Continuous control of the nonlinearity phase for harmonic generations

TLDR: The continuous phase engineering of the effective nonlinear polarizability enables complete control over the propagation of harmonic generation signals, paving the way for highly compact nonlinear nanophotonic devices.

Abstract: The capability of locally engineering the nonlinear optical properties of media is crucial in nonlinear optics. Although poling is the most widely employed technique for achieving locally controlled nonlinearity, it leads only to a binary nonlinear state, which is equivalent to a discrete phase change of π in the nonlinear polarizability. Here, inspired by the concept of spin-rotation coupling, we experimentally demonstrate nonlinear metasurfaces with homogeneous linear optical properties but spatially varying effective nonlinear polarizability with continuously controllable phase. The continuous phase control over the local nonlinearity is demonstrated for second and third harmonic generation by using nonlinear metasurfaces consisting of nanoantennas of C3 and C4 rotational symmetries, respectively. The continuous phase engineering of the effective nonlinear polarizability enables complete control over the propagation of harmonic generation signals. Therefore, this method seamlessly combines the generation and manipulation of harmonic waves, paving the way for highly compact nonlinear nanophotonic devices.

Figures

Figure 1| Illustration of geometric phase controlled nonlinear metamaterials. a, Rotating a nanostructure by an angle θ with respect to the laboratory frame will introduce a geometric phase. Hereby each nanostructure introduces a nonlinear geometric phase with a phase variation of (n-1)σθ or (n+1)σθ to the nth harmonic generation for the same or the opposite circular polarization to that of the fundamental wave, respectively. b, For a C2 nanostructure, THG of both circular polarizations with phases of 2σθ and 4σθ are generated in the forward direction. c, For a C4 nanostructure, only THG with circular polarization opposite to that of fundamental wave is generated, which has a phase of 4σθ.

Figure 4| THG signals from metasurfaces with a phase gradient of the nonlinearity. a, Illustration of phase controlled diffractions of third harmonic generation (THG) for right circularly polarized (RCP) light at the fundamental frequency. The C2 symmetry metasurfaces diffract right- (RCP) and left- (LCP) circularly polarized THG signals to the first- and seconddiffraction orders respectively; in comparison, the C4 symmetry metasurface only diffracts the opposite circularly polarized THG at first diffraction order direction. b, Measured diffraction pattern of the THG signals fromtheC2/PFO and C4/PFO metasurfaces for circular polarization states of the fundamental and the THG waves.

Figure 3| Diffraction of the fundamental wave on metasurfaces with linear phase gradients. a, b, Scanning electron microscopy images of metasurfaces consisting of C2 and C4 symmetry nanostructures. The lattice constant of the metasurfaces in both directions is a = 400 nm. Along the x direction (H) the nanostructures are rotated by an angle of π/8 for the C2 structures and π/16 for the C4 structures, respectively, resulting in the same superlattice period P=3.2 µm. scale bar: 400 nm. c, Measured diffraction pattern of the fundamental waves on theC2 (λ = 1250 nm) and theC4 (λ = 1200 nm) metasurfaces. The first and second rows were recorded using right circularly polarized (RCP) laser, while the third and fourth rows correspond to left circularly polarized (LCP) illumination.

Figure 2| Experimental verification of the nonlinear phase with nonlinear phase gratings. a-c, Scanning electron microscopy images of Sample A, B, and C fabricated by electron beam lithography. The width and length of each gold nanorod are 50 nm and 240 nm, respectively. The lattice size for the three metasurfaces are a= 400 nm along x- (H) and y- (V) axis directions; thickness of gold layer is t = 30 nm. The scale bar shows 400 nm. d, Measured and simulated transmission spectra for the PFO coated hybrid metasurfaces showing the plasmonic resonance of the gold nanostructures for horizontal (H-) polarized illumination (see inset for the orientation of the axes). e, Measured diffraction pattern of THG signals from metasurfaces A, B, and C. The THG signals of the first and second rows correspond to right circularly polarized (RCP) pumping laser; the third and fourth rows corresponds to the left circularly polarized (LCP) pumping laser.

Full text

University of Birmingham
Continuous control of the nonlinearity phase for
harmonic generations
Li, Guixin; Zhang, Shuang; Chen, Shumei; Pholchai, Nitipat; Reineke, Bernhard; Wong, Polis
Wing Han; Yue Bun Pun, Edwin; Cheah, Kok-wai; Zentgraf, Thomas
DOI:
10.1038/nmat4267
License:
None: All rights reserved
Document Version
Peer reviewed version
Citation for published version (Harvard):
Li, G, Zhang, S, Chen, S, Pholchai, N, Reineke, B, Wong, PWH, Yue Bun Pun, E, Cheah, K & Zentgraf, T 2015,
'Continuous control of the nonlinearity phase for harmonic generations', Nature Materials, vol. 14, no. 6, pp. 607-
612. https://doi.org/10.1038/nmat4267
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Continuous control of nonlinearity phase for harmonic generations
Guixin Li
1,2
†
, Shumei Chen
1,2
†
, Nitipat Pholchai
3,4,5
, Bernhard Reineke
3
, Polis Wing Han Wong
6
,
Edwin Yue Bun Pun
6
, Kok Wai Cheah
2
*, Thomas Zentgraf
3
*, and Shuang Zhang
1
*
1
School of Physics & Astronomy, University of Birmingham, Birmingham, B15 2TT, UK
2
Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
3
Department of Physics, University of Paderborn, Warburger Straße 100, D-33098 Paderborn, Germany
4
Department of Industrial Physics and Medical Instrumentation, King Mongkut's University of
Technology North Bangkok, 1518 Pibulsongkram Road, Bangkok 10800, Thailand
5
Lasers and Optics Research Group, King Mongkut's University of Technology North Bangkok, 1518
Pibulsongkram Road, Bangkok 10800, Thailand
6
Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Ave, Hong Kong
*email: kwcheah@hkbu.edu.hk; thomas.zentgraf@uni-paderborn.de; s.zhang@bham.ac.uk;
The capability of locally engineering the nonlinear optical properties of media is crucial in
nonlinear optics. While poling is the most widely employed technique for achieving locally
controlled nonlinearity, it only leads to a binary nonlinear state, which is equivalent to a discrete
phase change of π in the nonlinear polarizability. Here, inspired by the concept of spin rotation
coupling, we experimentally demonstrate nonlinear metasurfaces with homogeneous linear
optical properties but spatially varying effective nonlinear polarizability with continuously
controllable phase. The continuous phase control over the local nonlinearity is demonstrated for
second and third harmonic generations by using nonlinear metasurfaces consisting of
nanoantennas of C3 and C4 rotational symmetries, respectively. The continuous phase
engineering of the effective nonlinear polarizability enables complete control over the
propagation of harmonic generation signals. Therefore, it seamlessly combines the generation
and manipulation of the harmonic waves, paving the way for highly compact nonlinear
nanophotonic devices.

The local phase of the nonlinear polarizability determines how the generated nonlinear light in
the material will interfere during its generation and propagation processes. In general one is
interested in a constructive conversion of the fundamental to the nonlinear light while a wave is
propagating through the material. However, the chromatic dispersion prevents from an efficient
conversion due to the different propagation velocity of light for the different wavelengths. If the
phase of the induced nonlinear material polarization can be controlled locally without modifying
the linear properties such a mismatch can be avoided and the nonlinear process would be more
efficient. Up to date there has been no demonstration of a material that allows continuous and
arbitrary phase control for the local nonlinear polarizability. Such a nonlinear material would
enable exact phase matching conditions for nonlinear optical processes, in contrast to the widely
utilized quasi-phase matching scheme in which only the sign of the nonlinear polarizability can
be manipulated
1-6
. It may remove additional undesired nonlinear processes which are introduced
by the higher Fourier components of the nonlinear susceptibility in a periodically poled system.
Metamaterials on the other hand provide a high degree of freedom for tailoring the local optical
properties on a subwavelength scale. Nevertheless, they were mostly used for tailoring the linear
optical properties.
The phase control over the nonlinear polarizability of the metamaterial is inspired by the
concept of spin rotation coupling of light which has been utilized to control the wavefront of
light in the linear regime
7-11
. This novel concept has been applied to the design of various types
of functional metasurfaces
12-16
. These types of metasurfaces, which consist of plasmonic
structures with subwavelength feature size (sometimes called “artificial atoms”), can be
engineered to show rotation controlled local geometric phase shifts. This concept has been
employed for flat lens imaging, generation of vortex beams, three-dimensional holography, and
optical spin-orbital interaction
12-17
. Here we apply the concept of spin rotation coupling of light
to the nonlinear regime leading to a nonlinear material polarization with arbitrarily controllable
phase profile. For demonstration we show that this concept can be implemented by metasurfaces
containing plasmonic antennas. However, we like to note that the concept of continuous phase
control is universal and can be applied also to dielectric and bulk-like metamaterials.
We start by considering a single subwavelength plasmonic or dielectric nanostructure
(resembling a dielectric dipole) embedded in an isotropic nonlinear medium (Fig. 1a). We will
show that when excited by a circularly polarized fundamental beam, the phase of the nonlinear
polarization of the artificial atom can be controlled geometrically by the orientation of the
structure through a spin rotation coupling. For an incident fundamental beam with circular
polarization state
σ
propagating along +z direction, the electric field can be expressed
as:
0
()/2
xy
EEeie
σ
σ
=+
, where
1
σ
=±
represents the state of left- or right- handed circular
polarization, respectively. The excitation of the nanostructures (e. g. the plasmonic nanorods)
together with the nonlinear medium in close vicinity of the structure where the field can be
strongly enhanced, locally forms an effective nonlinear dipole moment:
()
nn
pE
ωσ
θθ
α
=
(1)
where
θ
α
is the n
th
harmonic nonlinear polarizability tensor of the nanostructure with orientation
angle of
θ
. We employ a coordinate rotation to analyze the dependence of the nonlinear dipole
moment on the orientation angle of the nanostructure. In the local coordinate of the nanostructure
(referred to as local frame) as shown in Fig. 1a where the local coordinate (x’, y’) axes are

rotated by an angle of
θ
with respect to the laboratory frame (x, y), the fundamental wave
acquires a geometric phase due to the rotation spin coupling effect
e
i
L
EE
σσσθ
=
(2)
where the index ‘L’ denotes the nanostructure’s local coordinate frame. The n
th
harmonic
nonlinear polarizability in the structure’s local frame is simply
α
0
=
α
θ
⏐
θ
=0
. Thus, the n
th
harmonic
nonlinear dipole moment in the local frame is given by
,0 0
() ()
n n n in
LL
pE Ee
ωσ σσθ
θ
αα
==
(3)
The nonlinear dipole moment can be decomposed into two in-plane rotating dipoles
(characterized by the circular polarization states
σ
and -
σ
) as
nn n
,L ,L , ,L ,
pp p
ωω ω
θθσθσ
−
=+
with
,, ,,
,
nn in
LL
pp e
ωω σθ
θσ θ σ
−
∝
(4)
After transforming back to the laboratory frame the two rotating dipole moments are given by
(1)
,,,
(1)
,,,
nni ni
L
nnini
L
ppe e
ppee
ωωσθ σθ
θσ θ σ
ωωσθ σθ
θσ θ σ
−−
+
−−
=∝
=∝
(5)
The nonlinear polarizabilities of the nanostructure can therefore be expressed as,
(1)
,,
(1)
,,
nni
nni
e
e
ωσθ
θσσ
ωσθ
θσσ
α
α
−
+
−
∝
∝
(6)
Thus, geometric phases of
(1)n
σθ
−
or
(1)n
σθ
+
are introduced to the nonlinear polarizabilities
of the n
th
harmonic generation with the same or the opposite circular polarization to that of the
fundamental wave, respectively. According to the selection rules for harmonic generation of
circular polarized fundamental waves, a single nanostructure with m-fold rotational symmetry
only allows harmonic orders of
1n lm=±
, where
l
is an integer, and the ‘+’ and ‘-‘ sign
correspond to harmonic generation of the same and opposite circular polarization, respectively
18-
21
. The phases of the nonlinear polarizability for an incident fundamental wave of circular
polarization, for various orders of harmonic generation and nanostructures of various rotational
symmetries, are given in the Supplementary Materials (Table S1).
Hence for a nanorod structure with two-fold rotational symmetry (C2), THG signals with
both the same and opposite circular polarizations as that of the fundamental wave can be
generated. According to equation (6), they have a spin dependent phase of 2
σθ
and 4
σθ
,
respectively (Fig. 1b). On the other hand, a nanostructure with four-fold rotational symmetry
(C4) does not allow a THG process for the same polarization state as the incident polarization.
Hence, only a single THG signal, that of the opposite circular polarization, is generated with a
geometric phase of
4
σθ
(Fig. 1c). Importantly, due to the local isotropic response of the C4
structure, both the polarization state and the propagation of the fundamental wave will not be
affected when transmitting through a metamaterial consisting of such C4 nanostructures of
arbitrary orientations. Thus, by assembling the C4 nanostructures with spatially varying
orientations in a 3-D or 2-D lattice, a nonlinear metamaterial or metasurface can be formed
which show homogeneous linear properties, but locally a well-defined nonlinear polarizability
distribution for a circularly polarized fundamental wave.

We verify this concept of spin-rotation induced nonlinear phase by designing and
fabricating three nonlinear binary phase gratings consisting of 2D arrays of plasmonic
nanocrosses with a local C4 rotational symmetry (Fig. 2a-c). These binary nonlinear phase
gratings can be easily used to characterize the relative phase of the nonlinear polarization
between C4 nanostructures of different orientations. Sample A consists of nanocrosses of
identical orientation along the entire lattice and a periodicity of a = 400 nm in both x- and y- axis
directions, while Sample B and Sample C consist of supercells of nanocrosses with two different
orientations and a superperiod of P = 3.2 µm (eight crosses per unit cell). The difference between
the orientation angles of the two subsets of nanocrosses within a unit cell is π/8 and π/4 for
Sample B and C, respectively. For the C4 structures, only THG of the opposite circular
polarization as that of the incident fundamental beam is generated with a geometric phase of
4
σθ
. Thus, the introduced nonlinear phase for Sample B and Sample C correspond to nonlinear
phase gratings with phase difference of π/2 and π between the two subunits, respectively. For the
binary nonlinear grating with period larger than the THG wavelength, the distribution of the
nonlinear signal in different diffraction orders is solely determined by the phase difference
between the two subsets. Thus, by experimentally measuring the ratio between the 0
th
and the 1
st
order THG signals, we are able to verify the orientation controlled induced phase difference
between the two subsets of nanocrosses within a unit cell in Sample B and C.
First, we coat a nonlinear active medium (PFO) on top of these three phase gratings to
form a gold/PFO hybrid metasurface (see Methods Section). The linear optical properties of the
hybrid phase gratings are characterized by using Fourier transformation infrared spectrometry.
From the measured transmission spectra (Fig. 2d), we identify that the localized Plasmon
resonance is around at wavelength: λ ~ 1230 nm for all three samples, and this also confirmed by
the numerical simulation. Subsequently, we measured the THG signal from these three samples
for a fundamental wavelength at the resonance dip to maximize the Plasmon enhanced nonlinear
response from the samples. As shown previously
22-30
, the combination of the strong field
enhancement and large nonlinear polarizability of the plasmonic structures gives rise to a large
nonlinear optical effect in such plasmonic system, making it easy to detect the generated
nonlinear signal.
In the next step we illuminated the samples with short laser pulses at 1240 nm and
measured the THG signal on a CCD camera. Our THG measurements with circular polarized
incident light confirm the selection rules for the C4 symmetry structure as the THG signals with
identical polarization states as the fundamental wave (RCP-RCP and LCP-LCP) are very weak
for all three samples and therefore not visible in Fig. 2e. Sample A, with spatially homogenous
orientation of the nanocrosses, only gives rise to a 0
th
order THG. On the other hand, Sample B
and Sample C, with their superlattice periodicity greater than the wavelength of the THG light,
emit THG into the first diffraction orders. For Sample B where the phase difference between the
third order nonlinearity of the two subsets of nanocrosses is π/2 the ratio between the 0
th
order
and the 1
st
order THG signals is predicted to be ~ 2.4:1 (see Supplementary Materials Fig. S1
and the corresponding discussion). The measured ratio is around 2.8:1, which agrees very well
with the theoretical prediction. The π phase difference between the nonlinear polarizabilities of
the two subsets of the nanocrosses in Sample C is expected to result in a complete destructive
interference for the 0
th
order THG signal, which is confirmed by the measurement shown in the
most right panel of Fig. 2e. The orientation angle dependent phase of nonlinearity is further
confirmed by numerically calculating the contribution to the far field THG radiation from each

Frequently asked questions

Q1. What is the widely used technique for achieving nonlinearity?

While poling is the most widely employed technique for achieving locally controlled nonlinearity, it only leads to a binary nonlinear state, which is equivalent to a discrete phase change of π in the nonlinear polarizability. 

Q2. What are the contributions in "Continuous control of nonlinearity phase for harmonic generations" ?

The concept of fair dealing under the Copyright, Designs and Patents Act 1988 ( CCDPA ) was introduced in this paper. 

Q3. What is the key application of the 2-D nonlinear metasurfaces?

One key application of the 2-D nonlinear metasurfaces is the realization of nonlinear holograms, where a beam at fundamental wavelength can be converted to arbitrary beam profile at different wavelength. 

Q4. what is the nth harmonic nonlinear generation in the nanostructure?

According to the selection rules for harmonic generation of circular polarized fundamental waves, a single nanostructure with m-fold rotational symmetry only allows harmonic orders of 1n lm= ± , where l is an integer, and the ‘+’ and ‘-‘ sign correspond to harmonic generation of the same and opposite circular polarization, respectively1821. 

Q5. What is the phase of the incident beam?

While the 0th order has the same circular polarization as the incident beam, the beam with the opposite circular polarization is diffracted into the +1 or -1 order, depending on the incident circular polarization state. 

Q6. What is the polarity of the nonlinear material?

Such a nonlinear material would enable exact phase matching conditions for nonlinear optical processes, in contrast to the widely utilized quasi-phase matching scheme in which only the sign of the nonlinear polarizability can be manipulated1-6. 

Q7. What order of diffraction is generated for the C2 sample?

For the C2 sample (Fig. 4b), the THG signal is generated in four different diffraction orders, -2, -1, +1 and +2, which correspond to the incident/transmitted polarization of RCP/LCP, RCP/RCP, LCP/LCP and LCP/RCP, respectively. 

Q8. What is the polarization of the C4 metasurface?

On the other hand, for the C4 metasurface, only THG with polarization opposite to that of the incident beam is generated with the phase gradient of4σθ , which corresponds to the first diffraction orders for the C4 metasurfaces with period of Δθ =π/2 for the orientation angle. 

Q9. What is the diffraction effect of the nanocrosses?

This diffraction effect arises from the spin dependent geometric phase in the linear regime, which has been investigated previously12. 

Q10. What is the phase of the nonlinear polarizability?

The local phase of the nonlinear polarizability determines how the generated nonlinear light in the material will interfere during its generation and propagation processes. 

Q11. What is the diffraction angle of the C2 metasurface?

For a circularly polarized dipole in the x-y plane, the emitted THG into a diffraction angle of ϕ in general contains both LCP and RCP components, with the ratio given by ).cos1/()cos1( ϕϕ +− 

Q12. What is the orientation angle of the C4 metasurface?

the C4 metasurface can be considered as a homogeneous optical surface in the linear regime, despite the spatial variation of the orientation angle along the surface. 

Q13. What is the phase difference between the two subunits of the sample?

the introduced nonlinear phase for Sample B and Sample C correspond to nonlinear phase gratings with phase difference of π/2 and π between the two subunits, respectively. 

Q14. what is the nth harmonic nonlinear polarization in the structure?

These binary nonlinear phase gratings can be easily used to characterize the relative phase of the nonlinear polarization between C4 nanostructures of different orientations. 

Q15. What is the phase difference between the two subsets of the nanocrosses?

The π phase difference between the nonlinear polarizabilities of the two subsets of the nanocrosses in Sample C is expected to result in a complete destructive interference for the 0th order THG signal, which is confirmed by the measurement shown in the most right panel of Fig. 2e. 

Q16. What is the method for forming a gold/PFO hybrid metasurface?

the authors coat a nonlinear active medium (PFO) on top of these three phase gratings to form a gold/PFO hybrid metasurface (see Methods Section).