3 = n1 n2 n3χ(2)(ω ,ω ,ω ) (25)
sǫ0ǫ2(ωn3)V3 n1 n2 n3
Tn1,n2,n3δ kn1 +kn2 kn3
× k k − k
1(cid:16) (cid:17)
P πδ(∆kn1,n2,n3) ,
× i∆kn1,n2,n3 − z
z
(cid:20) (cid:18) (cid:19) (cid:21)
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a) Magnetic Field z Electric Field
TEv TMu
ε(ω)
1
z=0 x
ε (ω)
2
b)
)
U.
A.
(
al
n
g
Si
c
ni
o
m
r
a
H
d
n
o
c
e
S
800 850 900 950 1000 1050
Harmonic Wavenumber (1/cm)
c)
1.0
Reststrahlen Band
7 0.5
-0
1
/
)
ω
ω,2 0
ω,
(
2)
χ(
-0.5
-1.0
800 850 900 950 1000 1050
Harmonic Wavenumber (1/cm)
FIG. 1. a) Sketch of the geometry considered for the generation of a second harmonic TM signal
(red)bytwoTEharmonicpumps(blue). b)Experimentaldata(redcircles)presentedbyPaarmann
et al [23] and theoretical fit (solid blue line) for intensity of a TM polarised second harmonic by TE
polarised harmonic pumps. c) Second harmonic χ(2)(ω,ω,2ω) for β-SiC. The real component is
indicated by the solid blue line and the imaginary by the dashed red line. The Reststrahlen region
is shaded in grey. 9
where P indicates the principal value and the choice of a TMu second harmonic is fixed by
the requirements of having a single rays escaping toward z > 0 and a resonance frequency
larger than ωLO. The mismatch in the out-of-plane wavevectors in Eq. 25 is given by
∆kn1,n2,n3 = k kn1,kn1 +k kn2,kn2 +kn3, (26)
z 2,z k 1,z 2,z k 1,z 2,z
(cid:16) (cid:17) (cid:16) (cid:17)
and the Fresnel tensor has the form
kn3
Tn1,n2,n3 = 2 k en1 e en1 e (27)
kn3 k · x k · y
0
(cid:16) (cid:17)(cid:16) (cid:17)
L kn1,kn1 L kn2,kn2 ,
× yy k 1,z yy k 1,z
(cid:16) (cid:17) (cid:16) (cid:17)
with
2k
L k ,k = 1,z , (28)
yy k 1,z
k +k k ,k
1,z 2,z k 1,z
(cid:0) (cid:1)
and the non-indexed out-of-plane wavevectors and th(cid:0)e freque(cid:1)ncies are calculated through
the Helmholtz equation in Eq. 8. Utilising Fermi’s golden rule and summing over final states
with different out-of-plane wavevectors we can now calculate the scattering rate in the TM
upper polariton mode of frequency ω
~N N ω ω A
W = 1 2 n1 n2 dω ωρ(ω, kn1 +kn2 ) (29)
16π4ǫ V2 | k k |
0
Z
χ(2)(ω ,ω ,ω) Tn1,n2,n3 2
n1 n2 δ(ω +ω ω),
×(cid:12) i∆kzn1,n2,n3 ǫ2(ωn3)(cid:12) n1 n2 −
(cid:12) (cid:12)
where Nj is the photon nu(cid:12)(cid:12)mber in the pumpp beam j(cid:12)(cid:12)and
k 2 +k2
∂k2,z 2,z k 1
ρ ω,k = = , (30)
k ∂ω q k v (ω)
2,z g
(cid:0) (cid:1)
with v (ω) the group velocity. Note that in Eq. 29 the term proportional to δ(∆kn1,n2,n3)
g z
disappears because for the second harmonic generation we wish to consider ∆kn1,n2,n3 = 0
z
cannot be satisfied together with the conservation of energy enforced by δ(ω +ω ω).
n1 n2 −
Specialising Eq.29tothecaseofsecond harmonicgenerationwithharmonicpumpfrequency
ω , and defining the energy density for each pump F = ~ω N /V, we finally obtain the
0 j 0 j
emission rate per unit of surface
W F F ω χ(2)(ω ,ω ,2ω ) Tn1,n2,n3 2
1 2 0 0 0 0
= ρ(2ω ) . (31)
A 8π4~ǫ0 0 (cid:12) i∆kzn1,n2,n3 ǫ2(2ω0)(cid:12)
(cid:12) (cid:12)
(cid:12) p (cid:12)
(cid:12) (cid:12)
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