Absolute Values Choy Mm and Byer Rl (1976) Physical Review B14 1693

Second-Order Nonlinear Optical Susceptibility

Nonlinear Optical Properties

Georges H. Wagnière , Stanislaw Woźniak , in Encyclopedia of Spectroscopy and Spectrometry (Third Edition), 2017

Noncentrosymmetric Crystals

Inorganic crystals are widely applied for second-harmonic generation and for optical parametric processes. Some ofttimes used materials: for 2d-harmonic generation from the near-IR into the visible and across KH₂PO₄ (KDP), KD₂PO_iv (KD*P) and more recently β-BaB_twoO₄(BBO). For optical parametric amplification into the mid-IR: AgGaSe₂, GaSe; the visible and about-IR: LiNbO₃, KTiOPO₄, KNbO₃; and into the visible and UV: β-BaB₂O_four and LiB₃O₅.

Table 8 shows experimental second-gild nonlinear optical susceptibilities for dissimilar tensor components d _il and various fundamental wavelengths. The quantities d _il are defined every bit follows:

Tabular array 8. Experimental second-order nonlinear optical susceptibilities d _il of inorganic crystals

Materials	Symmetry		dil(ten−12chiliad   V−1)	dilλ(μm)	Reference
Quartz (α-SiOii)	32=Dthree	d 11	0.46	one.06	a
		d xiv	0.009	ane.0582	b
LilO3	half-dozen=C6	d 31	6.43	2.12	a
			7.xi	1.06
			8.14	0.6943
		d 33	six.41	2.12
			half dozen.75	i.318
			seven.02	one.06
LiNbO3	iiiyard=C3v	d 31	five.77	1.15	a
			five.95	1.06
		d 33	29.1	2.12
			31.8	1.318
			34.four	1.06
		d 22	3.07	i.0582	b
KNbO3	mm2=C2v	d 31	−15.eight	1.064	g
		d 32	−18.three
		d 33	−27.4
		d 24	−17.one
		d xv	−16.5
BatwoNaNbfiveOxv	mm2=C2v	d 31	−14.55	1.0642	b
		d 32	−14.55
		d 33	−20
BaTiO3	ivmm=C4v	d fifteen	−17.two	1.0582	b
		d 31	−eighteen
		d 33	−6.half dozen
NH4H2PO4(ADP)	4̄2m=D2d	d 14	0.48	0.6943	b
		d 36	0.485
KH2PO4(KDP)	4̄2grand=D2d	d xiv	0.49	one.0582	b
		d 36	0.599	1.318	a
			0.630	1.06
			0.712	0.6328
KDiiPOfour(KD∗P)	4̄2m=D2d	d fourteen	0.528		c
		d 36	0.528
GaP	4̄3thousand=Td	d 14	35	3.39	b
		d 36	58.i	ten.vi	a
			77.5	2.12
			99.7	1.06
GaAs	4̄31000=Td	d xiv	188.5	10.6	b
		d 36	151	10.six	a
			173	2.12
AgGaSe2	4̄21000=D2d	d 36	57.7	x.6	a
			67.7	ii.12
AgSbS3	threem=C3v	d 31	12.6		c
		d 22	13.four
Ag3AsSiii	3m=C3v	d 31	15.i		c
		d 22	28.v
CdS	6mm=C6v	d 33	36.0		c
		d 31	37.7
		d 36	41.9
CdSe	half dozenmm=C6v	d xv	31	10.six	b
		d 31	28.5
		d 33	55.3	10.6	a
			65.4	ii.12
Te	32=D3	d 11	5×x3	10.half dozen	b
β-BaB2O4(BBO)	3m=C3v	d xi	ane.6	ane.064	d
		d 22,d 31	<0.08
LaBGeO5Nd3+		d eff	0.296	1.064	e
KTiOPOiv	mm2=C2v	d 15	1.91	1.064	h
(KTP)		d 24	iii.64
		d 31	2.54
		d 32	4.35
		d 33	16.ix
RbTiOPO4	mm2=C2v	d 15	half dozen.1	i.064	f
		d 24	7.6
		d 31	6.5
		d 32	five.0
		d 33	thirteen.seven
ZnO	6mm=C6v	d 31	2.ane	one.0582	b
		d 15	4.3
		d 33	−7.0

a

Absolute values: Choy MM and Byer RL (1976) Concrete Review B14: 1693.

b

Shen Yr (1984) The Principles of Nonlinear Optics. New York: Wiley.

c

Boyd RW (1992) Nonlinear Optics. Boston: Academic Press.

d

Eimerl D, Davis L, Velsko S, Graham EK, and Zalkin A (1987) Journal of Applied Physics 62: 1968.

e

For type I stage-matched SHG; Capmany J and Garcia Sole J (1997) Applied Physics Messages seventy: 2517.

f

Zumsteg FC, Bierlein JD, and Gier TE (1976) Journal of Applied Physics 47: 4980.

chiliad

Biaggio I, Kerkoc P, Wu L-South, Günter P, and Zysset P (1992) Journal of the Optical Club of America B9: 507.

h

Vanherzeele H and Bierlein JD (1992) Optics Letters 17: 982.

[13] ${\chi }_{\mathit{ijk}}^{\left(\mathrm{ii}\right)}\left(2\omega \right)=\mathrm{two}{d}_{\mathit{ijk}}\left(2\omega \right)$

The second and tertiary indices of d _ijk are and so replaced by a single symbol 50 according to the piezoelectric contraction:

jk:	11	22	33	23,32	31,thirteen	12,21
fifty:	one	2	iii	4	5	6

The nonlinear susceptibility tensor can and then be represented as a iii×6 matrix containing 18 elements. In the transparent region, i.e. outside of absorption bands, i may assume the validity of the Kleinman symmetry condition, which states that the indices i, j, one thousand may exist freely permuted:

[14a] $d_{\mathit{ijk}}=d_{\mathit{kij}}=d_{\mathit{jki}},\text{etc}.$

1 and then finds, for example,

[14b] $d_{123}=d_{321},d_{\mathrm{fourteen}}=d_{36}$

In this example in that location are simply 10 independent elements for d _il .

Table 8 shows that the values for d _il may vary over several orders of magnitude, and that it is not necessarily the crystals with the highest values that are nigh commonly used. The technical applicability is partly also determined past other qualities, such every bit phase-matching properties, ease of crystal growth, mechanical strength, chemic inertness, temperature stability and low-cal-impairment threshold.

A quantity often used to characterize the optical properties of nonlinear optical materials is the Miller alphabetize:

[xv] ${\delta }_{\mathit{ijk}}=\frac{d_{\mathit{ijk}}\left(2\omega \right)}{{\chi }_{\mathit{two}}^{\left(1\right)}\left(2\omega \right){\chi }_{\mathit{jj}}^{\left(\mathrm{one}\right)}\left(\omega \right){\chi }_{\mathit{kk}}^{\left(1\right)}\left(\omega \right)\cdot {\epsilon }_0}$

where χ ^(one)(2ω) represents the linear susceptibility for the doubled frequency 2ω, χ ⁽ⁱ⁾(ω) that for the central frequency ω. One finds that for nigh materials δ is non far from a mean value of almost 2×ten⁻ⁱⁱ  yardⁱⁱ  C⁻¹, suggesting that in a given substance nonlinear and linear susceptibilities are closely related.

Noncentrosymmetric crystals show other properties in addition to frequency conversion, for case the linear electro-optic or Pockels upshot: the linear change of the refractive index induced past an applied DC electric field. Furthermore, the signal groups C _n and C _nv allow for the existence of a permanent electric dipole moment. Indeed, crystals such equally LiNbO₃ (C_3v) and BaTiO_three (C_4v) are well known for their ferroelectric backdrop. Crystals transforming co-ordinate to point groups containing only rotations, such as C _n , D _n , T and O, are chiral and therefore optically active. In Table 8 we find quartz α-SiO₂(D₃), LiIO₃(C₆) and Te(D₃).

Read full affiliate

URL:

https://world wide web.sciencedirect.com/science/article/pii/B9780128032244002326

TOMOGRAPHY | Tomography and Optical Imaging

Z. Chen , in Encyclopedia of Modernistic Optics, 2005

Second Harmonic OCT

SH-October combines SHG with coherence gating for high-resolution tomographic imaging of tissue structures with molecular contrast. SHG is the lowest-guild nonlinear optical procedure where the second-order nonlinear optical susceptibility is responsible for the generation of light at second-harmonic frequency. Because second-order nonlinear optical susceptibility is very sensitive to electronic configurations, molecular structure and symmetry, local morphology, and ultrastructures, SHG provides molecular contrast for the coherence epitome.

The schematic diagram for SH-Oct is shown in Figure 14. A broadband short pulse laser is used every bit a light source in a Michelson interferometer. The lite is split into the reference arm and the sample arm. In the reference arm, a thin nonlinear crystal is used to convert the input radiation to SH photons. Both SH and key waves are and then reflected by a metal mirror (M1) mounted on a motorized translation stage, which acts as the delay line in this SH-OCT system. A backscattered SH moving ridge from the sample recombines with the SH moving ridge from the reference arms to form interference fringe. The SH interference fringe signal is detected by a photomultiplier tube (PMT) after passing through a brusk-pass filter (F1). The fundamental interference fringe signal is detected by a photodiode (PD) afterwards passing through a long-pass filter (F2).

Figure 14. Schematic of experiment ready-up for SH-Oct. M1, mirror; NLC, nonlinear crystal; BS, broadband nonpolarization beamsplitter; DBS, dichroic beamsplitter; L1, lens; F1–F2, filters; PD, photograph diode; PMT, photomultiplier tube.

A SH-OCT prototype of a rat tendon is shown in Figure xv. Compared with conventional October performed at fundamental wavelengths, SH-OCT offers enhanced molecular contrast and spatial resolution. It is as well an improvement over existing SHG scanning microscopy technology equally the intrinsic coherence gating mechanism enables the detection and discrimination of SH signals generated at deeper locations. The enhanced molecular contrast of SH-OCT extends conventional OCT's adequacy for detecting small changes in molecular structure. SH-OCT is promising for the diagnosis of cancers and other diseases at an early stage when changes in tissue and molecular structure are minor.

Effigy xv. SH-OCT image of a rat tendon; image size 250   μm×250 μm. Scale bar 100 μm. Prototype reprinted with permission from Jiang Y et al. (2004) Proceedings of the 26th Almanac International Conference of the IEEE Engineering in Medicine and Biology Order.

Read total chapter

URL:

https://world wide web.sciencedirect.com/science/article/pii/B0123693950007193

Nonlinear Optical Materials

J. Xu , ... M.L. Fischer , in Reference Module in Materials Science and Materials Engineering, 2016

1.1 Noncentrosymmetric Crystals

Insulating crystals form an of import class of 2d-order nonlinear optical materials. It is well established that merely crystals that lack a center of inversion symmetry can possess a nonvanishing second-order nonlinear optical susceptibility. This requirement limits the option of crystals to those of certain symmetry classes. An additional requirement on materials properties is gear up by the fact that second-club nonlinear optical processes can occur with good efficiency only if a standard phase matching status is satisfied. This condition requires that the spatial variation of the nonlinear polarization be synchronous with that of the generated field, or mathematically that Δ k=k ₃−k ₂−k ₁ be much smaller than the changed of the length, L, of the interaction region. Hither thou ₃ is the wavevector of the highest frequency moving ridge, and m ₁ and k _two are those of other waves. Because of the frequency dependence (dispersion) of the refractive indices, the phase matching condition has often been satisfied by using birefringent materials and past allowing the birefringence to compensate for dispersion. However, not all crystals with big 2d-order nonlinearities possess birefringence adequately large for this method to be used, and thus phase matching past ways of birefringence imposes further restrictions on the choice of crystals for use in 2d-order NLO. The optical backdrop of some of import crystals for use in second-order NLO are reviewed in Tabular array 1. More than extensive lists of 2nd-order NLO crystals and their properties tin can be found in standard reference works (Sutherland, 1996), in manufacturers' specifications (due east.thou., Cleveland Crystals Inc., Cleveland, OH, provides data sheets, which may too be obtained at http://www.clevelandcrystals.com), and in survey books (Nikogosyan, 2005).

Table 1. Properties of various second-society nonlinear optical materials

Crystal (class)	Transmission range (µm)	Refractive index (at 1.06   µm)	Nonlinear coefficient (pmV−one)	Impairment threshold (GWcm−ii)
Silver gallium selenide, AgGaSe2 ( $\overline{4}$ 2m)	0.78–18	n o=2.7010	d 36=33 (at ten.vi   µm)	0.25 for ten   ns
		n e=ii.6792
β-Barium borate, BBO (3thousand)	0.21–2.1	n o=1.6551	d 22=2.3	4.6 for 1   ns, 15 for 0.1   ns
		northward e=ane.5425	d 24=d 15⩽0.1
Lithium iodate, LiIOiii (half-dozen)	0.31–5	north o=one.8517	d 31=−7.11	~0.5
		northward e=one.7168	d 33=–seven.02
			d 14=0.31
Lithium niobate, LiNbO3 (iiim)		due north o=2.234	d 31=−5.95
		n east=ii.155	d 33=−34.4
Potassium dihydrogen phosphate, KHiiPOfour (KDP)	0.18–ane.55	n o=ane.4944	d 36=0.63
		due north eastward=one.4604
KTiOPO4, KTP (mmii)	0.35–4.5	due north x =1.7367	d 31=half dozen.five
		n y =1.7395	d 32=5.0
		n z =1.8305	d 33=13.vii
			d 24=6.six
			d 15=6.one

From a diversity of sources. By convention, $d=\frac{1}{\mathrm{two}}{\chi }^{\left(3\right)}$ . The tensor nature of the nonlinear coefficients is expressed in contracted annotation, in which the first index of d _il represents whatsoever of the three Cartesian indices and the second index, l, represents the product of two Cartesian indices according to the rule l=1 implies xx,=2 implies yy,=3 implies zz,=four implies yz or zy,=v implies zx or xz, and=half dozen implies xy or yx. To convert d _il to the Gaussian c.g.s. units of cmstatvolt⁻¹, each entry should exist divided past 4.189×ten^−iv.

Metal–organic frameworks (MOFs), a novel type of organic–inorganic hybrid material which has well-defined geometry of metal centers and highly directional metal–ligand coordination interactions, have been extensively engineered as noncentrosymmetric crystals for second-order NLO, as reviewed in Wang et al. (2012a,b). Due to the porous nature of MOFs, their bridging ligands can be rationally designed to reversibly change their configurations which result in the on/off NLO switch in the solid country. For instance, MOFs NH₂-MIL-53(Al) (Serra-Crespo et al., 2012) and (Hdabco⁺)(CF₃COO⁻) (Dominicus et al., 2013) exhibit loftier NLO switch contrasts of 38 and 35, respectively.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128035818024048

SPECTROELECTROCHEMISTRY

J.A. Crayston , in Encyclopedia of Analytical Science (2d Edition), 2005

2d Harmonic Generation

When an intense light beam is directed to a noncentrosymmetric crystal then light may be reflected not only at the incident frequency, ω, only besides at the 2nd harmonic frequency twoω. Such second harmonic generation (SHG) is an case of nonlinear behavior, and is the basis for optical components such as frequency doublers. Since a surface or interface is inherently noncentrosymmetric, low-cal from the electrode–liquid interface may show evidence for SHG (Figure 5 ). The intensity of the SHG lite is dependent on the square of the incident intensity, and also on the foursquare of the second-order nonlinear optical susceptibility, which in plough depends on the disproportion of the interface. The event is both angle and wavelength dependent. Although the effect is weak, typically x ⁻¹²% of the incident intensity, requiring pulsed laser excitation and gated detection methods, the fact that the iiω signal can only arise from the surface region lends the technique the advantage of surface specificity and blindness to the bulk solution. The twoω signal intensity is sensitive to the presence of absorbed species and, in particular, the rearrangement of adsorbed molecules (e.g., CO) or ions on the single-crystal electrode surfaces. It is also not particularly demanding of the nature of the metallic electrode (in contrast to SERS or SPR) then metals such as platinum or mercury can be used. Furthermore, since the excitation source is visible calorie-free, less light is captivated by the solvent and so longer path length cells can be used. In improver to simply measuring the intensity of the SHG beam as a function of potential or time, information technology is possible to derive molecular orientation information from the polarization dependence of the intensity, and symmetry data (if dissimilar from the surface) by rotating the sample or aeroplane of polarization well-nigh the surface normal.

Figure 5. Schematic SHG setup. Key: BS=beamsplitter. L/2=one-half-wave plate, P=polarizer, L=lens, F=filter, MC=monochromator, PMT=photomultiplier tube, GI=gated integrator, DA=data acquisition, PD=photodiodes). (Reprinted with permission from Marrucci L, Paparo D, Cerrone G, et al. (2002) Optics and Lasers in Engineering 37: 601–610. © 2002, with permission from Elsevier.). The jail cell mostly uses a reflection cell similar to that in Figure two with a Pigeon prism (Figure 4A (middle)).

Read full affiliate

URL:

https://www.sciencedirect.com/scientific discipline/article/pii/B0123693977005720

Excited-Country Dynamics of Organic Dyes at Liquid/Liquid Interfaces

G. Licari , E. Vauthey , in Encyclopedia of Interfacial Chemical science, 2018

Introduction

Ubiquitous in nature, interfaces between 2 immiscible liquids play crucial roles in many areas of sciences and technology, going from biological processes at membranes, to mass transfer across interfaces or solar energy conversion. In chemistry, several organic reactions of hydrophobic molecules have been plant to be more efficient in the presence of h2o, pointing to the involvement of organic solvent/water interfaces. Despite the relevance of liquid interfaces, our cognition about their backdrop and about how the chemical reactivity of molecules located in these anisotropic environments differ from that of molecules dissolved in majority media is still very limited. Ane of the main reasons is that nearly spectroscopic techniques are not interface-specific and, thus, the interfacial response is totally hidden by the majority response. I style to overcome this trouble is to mensurate the second-guild nonlinear optical susceptibility , ${\overleftrightarrow{\chi }}^{\left(2\right)}$ , which vanishes in centrosymmetric materials but differs from zero at the interface between two isotropic media. Moreover, ${\overleftrightarrow{\chi }}^{\left(\mathrm{ii}\right)}$ depends on the frequency of the incoming optical fields and exhibits resonances when these frequencies coincide with one- or two-photon transitions of the cloth. The virtually common nonlinear optical techniques for measuring ${\overleftrightarrow{\chi }}^{\left(2\right)}$ at interfaces are surface second-harmonic generation (SSHG) and surface sum-frequency generation (SSFG). In general, SSHG is performed in the visible–near-IR range and is thus sensitive to electronic resonances, whereas SSFG is rather carried out with one of the optical fields in the IR region and is thus used to record interfacial vibrational spectra. To be SSH(F)G active, molecules should fulfill two weather: (1) to be noncentrosymmetric, that is to have a nonzero hyperpolarizability, $\overleftrightarrow{\beta }$ , and (2) to have an anisotropic orientation at the interface, that is to have a nonzero ${\overleftrightarrow{\chi }}^{\left(2\right)}$ .

Here, we focus on the time-resolved (TR) SSHG technique, which allows measuring the dynamics of photoinduced processes at interfaces. As shown below, TR-SSHG can exist used to obtain information on the backdrop of liquid interfaces, such as friction or hydrogen bonding. It can be also used to investigate how the excited-state dynamics of a molecule, hence its photochemical reactivity, may change when going from the bulk stage to the interface.

Read full affiliate

URL:

https://world wide web.sciencedirect.com/science/commodity/pii/B9780124095472132652

Second harmonic generation (SHG) as a label technique and phenomological probe for organic materials

K.D. Vocalist , Y. Wu , in Handbook of Organic Materials for Optical and (Opto)electronic Devices, 2013

16.v.3 Orientational dynamics

Poled polymer systems usually showroom orientational relaxation which decreases the NLO susceptibility over time. The ultimate applicability of polymers for 2d-club nonlinear optics may depend on the ability to produce polymers whose orientation has sufficient stability for extended employ in photonic and electronic systems.

In the poled polymer movie, although the flick is already in solid country when the external field is removed, the molecular orientation relaxes because it is in a metastable drinking glass state. In gild to empathize the relaxation mechanisms, SHG has been used as a probe (Hampsch et al., 1990). In fact, the relaxing SHG from such system can be used in studying the dynamics of the molecules (Singer and King, 1991; Sugihara et al., 1996; Dureiko et al., 1998; Herman and Cline, 1998 ). Early studies on the isothermal decay of the 2nd-club nonlinear optical susceptibility of a guest–host azo–dye (disperse ruddy 1) poly methy methacrylate (PMMA) flick show that a mutual underlying process involving a distribution of local relaxations governs the film behavior during temperature changing ( Singer and King, 1991).

The relaxation of orientation of chromophores in a polymer is closely related to the general relaxation mechanisms in the polymers, which have been the subject of much report using both mechanical and dielectric spectroscopy. Nonlinear optics provides some other approach to these spectroscopies, which straight probes the orientational dynamics of the NLO chromophores by second harmonic generation. These measurements, in the frequency domain are chosen chielectric spectroscopy (Sugihara et al., 1996; Dureiko et al., 1998; Herman and Cline, 1998). Temperaturedependent chielectric spectra provide information on the relevant time and energy scales for relaxation processes and were establish to be consistent with previously derived models describing relaxation in thermoplastic polymers. The time dependence of the isothermal decay is modeled by use of chromophore-reorientation models. The statistical physics is applied in deriving the temperature dependence of the chromophore-reorientation parameters (Dureiko et al., 1998). In practice, the Dissado–Hill (D-H) model was chosen for the chromophore reorientation and the Adam– Gibbs model for the relaxation. The temperature dependence of the parameters that describe the chromophore reorientation is obtained through a fit to the isothermal fourth dimension dependent information. Chielectric spectroscopy almost the drinking glass transition temperature of the host polymer has also been shown and agreement between theory and experiment was constitute (Dureiko et al., 1998).

Photorefractive (PR) polymers are potentially interesting for a number of image processing and display technologies (Christenson et al., 2010). A general requirement for a PR material is that the material must showroom a linear electro-optic consequence, that is, that the material lacks inversion symmetry (Moerner et al., 1997). The figure of merit of PR material is the diffraction efficiency. The centrosymmetry of the polymer is broken through applying an electrical field above the drinking glass transition temperature of the polymer resulting electro-optic response. NLO chromophores will rotate inside the polymer liquid. The Debye rotational diffusion model was adjusted to describe the reorientation of chromophores, which can exist treated as a rigid dipolar molecule in an isotropic medium. Both the EFISHG and FWM (4-wave mixing) experimental data show that onset (i.e., external field turned on) in the diffraction efficiency has contributions from photoconductive (induces charge distribution, i.east. electric field) and molecular reorientation (breaks symmetry) (Ostroverkhova et al., 2002). The EFISHG indicate is quadratically proportional to the electric field. Thus, the average signal is sensitive to space charge distribution. Polymers with dissimilar glass transition temperature were studied with EFISHG technique (Ostroverkhova et al., 2002).

Read full chapter

URL:

https://www.sciencedirect.com/science/commodity/pii/B9780857092656500168

Molecular crystals and crystalline thin films for photonics

M. Jazbinsek , P. Günter , in Handbook of Organic Materials for Optical and (Opto)electronic Devices, 2013

half-dozen.iii.one THz-wave generation by difference-frequency mixing

Generating THz waves by optical difference-frequency generation (DFG) requires a pump source consisting of two frequencies ω _i and ω _two that are very close to each other, so that their deviation frequency lies in the THz range: ω _THz = ω _{i +} ω ₂. The phase matching condition should exist also satisfied

[6.6] $\Delta \mathbf{thousand}={\mathbf{k}}_{\mathrm{THz}}-\left({\mathbf{k}}_1-{\mathbf{grand}}_{\mathrm{ii}}\right).$

For collinear DFG and assuming that the optical frequencies are close together, so that in start approximation the dispersion in the optical range tin be considered as due north ₂ = n_ane + (∂due north/∂λ)Δλ, Δλ = λ _two − λ _i, this leads to

[6.7] $\Delta \mathit{g}=\frac{{\omega }_{\mathrm{THz}}}{c}\left(n_{\mathrm{THz}}-n_g\right)$

and the following coherence length for THz generation

[6.8] ${\mathrm{50}}_c=\frac{{\lambda }_{\mathrm{THz}}}{2\left(n_{\mathrm{THz}}-n_g\right)},$

where northward_grand = n − (∂n/∂λ)λ is the group alphabetize of the optical wave. Equation (6.8) is valid for a relatively small dispersion in the optical range, i.e. up to several THz if we use infrared pump calorie-free. For larger THz frequencies, it should be calculated as

[6.9] $l_c=\frac{1}{\mathrm{two}}\left(\frac{n_{\mathrm{THz}}}{{\lambda }_{\mathrm{THz}}}-\frac{{\mathrm{due\; north}}_{\mathrm{ane}}}{{\lambda }_1}+\frac{{\mathrm{northward}}_2}{{\lambda }_2}\right).$

For the efficient generation of THz waves by difference-frequency generation, besides a high second-order nonlinear optical susceptibility, the most important parameter is the low refractive index mismatch ∆ n = n_THz − due north_{one thousand} between the generated THz and the pump optical waves. This is where organic materials are of a big advantage compared to standard inorganic materials such as LiNbO₃. Because of the relatively depression contribution of the lattice phonon vibrations to the dielectric constant, the dispersion of the refractive index between the optical and the THz frequency range is low and therefore the stage matching condition is most naturally satisfied, while for inorganic materials such as LiNbO3 special phase matching configurations are needed.

For DFG in the case of stage-matching and neglecting the pump-low-cal assimilation, the visible-to-THz conversion efficiency is given past (Sutherland, 2003)

[6.10] ${\eta }_{\mathrm{THz}}=\frac{{\omega }_{\mathrm{THz}}^2{d}_{\mathrm{THz}}^{\mathrm{two}}{L}^{\mathrm{ii}}{I}_0}{2{\epsilon }_0{c}^3{n}_0^2{n}_{\mathrm{THz}}}exp\left(-{\alpha }_{\mathrm{THz}}\mathrm{50}/\mathrm{two}\right)\frac{{sinh}^{\mathrm{two}}\left({\alpha }_{\mathrm{THz}}L/4\right)}{{\left({\alpha }_{\mathrm{THz}}L/4\right)}^2},$

is the nonlinear optical susceptibility for THz-wave generation, ω _THz the angular frequency of the generated THz moving ridge, L the length of the THz-generation materials, I ₀ the pump intensity, α _THz the absorption constant at the THz frequency, r the electro-optic coefficient, n ₀ and due north _THz the refractive indices at the pump optical and the generated THz frequencies, respectively. Too phase-matching and minimal THz assimilation, the main material figure of merit for THz generation (FM_THz) co-ordinate to (Eq. half-dozen.10) is

[6.12] ${\mathrm{FM}}_{\mathrm{THz}}=\frac{d_{{}_{\mathrm{THz}}}^2}{n_0^{\mathrm{ii}}{n}_{{}_{\mathrm{THz}}}}=\frac{n_0^6{r}^2}{\mathrm{sixteen}{\mathrm{north}}_{{}_{\mathrm{THz}}}}.$

Tabular array 6.ii shows most of these parameters for a serial of inorganic and organic crystals, equally well every bit for an electro-optic polymer. As it can exist seen in this tabular array, the organic crystals OH1, DSTMS and OH1 show the largest figure of merit and tin can be besides phase matched using pump lasers at telecommunication wavelengths i.3-1.55   μm. OH1 in addition shows a very minor absorption constant at THz frequencies, thus allowing large interaction lengths to be used (Brunner et al., 2008). The optical damage threshold of these organic crystals mainly depends on the optical quality, both of the bulk crystal quality and the quality of surface polishing. Very slow cooling growth with loftier temperature stability of (±   0.002 °C) has to be used for high damage threshold materials reaching I _damage > 150 GW/cmⁱⁱ for 150   fs pulses at 1550   nm (Rainbow Photonics, 2012).

Table half dozen.ii. Organic and inorganic nonlinear optical materials that have been investigated for optical-to-THz frequency conversion and their most relevant parameters*. Where possible, the parameters close to the velocity-matched optical wavelengths and THz frequencies are given

	due northo	due northm	northTHz	r (pm/V)	d THz 1 (pm/V)	FMTHz 2 (pm2/V2)	vphonon (THz)	αTHz(cm—   1)	λ (nm)
DAST	2.13	2.3 iii	2.26	47	240	5600	22	twenty	1500
DSTMS	2.13	2.3	2.26	49	250	6100	22	15	1500
OH1	two.16	2.33	2.28 4	52	280	7400	8	two six	1350
LAPC 5	1.half-dozen	ane.8	1.7	52	85	1700	>   17	15	1500
GaAs	3.37	3.61	3.63	one.6	52	66	7.6	0.v	1560
ZnTe	2.83	two.18	3.16	4	64	160	5.three	1.3	840
InP	iii.two	3.16	3.54 6	ane.45	38	40	10
GaP	3.12		three.34	ane	24	17	10.8	0.2	1000
ZnS	ii.3		2.88	1.5	10	vii	9.eight
CdTe	2.82		3.24	6.8	110	470		four.8
LiNbO3	2.two	2.18	iv.96	28	160	1100		17

1

$d_{\mathrm{Thz}}=\frac{\mathrm{i}}{4}{n}_0^{4}r$ .

2

${\mathrm{FM}}_{\mathrm{THz}}=\frac{d_{\mathrm{THz}}^2}{n_0^2{\mathrm{northward}}_{\mathrm{THz}}}=\frac{n_0^{\mathrm{vi}}{r}^2}{\mathrm{xvi}{n}_{\mathrm{THz}}}$

3

five > one.5 THz.

iv

ν < i.9 THz.

5

LAPC guest-host polymer (Zheng et al., 2007).

vi

ν ≈ 1 THz

*

Refractive index north_o at the pump optical wavelength λ; group index n_g at λ; refractive index northward _THz in the THz frequency range; the electro-optic coefficient r; the susceptibility d _THz for THz-moving ridge generation; effigy of merit FM_THz for THz generation by optical rectification; optical phonon frequency of the cloth v _phonon in the THz range; the absorption α _THz in the THz frequency range.

Read full affiliate

URL:

https://www.sciencedirect.com/scientific discipline/article/pii/B9780857092656500065

Amorphous and Glassy Semiconducting Chalcogenides

M. Frumar , ... G.1000. Sujan , in Reference Module in Materials Science and Materials Engineering, 2016

6.four Nonlinear Optical Effects

These effects are large in AGC containing highly polarizable atoms or ions (Frumar et al., 2003a,b; Ticha and Tichy, 2002). The effect is strong also in chalcogenide crystals; the AGC have some advantage over crystals in terms of lower cost, easier processing, and in the possibility to include them into optical cobweb systems and planar circuits.

The refractive alphabetize, n, can exist expressed every bit northward=n ₀+northward ₂〈Eastward ²〉, where north ₀≫n ₂. The 〈Due east ²〉 is the mean square of electric field of the calorie-free. The nonlinear part of index of refraction, due north ₂, is connected with nonlinear electron polarizability, P _NL:

[34] $P={\mathrm{\chi }}^{\left(1\right)}\mathrm{East}+P_{\mathrm{NL}}$

where

[35] $P_{\mathrm{NL}}={\mathrm{\chi }}^{\left(\mathrm{ii}\right)}{E}^2+{\mathrm{\chi }}^{\left(3\right)}{\mathrm{Due\; east}}^3$

where P is the polarizability, χ⁽¹⁾ is the linear optical susceptibility, χ⁽²⁾ and χ⁽³⁾ are 2nd- and third-club nonlinear optical susceptibilities that are related to electronic and nuclear structure of the material. For crystals with a center of symmetry and also for optically isotropic glasses, the second-gild nonlinear susceptibility χ⁽²⁾ is naught (Lines, 1990). The third-society susceptibility χ⁽ⁱⁱⁱ⁾ is of import because its existent and imaginary parts give ascent to the nonlinear refractive alphabetize and nonlinear absorption coefficient. The tertiary-order nonlinearity of most transparent materials results from anharmonic terms of the polarization of leap electrons. The fast component of northward _two and χ⁽³⁾ is believed to ascend from pure electronic effects. The nonlinear effects are influenced too by nuclear contributions. The nuclear contribution gives just 12–13% of nonlinearities in sulfide glasses as it was evaluated in Kang et al. (1996). The response of nuclei is much slower and the ii effects can be separated.

The covalency or ionicity of chemical bonds strongly influences the magnitude of nonlinearity. Single crystals of highly covalent Ge reached surprising χ⁽ⁱⁱⁱ⁾=ten⁻⁹  esu, while highly ionic materials such equally NaF just ix.4×x^−fourteen  esu.

The linear optical susceptibility in an isotropic medium can be expressed past (Ticha and Tichy, 2002):

[36] ${\mathrm{\chi }}^{\left(\mathrm{one}\right)}=\left(n^{\mathrm{two}}-\mathrm{i}\right)/\mathit{4}\pi$

The dependence of (due north ²–one)^−one on (ℏω)^two is oft linear and this dependence can exist described past the unmarried oscillator formula of Wemple and DiDomenico as it was already mentioned:

[37] $n^{\mathrm{ii}}\left(\omega \right)-1={\mathrm{East}}_0{E}_d/\left({\mathrm{Eastward}}_0^2-{\left(\hslash \omega \right)}^2\right)$

From eqns [36] and [37], we obtain

${\mathrm{\chi }}^{\left(\mathrm{one}\right)}=\frac{{\mathrm{East}}_{\mathrm{d}}{E}_0}{4\pi \left(E_0^2-{\left(\hslash \omega \right)}^2\right)}$

For long wavelengths (ℏω→0), and so χ^(one)=Eastward _d /fourπE ₀.

The χ⁽³⁾ as a dominant nonlinearity in all glassy materials is produced by excitation in the transparent frequency region, well below the band gap E _grand ^opt. The χ⁽³⁾ of AGC can be adamant experimentally by several methods, for case, by degenerate 4-moving ridge mixing method, past Z-scan method, past 3rd-harmonic generation, etc. (Zakery and Elliott, 2007). The χ⁽³⁾ tin be also roughly evaluated from linear refractive alphabetize (n ₀), or from optical susceptibility χ^(ane) (see east.chiliad., Nasu et al., 1994, and references therein). The use of physically-based Milleŕs rule is one of the user-friendly ways for rough evaluation of χ^(three), especially for virtually-infrared frequencies. In the spectral region far from resonance, the χ⁽³⁾ is approximately equal to χ^(three)=A(χ ⁽ⁱ⁾)⁴; then:

[38] ${\mathrm{\chi }}^{\left(\mathrm{iii}\right)}=A{\left[{\mathrm{East}}_{\mathrm{d}}{E}_0/4\pi {\left(E_0^2-\hslash \omega \right)}^2\right]}^{\mathrm{iv}}$

for ℏ ω→0, 1 obtains

[39] ${\mathrm{\chi }}^{\left(\mathrm{iii}\right)}=\frac{A}{{\left(4\pi \right)}^4}{\left(\frac{E_{\mathrm{d}}}{E_0}\right)}^4=\frac{A}{{\left(\mathrm{four}\pi \right)}^4}{\left(n_0^2-1\right)}^4$

where n ₀ is the linear alphabetize of refraction for ℏ ω→0. The mean value of the constant A, as evaluated from 97 experimentally found values, is nearly ane.vii×x⁻¹⁰ (for χ^(three) in esu) and χ⁽³⁾=half-dozen.eight×10⁻¹⁵(Eastward _d /E ₀)^four (esu) (Ticha and Tichy, 2002). The accuracy of Milleŕs rule is generally improve than the order of magnitude for many covalent or ionic compounds. For many halides, oxides, and sulfides values calculated past Milleŕs rule agree with experimental values within a factor of ii.

The value of χ⁽³⁾ for As_iiS₃ glass was found to be χ^(three)=(one.48–2.2)×x⁻¹²  esu (measured by different methods), while for GeS₂ glass χ⁽³⁾=one×10⁻¹²  esu, and for SiO_ii glass χ^(three)=two.8×10⁻¹⁴  esu for λ=1900   nm (Vogel et al., 1991). The nonlinear susceptibility χ⁽³⁾ of chalcogenide glasses was therefore found to exist several orders of magnitude college than those of SiO₂ glass. The AGCs have generally loftier values of due north ₀; they are promising candidates for optical switching and other nonlinear applications. The high nonlinear role of the index of refraction of AGC tin be used in active parts of optical integrated circuits.

Very prospective nonlinear optical materials are Air conditioning containing nanoparticles of Ag, Au, CdS, CdSe, CdTe, PbS, CuCl, etc. Their third-order optical nonlinear susceptibility can be much higher than in pure AGC.

The nonlinear effects in AGC could have, yet, a limitation for their application, because loftier intensities of low-cal should be used for nonlinear effects. High light intensities can induce formation of intrinsic defects that lower the transparency. When the AGCs are used in their loftier-transparency IR region, the formation of such effects tin can be negligible.

Read total chapter

URL:

https://www.sciencedirect.com/science/commodity/pii/B9780128035818098040

Second harmonic generation in chalcogenide glasses

Five. Nazabal , I. Kityk , in Chalcogenide Spectacles, 2014

sixteen.2.1 Bones notions of nonlinear polarization

The nonlinear optical processes express the property of dielectric materials to achieve, under an intense electromagnetic field, a polarization that depends nonlinearly on the electric field amplitude (Courtois, 1996). The polarization plays a cardinal role in the description of nonlinear optical phenomena, considering a time-varying polarization acting as the source of new components to the electromagnetic field. Under the activeness of the electromagnetic field of a laser wave, aquiver at a frequency of about 10^xiii–x¹⁵  Hz, dielectric charges are subject to frequency oscillation forming a set of oscillating dipoles. The effect of magnetic field on charged particles is, in plow, much smaller and can exist neglected. In improver, due to the large mass of positive ions as compared to that of electrons, ane can consider with good approximation that electron movements only must be taken into account. Thus, in the presence of such electric field, the charged particles of the dielectric fabric, strongly related to each other although they are linked with some 'elasticity', show just transitory movements and deviate slightly from their original positions. This state of affairs results in induced electrical dipole moments, i.e. polarization. With pulsed light amplification by stimulated emission of radiation, the applied electrical field can reach an aamplitude comparable to feature inter-atomic electric field strength that is typically greater than 100–10⁴ MV/chiliad. The usual notion of linear optical response characterized by a constant refractive index ceases to exist valid. In near cases, the physical origin of nonlinear backdrop is strongly affected, at the microscopic level of dielectric materials, by the nature of atoms or molecules, of bond blazon and orientation, of crystalline structure, etc. In the perturbation limit, where the laser field magnitude is even so minor compared with that of the intra-atomic field, the resulting polarization including nonlinear phenomena tin be described as a power of field strength, $\overrightarrow{E}\left(\omega \right)$ which is of the form:

[16.1] $\overrightarrow{P}={\epsilon }_0\left({\chi }^{\left(1\right)}\overrightarrow{\mathrm{East}}\left(\omega \right)+{\chi }^{\left(2\right)}\overrightarrow{\mathrm{Due\; east}}\left(\omega \right)\overrightarrow{E}\left(\omega \right)+{\chi }^{\left(\mathrm{iii}\right)}\overrightarrow{E}\left(\omega \right)\overrightarrow{E}\left(\omega \right)\overrightarrow{\mathrm{East}}\left(\omega \right)+\dots \right)$

where the constant of proportionality, χ ⁽¹⁾, is the linear susceptibility, and χ ⁽²⁾ and χ ⁽³⁾ are the second- and third-order nonlinear optical susceptibilities, respectively, (χ⁽ⁿ⁾ is an n  +   1 order tensor), ε₀ is the permittivity of complimentary space and the electrical field force Eastward(ω) is oscillating at the pulsation ω.

The quadratic polarization ${\overrightarrow{P}}^{\left(2\right)}={\epsilon }_0{\chi }^{\left(2\right)}\overrightarrow{E}\left(\omega \right)\overrightarrow{E}\left(\omega \right)$ causes singular effects, essential for mixing wave phenomena, based on the generation of sum and difference frequencies (Butcher and Cotter, 1990). As a general dominion, the aamplitude of the nonlinear polarization component oscillating at pulsation ω₃ is defined by the human relationship:

[16.2] $P_i^{\left(\mathrm{two}\right)}\left({\mathrm{\omega }}_3\right)=\frac{{\mathrm{\epsilon }}_0}{\mathrm{two}}{\mathrm{\chi }}_{\mathrm{ijk}}^{\left(2\right)}\left(-\mathrm{\omega };{\mathrm{\omega }}_{\mathrm{one}},{\mathrm{\omega }}_{\mathrm{ii}}\right)E_{\mathrm{j}}\left({\mathrm{\omega }}_1\right){\mathrm{East}}_{\mathrm{one\; thousand}}\left({\mathrm{\omega }}_2\right)$

These wave-mixing phenomena lead to many applications such as tunable sources, parametric amplifiers and oscillators. In the process of difference frequency generation (Fig. 16.i), the presence of radiations at frequency ω₂ or ω₃ tin can stimulate the emission of additional photons at these frequencies. Optical parametric oscillators based on this issue are mainly used as tunable infrared sources.

Fig. xvi.1. Scheme of difference frequency generation and optical parametric oscillator.

The process of 2d harmonic generation (SHG) can exist described by considering a nonlinear fabric characterized past a 2nd-order susceptibility assumed to exist purely real, which is illuminated past a plane wave of pulsation ω and amplitude E(ω) propagating along the z centrality. The electrical field associated with the optical wave, without taking into account the polarization of the field, is of the form:

[sixteen.3] $\mathrm{Eastward}\left(z,t\right)=E\left(\omega \right){\mathrm{east}}^{\left(-\mathit{i\omega t}+i{\mathrm{chiliad}}_{\omega }z\right)}+c.c$

In the special case of SHG (where ω ₂  = ω ₁  = ω), and considering Eq.[16.1], the polarization is composed of ii terms. The get-go term is given by Eq.[16.four] with a contribution at 2ω frequency and the 2d is given by Eq.[xvi.5] with a contribution at zero frequency:

[sixteen.4] $P_i^{\left(2\right)}\left(2\omega \right)=\frac{{\epsilon }_0}{\mathrm{two}}{{\chi }_{\mathit{ijk}}}^{\left(2\right)}\left(-2\omega ;\omega ,\omega \right)E_j\left(\omega \right)E_{\mathrm{k}}\left(\omega \right)$

This component of the polarization is at the origin of the frequency-doubling procedure or SHG. The generation of 2nd harmonic at pulsation of 2ω was showtime observed past Franken et al. (Franken, Weinreich et al., 1961); their work opened the era of experimental nonlinear eyes.

[16.5] $P_i^{\left(2\right)}\left(0\right)=\frac{{\epsilon }_0}{2}{{\chi }_{\mathit{ijk}}}^{\left(\mathrm{two}\right)}\left(0;\omega ,-\omega \right)E_j\left(\omega \right)E_{\mathrm{k}}\left(\omega \right)$

Equation[16.5] is associated with the phenomenon of optical rectification, resulting in the cosmos of a static electrical field across the nonlinear material.

The components of the susceptibility tensor exhibit invariance in the permutation of j and k indices as both electrical fields associated to the optical wave are indistinguishable. Since the medium is necessarily lossless when the applied field frequencies, ω_i , are very much smaller than the resonance frequency ω ₀, Kleinman has shown that the χ^(two) tensor is symmetric with respect to total permutations of the three indices, ijk (Kleinman, 1962a, 1962b). This condition is valid when dispersion of the susceptibility can be neglected. The components of the polarization at iiω in the Cartesian coordinate system (ten, y, z) can be written in reduced matrix class:

[16.6] $\begin{array}{l}\left[\begin{array}{cc}\hfill p_x^{\left(2\right)}\hfill & \hfill \left(2\omega \right)\hfill \\ \hfill \begin{array}{l}{p}_y^{\left(2\right)}\\ p_z^{\left(\mathrm{two}\right)}\end{array}\hfill & \hfill \begin{array}{l}\left(2\omega \right)\\ \left(2\omega \right)\end{array}\hfill \end{array}\right]=\frac{{\epsilon }_0}{2}\left[\begin{array}{ccc}\hfill {\chi }_{111}\hfill & \hfill {\chi }_{122}\hfill & \hfill {\chi }_{133}\hfill \\ \hfill {\chi }_{211}\hfill & \hfill {\chi }_{222}\hfill & \hfill {\chi }_{233}\hfill \\ \hfill {\chi }_{311}\hfill & \hfill {\chi }_{322}\hfill & \hfill {\chi }_{333}\hfill \end{array}\begin{array}{ccc}\hfill {\chi }_{123}\hfill & \hfill {\chi }_{113}\hfill & \hfill {\chi }_{112}\hfill \\ \hfill {\chi }_{223}\hfill & \hfill {\chi }_{213}\hfill & \hfill {\chi }_{212}\hfill \\ \hfill {\chi }_{323}\hfill & \hfill {\chi }_{313}\hfill & \hfill {\chi }_{312}\hfill \end{array}\right]\\ \left[\begin{array}{c}\hfill E_x^2\left(\omega \right)\hfill \\ \hfill {\mathrm{Due\; east}}_y^2\left(\omega \right)\hfill \\ \hfill \begin{array}{l}{E}_z^2\left(\omega \right)\\ 2{\mathrm{Eastward}}_y\left(\omega \right){\mathrm{East}}_z\left(\omega \right)\\ 2E_x\left(\omega \right){\mathrm{East}}_z\left(\omega \right)\\ 2{\mathrm{East}}_{\mathrm{10}}\left(\omega \right)E_y\left(\omega \right)\end{array}\hfill \end{array}\right]\end{array}$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780857093455500167

reardonfichames1940.blogspot.com

Source: https://www.sciencedirect.com/topics/chemistry/second-order-nonlinear-optical-susceptibility