Matrix formalism for radiating polarization sheets in
multilayer structures of arbitrary composition
Hui Shi (石 卉), Yu Zhang (张 雨), Hongqing Wang (王洪庆), and
Weitao Liu (刘韡韬)*
Physics Department, State Key Laboratory of Surface Physics, Key Laboratory of Micro and Nan o Photonic
Structures (MOE), Collaborative Innovation Center of Advanced Microstructures (Nanjing), Fudan University,
Shanghai 200433, China
*corresponding author: wtliu@fudan.edu.cn
Received February 21, 2017; accepted April 21, 2017; posted online May 15, 2017
In optical studies on layered structures, quantitative analysis of radiating interfaces is often challenging due to
multiple interferences. We present here a general and analytical method for computing the radiation from
two-dimensional polarization sheets in multilayer structures of arbitrary compositions. It is based on the
standard characteristic matrix formalism of thin films, and incorporates boundary conditions of interfacial
polarization sheets. We use the method to evaluate the second harmonic generation from a nonlinear thin film,
and the sum-frequency generation from a water/oxide interface, showing that the signal of interest can be
strongly enhanced with optimal structural parameters.
OCIS codes: 190.4400, 260.3160, 310.4165, 310.6860.
doi: 10.3788/COL201715.081901.
Multilayer structures of thin films are indispensable in
modern technology and scientific research
[1–5]
. In such
structures, interfaces often play a key role: they give rise
to desirable electronic and optoelectronic functions
[6]
, and
are the host of many novel phenomena
[7]
. Across the inter-
faces, the broken symmetry often causes a net polar order-
ing, which is readily monitored by surface-specific optical
techniques. For example, second-order nonlinear optical
processes such as second harmonic generation (SHG)
and sum-frequency generation (SFG) can be highly sur-
face sensitive for centrosymmetric media, and are widely
employed in interfacial studies
[8–12]
. Experimentally, the
optical signal depends on both the interfacial polarizations
and local electric fields
[13,14]
. Yet, due to the coexistence of
multiple interfaces and the interference between multiply
reflected beams
[15]
, it is often challenging to perform quan-
titative analysis
[16,17]
.
In this study, we introduce a method for computing the
radiation from interfacial polarization sheets in multi-
layer structures. Our method i s based on the standard
characteristic matrix formalism of thin films
[16,17]
,and
incorporates boundary conditions of electromagnetic
fields due to such polarization sheets
[18]
.Ityieldsthe
contribution of each individual interface, and applies
to multilayer structures of arbitrary composition. With
this analytical approach, we can easily choose appropri-
ate structural parameters to selectively boost up or
suppress responses from specific l ocations. We present
here two practical examples showing that appropriate
geometries can strongly enhance the response from thin
filmsorinterfacesofinterest.Thismethodisnotlimited
to n onlinear optical studies, but is generally applicable
for optical probes of radiating polarization sheets in such
structures, for example, the photoluminescence from
two-dimensional (2D) transition metal dichalcogenides
in a field-effect transistor
[19]
.
We first consider radiation from the polarization sheet
on top of a thin film, as illustrated in Fig.
1(a). A thin film
of refractive index n
2
is sandwiched between two semi-
infinite media, n
1
and n
3
. Assuming an oscillating 2D
polarization sheet of polarization P
s
ðωÞ at frequency ω
is excited by beams incident from medium 1 and overlap-
ping at boundary I [inset of Fig.
1(a)]. To find the electric
field EðωÞ generated by P
s
ðωÞ in media 1 and 3, we employ
boundary conditions of electromagnetic fields by using a
2D polarization sheet
[18]
. At boundary I
ΔE
x
¼ σ
z
¼ −
4π
ϵ
0
ik
x
P
sz
;
ΔE
y
¼ 0;
ΔH
x
¼ σ
y
¼ −
4πi
c
ωP
sy
;
ΔH
y
¼ σ
x
¼
4πi
c
ωP
sx
; (1)
where ϵ
0
is the effective dielectric constant of the interfa-
cial layer
[13]
, i is the square root of −1, k ¼ ω∕c is the light
wave vector in a vacuum, and the lab coordinates (x; y; z)
are set with z parallel to the surface normal (
ˆ
n) and x–z is
the beam incident plane [inset in Fig.
1(a)]. σ is the dis-
continuity in the electromagnetic field caused by P
s
ðωÞ.
For simplicity, we use the same symbol to represent differ-
ent quantities in different cases. We define σ
E
≡ 0 and
σ
H
≡ σ
y
for TE waves, and σ
E
≡ σ
z
and σ
H
≡ σ
x
for TM
waves
[18]
. So, across bou ndary I , the relation between tan-
gential components of the fields can be written as
COL 15(8), 081901(2017) CHINESE OPTICS LETTERS August 10, 2017
1671-7694/2017/081901(5) 081901-1 © 2017 Chinese Optics Letters
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