Controlling light with metamaterial-based Photonic ... · dimensional periodic and quasi-periodic...

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2015.17

NATURE PHOTONICS | www.nature.com/naturephotonics 1

Supplementary Information:

Controlling Light by Metamaterial based Nonlinear

Photonic Crystals

Nadav Segal, Shay Keren-Zur, Netta Hendler, Tal Ellenbogen*

Department of Physical Electronics, Fleischman Faculty of Engineering, Tel-Aviv University,

Tel-Aviv 69978, Israel

Corresponding author e-mail address: [email protected]; [email protected]

Supplementary Section I: Numerical simulation based on a hydrodynamic model

The Hydrodynamic Model

The hydrodynamic model describes the nonlinear optical response of the charge carriers in the

metal. It is based on modeling the electron fluid by Euler’s equation where n(r,t) represent the

electron fluid density and v(r,t) represent the electron velocity field.

∂𝐯𝐯∂𝑡𝑡 + (𝐯𝐯 ∙ ∇)𝐯𝐯 + 𝛾𝛾𝐯𝐯 = 𝑒𝑒

𝑚𝑚𝑒𝑒∗(𝐄𝐄 + 𝐯𝐯 × 𝐁𝐁) − 𝛽𝛽2

𝑛𝑛 ∇𝑛𝑛 (1) 𝛾𝛾 − electron collision rate

𝛽𝛽 ∝ Fermi velocity 𝐯𝐯𝐹𝐹

∇ ∙ 𝐉𝐉 = −𝑒𝑒�̇�𝑛 (𝐉𝐉 = 𝑒𝑒𝑛𝑛𝐯𝐯) (2) 𝑚𝑚𝑒𝑒∗ − electron effective mass

After expanding all fields in a perturbative manner, it is possible to extract the nonlinear current

terms which depend on the bulk polarization (1, 2 and refs. within):

𝐉𝐉𝑁𝑁𝑁𝑁 = 𝑖𝑖𝑖𝑖𝑛𝑛0𝑒𝑒 [�̂�𝐭(𝑃𝑃1

⊥𝑃𝑃1∥) + �̂�𝐧 1

23𝑖𝑖+𝑖𝑖𝑖𝑖2𝑖𝑖+𝑖𝑖𝑖𝑖

(𝑃𝑃1⊥)2] (3)

Controlling light with metamaterial-based nonlinear photonic crystals

© 2015 Macmillan Publishers Limited. All rights reserved

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Supplementary Figure 1: Numerical simulation. (a) Illustrates one of the simulated SRR

geometrical configurations. The height of the SRR is 30nm. (b) Currents (arrows) and near-field

radiation patterns (color scale) from a uniform array for FH and (c) SH. The near-field radiation

patterns were taken 100nm above the SRRs. In the simulation we insert the obtained linear

polarization in b into Equation 3 to produce the currents and radiation at the SH shown in c. (d)

Inverted particles produce nonlinear radiation pattern with phase shift. This is the mechanism

used to construct the NLMPCs.

Supplementary Figure 2: Illustration of point-of-view of simulation results shown in Fig. 1

of the manuscript. (a) Simulation results of near-field at the FH and emitted SH for uniform

arrays. (b) Illustration of the simulation point-of-view.


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Supplementary Section II: Two-dimensional nonlinear metamaterials photonic crystals

and nonlinear Raman-Nath diffraction

The ability to control the phase of the nonlinear output is used here to generate one- and two-

dimensional periodic and quasi-periodic nonlinear metamaterial photonic crystals. This is

identical to conventional nonlinear photonic crystals 3,4

in which the manipulation of the material

symmetry is usually obtained by electric field poling of the nonlinear crystal structure or by

special molecular beam epitaxial growth4. In this work this is achieved by fabricating meta-

atoms with inverted symmetry (See Supplementary Figure 3a). As shown above and in the main

text inverted SRRs impose an opposite phase on the nonlinear generation. For example, taking a

periodic 1D structure of the SRRs with inversion period Λ will effectively multiply the generated

SH wave-function with the term 𝑒iΔ𝜙 = 𝑠𝑖𝑔𝑛 (𝑐𝑜𝑠2𝜋𝑥

Λ). By performing a Fourier transform this

modulation is transformed from the real space into the reciprocal space, with discrete RLVs,

𝒌𝛬 = 𝑚2𝜋

𝛬 . The RLVs in conventional nonlinear photonic crystals (NLPCs) are usually designed

to phase-match (or momentum-match) the nonlinear interaction. For example, for a second

harmonic generation process, full momentum conservation (nonlinear Bragg diffraction) leads to

2𝒌𝟏 + 𝑚 ∙ 𝑲𝛬 = 𝒌2, where 𝒌𝟏 is the incident wave vector, 𝑲𝛬 = 𝑚2𝜋

𝛬�̂�𝛬 is RLV with Λ being

the period of grating, and m is an integer number.

In the case of sufficiently thin NLPC (which is the case in the current work) only the

transverse component of nonlinear Bragg’s law 2𝒌∥,1 + 𝑚 ∙ 𝑲𝛬 = 𝒌∥,2 needs to be satisfied. This

regime is the nonlinear Raman-Nath diffraction regime5. For the more general case of oblique

incidence of the pump beam, the SH wave (with a wave vector k2) is emitted at angles of

propagation 𝜃2,𝑚 which satisfy 𝑠𝑖𝑛(𝜃2,𝑚) = 2𝒌1

𝒌2𝑠𝑖𝑛(𝜃1) + 𝑚

𝑲Λ

𝒌2= sin (𝜃1) + 𝑚

𝜆1

2Λ. The momentum

matching diagrams of nonlinear Raman–Nath diffraction are illustrated in Supplementary Figure

3.

Although the NLMPCs that are studied here experimentally have 1D or 2D modulations, the

concept of matching the momentum in the interaction by using RLVs of the NLMPCs can be

extended also to 3D periodic or quasi periodic NLMPCs in analogy to conventional NLPCs4 as

shown below in Supplementary Section IV.


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Supplementary Figure 3: Nonlinear Metamaterials Photonic Crystal and Raman-Nath

diffraction. (a) Exemplified real space and the appropriate reciprocal space of NLMPC with

one-dimensional modulation. (b) Phase-matching conditions for nonlinear Raman–Nath

diffraction for normal and (c) oblique incidence.

Supplementary Section III: Experimental setup

Supplementary Figure 4: Experimental setup. SP- shortpass filter; LP- Longpass filter; P-

polarizer; /2- Half waveplate; OPO- optical parametric oscillator.


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Supplementary Figure 5: Spectral measurement of the SH. Measurement of the SH emitted

from a uniform array which was illuminated with FH with 1200 nm wavelength. The

spectrometer was Andor Shamrock 303i with Newton 970 camera.

Supplementary Section IV: Increasing conversion efficiency by three-dimensional

NLMPCs

The ability to generate materials with artificial nonlinearity is exciting; however so far the

nonlinear metamaterials that were studied were made from extremely thin surfaces. One way to

increase the total nonlinear conversion efficiency is to use advanced fabrication methods6 to

construct three-dimensional metamaterials that provide longer interaction lengths.

However, in the case of multilayered metamaterials there are some considerations that have to

be taken into account, including phase matching of the nonlinear interaction, reflection,

transmission and absorption of the waves in the multilayered structures. These considerations

make the wave dynamics in nonlinear multilayered metamaterials much more complex in

comparison to conventional nonlinear materials.

In the following section we describe the problem in more details, analyze analytically and

numerically the major implications of stacking multiple layers for forming three-dimensional

nonlinear metamaterials and NLMPCs and show that by understanding the fundamental

problems of this configuration the total conversion efficiency can be increased substantially by


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using multilayered structures. In our studies we show about 500-fold increase of the total power

conversion efficiency in comparison to the performance of a single layer resonant SRR structure

which was studied before1. We believe that additional future studies can lead to even better

improvements in the conversion efficiencies of multilayered nonlinear metamaterials.

Quasi-phase-matching:

Momentum mismatch between the waves (for SHG Δ𝑘 = 𝑘𝑆𝐻 − 2𝑘𝐹𝐻), that is caused from

material dispersion, usually leads to accumulated phase mismatch as the waves propagate

resulting in a destructive interference of the generated nonlinear signal. Therefore, without phase

matching, the nonlinear interaction is inefficient even for thick, three-dimensional, nonlinear

materials.

In conventional materials the common solution to the phase matching problem is to use

birefringence7 in uniaxial or biaxial nonlinear crystals, or quasi-phase-matching (QPM) in poled

NLPCs8. NLPCs are widely used to control nonlinear optical interactions since in addition to

generating efficient nonlinear signal, they prevent walk-off of the interacting beams and enable

adjusting other aspects of the interaction such as the direction of the generated beams3 and their

spatial shape9,10

. Moreover, NLPCs can be used for active devices11

and efficient generation of

entangled photon pairs12

. Our work presented in the manuscript demonstrates experimentally

two-dimensional NLMPCs and gives motivation to create three-dimensional NLMPCs and to use

them for QPM and additional manipulations of nonlinear interactions.

Supplementary Figure 6 illustrates three different phase matching schemes in three-

dimensional metamaterials based on multiple SRRs layers. Perfect phase matching can be

achieved by placing different SRR layers of the same orientation spaced at coherence lengths

(𝐿𝑐 = 2𝜋/Δ𝑘 ) or by placing the SRR layers at half coherence lengths and flipping the

orientation of the SRRs (and phase of nonlinear output) in successive layers. QPM can be

achieved by closely packing the SRR layers and flipping different regions of the structures (at

half coherence length distance). This is the same method which is used for construction of

conventional NLPCs where the inversion of (2)

is used to impose π phase shifts on the locally

generated radiation. Using QPM for the multilayered metamaterial structure will result in much

higher conversion efficiency per propagation length, since it allows packing many more SRR

layers in the same region in space.


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Supplementary Figure 6: Illustration of different phase matching configurations with

multilayered nonlinear metamaterials. (a) Perfect phase matching with uniform multilayered

structures. (b) Perfect phase matching with NLMPC. (c) QPM with NLMPC.

As mentioned above, there is a major difference between conventional nonlinear materials

and the nonlinear metamaterials which were studied up to date. Conventional nonlinear materials

that are used to create NLPCs are transparent in the operation wavelength regime, whereas

metamaterials working in the plasmonic resonance regime are not transparent. The linear

interaction of FH and SH waves with each SRR layer involves partial transmittance (TFH, TSH),

reflectance (RFH, RSH) and absorbance. The partial transmittance leads to attenuation of the

propagating FH and generated SH. In addition, in the case of multilayered structures which are

not impedance-matched there are multiple reflections. As a result, depending on the spacing

between the layers, optical micro-cavities can be formed. These factors make the interaction

more complex compared to conventional nonlinear materials. Supplementary Figure 7 illustrates

the problem.


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Supplementary Figure 7: Illustration of the dynamics of propagating waves through the

multilayered structure: As the waves (FH- red; generated SH- green) propagate they are

discretely attenuated. In addition, a portion of the waves is reflected back and forth between the

layers, resulting in a cavity behavior (dashed lines) for the FH and SH independently.

Numerical study:

To study numerically the total generation of SH out of multiple layers of SRRs we use a

combination of finite element (FEL) simulation and a transfer matrices method13

. The parameters

that were taken from the FEL simulation are the linear transmission and reflection from each

SRR layer for the FH and the SH, and the nonlinear generation coefficient (using the

hydrodynamic model). We took into account both forward and backward generated SH waves.

The simulation process is illustrated and described in detail in Supplementary Figure 8.


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Supplementary Figure 8: Illustration of simulation process for N layers: FEL simulation

produces linear transmission and reflection coefficients and the generated SH fields from each

single layer. The linear coefficients are used in the transfer matrix formulation to calculate the

FH everywhere in the multilayered structure. The local FH in the nth

layer is used to calculate the

locally generated SH in the nth

layer. The locally generated SH is fed to the SH transfer matrix

formulation which takes into account all the layers before and after the locally generated SH and

calculates the contribution of the locally generated SH in each layer to the output SH. The total

output SH is calculated as the sum of contributions of the locally generated SH from each layer

after interacting with the entire multilayered structure.

Results of numerical study:

We started by examining the SHG from a single layer of SRRs with dimensions and unit-cell

size which were studied before by Ciraci et al.1, in a more realistic configuration for a

multilayered structure where the structure is embedded in silica glass14

. This added dispersion to

the problem (and need for phase matching) and led to a red shift of the resonances. The single

layer transmittance and reflectance of the FH (polarized along the base of the SRRs) and SH

(polarized along the arms of the SRRs) in this case are shown as the blue lines (F=1) in

Supplementary Figure 9a-d.


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Supplementary Figures 9 a-d: Spectral transmittance and reflectance from a single layer of

SRRs. Spectral transmittance of FH (a) and SH (b). Spectral reflectance of FH (c) and SH (d).

The FH is polarized along the base of the SRRs and the SH is polarized along the arms of the

SRRs. F is a scaling factor of the SRR dimensions (while maintaining the unit-cell size at 180

nm × 180 nm).

In this case (F=1), in agreement with previous works1,15

, the highest SHG correlates to the

resonances and to the dip in the transmission. This is shown as the blue line in supplementary

figure 9a. However due to the strong reflection and absorption at these specific geometrical

parameters (as shown in Supplementary Figure 9a-d) the power of light passing to the next layer

decreases drastically, and the overall efficiency of a multilayered structure is lower than that of a

single layer. Therefore in order to be able to accumulate SH through the multilayered structure

the transmittance of light through layers needs to be increased.

One way to do it is to use smaller SRRs. In addition to providing better transmittance, the

larger field enhancements of smaller SRRs enhance their effective conversion efficiency. It was

suggested before to use tiny nonlinear meta-atoms that exhibit strong effective conversion and


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pack them densely to obtain efficient total conversion, however it was shown that there is a limit

on the packing density due to collective effects which reduce the overall conversion efficiency16

.

Therefore, as we show next, the route to harness the maximal total conversion efficiencies of

multilayered structures is indeed to use layers of small nonlinear meta-atoms which are not

densely packed. This will cancel the collective effects that reduce the effective conversion and in

addition increase the layer transmittance and allow accumulation of the nonlinear signal.

Supplementary Figure 10: Normalized SHG power. (a) From single layer. (b) Maximal total

SHG from multilayered structure. The results are normalized to the maximum of the F=1 (SRR

dimensions studied by Ciraci et al1) single layer SHG. N denotes the number of layers that

yielded maximal SHG.

We studied the case of multilayered structures from layers where the SRRs dimensions were

rescaled to half (F=0.5) and quarter (F=0.25) of the original SRR dimensions (F=1).

Supplementary Figure 9 shows that the resonances become slightly blue shifted and that the total

transmission is substantially increased in the cases of F=0.5 (red) and F=0.25 (cyan). The

corresponding generated SH power from a single layer normalized to the case of a single (F=1)

SRR layer is shown in Supplementary Figure 10a. It can be seen that F=0.5 produces the

strongest SH fields from a single layer and that by making the particles smaller (F=0.25) the

SHG field that radiates from a single layer is reduced. However in the case of multilayered

structure which is phase matched (see Supplementary Figure 6a) the total SH is the strongest for

the case of F=0.25 where the SHG is about 500 times stronger than the starting case of F=1. The

maximal SHG values for F=1, F=0.5 and F=0.25 were obtained for a single layer, 3 phase

matched layers and 21 phase matched layers respectively.


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The simulated accumulation through the multilayered structure for F=0.25 is shown in

Supplementary Figure 11. The FH wavelength in the simulation was 1750 nm and Lc in the silica

matrix was 83 m. In the case of perfect phase matching the layers were placed at Lc separations.

In the case of non-phase matched and QPM cases the layers were placed in separations of Lc/8 in

uniform orientations and in alternating orientation respectively (alternating orientation

configuration is shown in Supplementary Figure 6c). The buildup of the SH waves with respect

to the number of layers resembles the known dynamics for perfect phase matching, QPM and

non-phase matched nonlinear interactions17

with added losses due to the partial transmittance.

Supplementary Figure 11b shows the SH field accumulation in perfect phase matching and QPM

stacked layer configurations (shown as configurations a b and c in Supplementary Figure 6). It

can be seen that in the QPM case the maximal total SHG is slightly reduced in comparison to the

perfect phase matched cases, however it’s maximum is obtained much faster which means that it

can be used to significantly reduce the total dimensions of the converting device.

Supplementary Figure 11: Phase matching with multilayered metamaterials. Total SHG

output field for F=0.25 (a) as function of number of layers and (b) as function of propagation

length in the structure.

From the obtained results we can understand that multilayered structures can be used to

significantly enhance the efficiency from nonlinear metamaterials. The best efficiency will be

achieved for small particles which are not too closely packed to avoid collective effects. QPM by

alternating the orientation of the structures is highly important since it allows getting efficient

interaction in much shorter lengths compared to placing layers at Lc or Lc/2. We believe that this


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study gives the motivation to use QPM in three-dimensional metamaterials and opens many new

directions to study nonlinear dynamics in three-dimensional metamaterials which need further

investigations and are out of the scope of this work.

Analytical calculations for the case of low reflection (small particles)

In order to analyze analytically the effect of propagation through multilayered structure we

consider the case of zero reflections which simplifies the picture. Therefore, we can express the

value of the propagating SH wave at the n-th layer, as described in the illustration in

Supplementary Figure 12

Supplementary Figure 12: Illustration of the effect of the n-th layer on the SHG wave. E0 is

the input FH field; tFH and tSH are the FH and SH transmission coefficients respectively; 𝜂 is the

non-linear conversion coefficient.

The total SHG field output (ESH,N) is the sum over the contributions from each of the layers (after

attenuation through the structure), giving the following expression:

2

2 2 2

, 0

2

1

1

N

SH

FHN

SH N FH

SH

FH

t

tE E t

t

t

In order to express the propagation through the dispersive medium, we add the accumulated

propagation phase of the interlayer distance L to each of the transmission coefficients and get the

following expression:


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2

2 2 2 22

, 0

2

1 e

e

1 e

FH

N

iN kLSH

N i N k L FH

SH N FH

i kLSH

FH

t

tE E t

t

t

Where

2SH FHk k k

As shown by the formula for the total SHG field output, and as described above, the interlayer

distance plays a major role as it comes to phase matching. Supplementary Figure 13a shows the

total SHG field output as a function of the inter-layer distance, as calculated by the simulation

(green line) for a 10 layers structure of SRRs with F=0.25, FH wavelength of 1750 nm and by

the analytical formula (blue line). The maximal values are achieved at LC (83 µm) or when the

layers are at the same position which is physically impossible. This is due to the phase mismatch

accumulated through the dispersive medium. In addition, up to the highest value at separation of

83 µm, 10 peaks are visible, matching the 10 layers in the structure. This is another consequence

of phase matching, but as a result of constructive interference only from a combination of

specific layers which are separated by multiples of the coherence length. Another important

feature is the modulation at the FH and SH wavelengths in the medium (shown as the

fluctuations in the green line), which is caused by the multiple reflections in the structures. This

feature does not appear at the analytical solution which does not take into account the reflections.

Supplementary Figure 13b shows the accumulation of SH field in a perfectly phase matched

structure with respect to the number of layers. The simulation produces higher results since it is

taking into account the multiple reflections, which contribute the total generation process in this

case.


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Supplementary Figure 13: Comparison between simulation and analytical results for small

particles (F=0.25) (a) Total SHG dependence of a structure of 10 layers of SRRs with uniform

orientations on inter-layer distance. (b) SH field accumulation Vs. layer number in a perfect

phase matched structure.

Supplementary Section V: Illustrations of the SH radiation emitted from the NLMPC

based FZP

Each pair of adjacent zones of the NLMPC-based FZP consists of SRRs with mirror inversion

symmetry and generate SH with an opposite phase. The generated SH creates constructive

interference propagates in two main directions, as illustrated in Supplementary Figure 14a,

corresponding to m=1 and m=-1 diffraction orders.

One diffraction order focuses at 1 mm in front of the lens whereas the other defocuses and

creates an imaginary focus 1 mm behind the lens as illustrated at Supplementary Figure 14b. In

addition the nonlinear FZP emits focused and defocused waves also backwards.


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Supplementary Figure 14: Emission of nonlinear FZP (a) Simulation of far-field SH emission

from a one-dimensional NLMPC on glass. The asymmetry of emission is due to emission to air

on the right and glass on the left. (b) Illustration of the NLMPC-based FZP SH radiation pattern

showing forward and backward focusing and defocusing of nonlinear emission.

References:

1. Ciracì, C., Poutrina, E., Scalora, M. & Smith, D. R. Origin of second-harmonic generation

enhancement in optical split-ring resonators. Phys. Rev. B 85, 201403 (2012).

2. Scalora, M. et al. Second- and third-harmonic generation in metal-based structures. Phys.

Rev. A 82, 043828 (2010).

3. Berger, V. Nonlinear Photonic Crystals. Phys. Rev. Lett. 81, 4136–4139 (1998).

4. Arie, A. & Voloch, N. Periodic, quasi-periodic, and random quadratic nonlinear photonic

crystals. Laser Photon. Rev. 4, 355–373 (2010).

5. Sheng, Y., Kong, Q., Wang, W., Kalinowski, K. & Krolikowski, W. Theoretical

investigations of nonlinear Raman–Nath diffraction in the frequency doubling process. J.

Phys. B At. Mol. Opt. Phys. 45, 055401 (2012).

6. Soukoulis, C. M. & Wegener, M. Past achievements and future challenges in the

development of three-dimensional photonic metamaterials. Nat. Photonics 5, 523–530

(2011).

7. Armstrong, J., Bloembergen, N., Ducuing, J. & Pershan, P. Interactions between Light

Waves in a Nonlinear Dielectric. Phys. Rev. 127, 1918–1939 (1962).


17

8. Fejer, M. M., Magel, G. A., Jundt, D. H. & Byer, R. L. Quasi-phase-matched second

harmonic generation: tuning and tolerances. IEEE J. Quantum Electron. 28, 2631–2654

(1992).

9. Ellenbogen, T., Dolev, I. & Arie, A. Mode conversion in quadratic nonlinear crystals. Opt.

Lett. 33, 1207–9 (2008).

10. Ellenbogen, T., Voloch-Bloch, N., Ganany-Padowicz, A. & Arie, A. Nonlinear generation

and manipulation of Airy beams. Nat. Photonics 3, 395–398 (2009).

11. Ellenbogen, T., Ganany-Padowicz, A. & Arie, A. Nonlinear photonic structures for all-

optical deflection. Opt. Express 16, 3077–82 (2008).

12. Tanzilli, S. et al. Highly efficient photon-pair source using periodically poled lithium

niobate waveguide. Electron. Lett. 37, 26 (2001).

13. Saleh, B. E. A. & Teich, M. C. in Fundam. Photonics 243–288 (Wiley, 2007).

14. Bass, M. et al. Handbook of Optics, Third Edition Volume IV: Optical Properties of

Materials, Nonlinear Optics, Quantum Optics (set): Optical Properties of Materials,

Nonlinear Optics, Quantum Optics (set). 1152 (McGraw Hill Professional, 2009).

15. Klein, M. W., Enkrich, C., Wegener, M. & Linden, S. Second-harmonic generation from

magnetic metamaterials. Science 313, 502–4 (2006).

16. Linden, S. et al. Collective Effects in Second-Harmonic Generation from Split-Ring-

Resonator Arrays. Phys. Rev. Lett. 109, 015502 (2012).

17. Boyd, R. W. Nonlinear Optics. Nonlinear Opt. 69–133 (Elsevier, 2008).

doi:10.1016/B978-0-12-369470-6.00002-2


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