Nonlinear optics - Knowledge (XXG)

4097:, which led the domination of the lower orders, does not converge anymore and instead a time based model is used. When a noble gas atom is hit by an intense laser pulse, which has an electric field strength comparable to the Coulomb field of the atom, the outermost electron may be ionized from the atom. Once freed, the electron can be accelerated by the electric field of the light, first moving away from the ion, then back toward it as the field changes direction. The electron may then recombine with the ion, releasing its energy in the form of a photon. The light is emitted at every peak of the laser light field which is intense enough, producing a series of 6162:

chromophore π-systems, adjusting bond length alternation, inducing intramolecular charge transfer, extending conjugation in 2D, and engineering multipolar charge distributions. Recently, many novel directions were proposed for enhanced nonlinearity and light manipulation, including twisted chromophores, combining rich density of states with bond alternation, microscopic cascading of second-order nonlinearity, etc. Due to the distinguished advantages, molecular nonlinear optics have been widely used in the biophotonics field, including bioimaging, phototherapy, biosensing, etc.

6281: 3614:). There are several schemes of choosing the polarizations for this crystal type. If the signal and idler have the same polarization, it is called "type-I phase matching", and if their polarizations are perpendicular, it is called "type-II phase matching". However, other conventions exist that specify further which frequency has what polarization relative to the crystal axis. These types are listed below, with the convention that the signal wavelength is shorter than the idler wavelength. 31: 4498: 4204: 3873: 6131: 974: 2700: 1505: 6262: 2620: 3848:. This results in the polarization response of the crystal to be shifted back in phase with the pump beam by reversing the nonlinear susceptibility. This allows net positive energy flow from the pump into the signal and idler frequencies. In this case, the crystal itself provides the additional wavevector 5969: 5303:

Note that the constant of proportionality between the signal and conjugate beams can be greater than 1. This is effectively a mirror with a reflection coefficient greater than 100%, producing an amplified reflection. The power for this comes from the two pump beams, which are depleted by the process.

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wave. In effect, all three incident beams interact (essentially) simultaneously to form several real-time holograms, resulting in a set of diffracted output waves that phase up as the "time-reversed" beam. In the language of nonlinear optics, the interacting beams result in a nonlinear polarization

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axes. In those crystals, type-I and -II phase matching are usually the most suitable schemes. In positive uniaxial crystals, types VII and VIII are more suitable. Types II and III are essentially equivalent, except that the names of signal and idler are swapped when the signal has a longer wavelength

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depends on the polarization and direction of the light that passes through. The polarizations of the fields and the orientation of the crystal are chosen such that the phase-matching condition is fulfilled. This phase-matching technique is called angle tuning. Typically a crystal has three axes, one

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The early studies of nonlinear optics and materials focused on the inorganic solids. With the development of nonlinear optics, molecular optical properties were investigated, forming molecular nonlinear optics. The traditional approaches used in the past to enhance nonlinearities include extending

4477:. In this case, the interacting beams simultaneously interact in a nonlinear optical material to form a dynamic hologram (two of the three input beams), or real-time diffraction pattern, in the material. The third incident beam diffracts at this dynamic hologram, and, in the process, reads out the 734: 805: 6276:

Certain molecular materials have the ability to be optimized for their optical nonlinearity at the microscopic and bulk levels. Due to the delocalization of electrons in π bonds electrons are more easily responsive to applied optical fields and tend to produce larger linear and nonlinear optical

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that is not parallel to the propagation vector. This would lead to beam walk-off, which limits the nonlinear optical conversion efficiency. Two other methods of phase matching avoid beam walk-off by forcing all frequencies to propagate at a 90° with respect to the optical axis of the crystal.

6149:. This has been observed both in photo-refractive, photonic lattices, as well as photo-reactive systems. In the latter case, optical nonlinearity is afforded by reaction-induced increases in refractive index. Examples of pattern formation are spatial solitons and vortex lattices in framework of 6873:

Bula, C.; McDonald, K. T.; Prebys, E. J.; Bamber, C.; Boege, S.; Kotseroglou, T.; Melissinos, A. C.; Meyerhofer, D. D.; Ragg, W.; Burke, D. L.; Field, R. C.; Horton-Smith, G.; Odian, A. C.; Spencer, J. E.; Walz, D.; Berridge, S. C.; Bugg, W. M.; Shmakov, K.; Weidemann, A. W. (22 April 1996).

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Note: Parametric generation and amplification is a variation of difference-frequency generation, where the lower frequency of one of the two generating fields is much weaker (parametric amplification) or completely absent (parametric generation). In the latter case, the fundamental

1341: 1092: 3950:. In common scenarios, such as mixing in dilute gases, the non-linearity is weak and so the light beams are focused which, unlike the plane wave approximation used above, introduces a pi phase shift on each light beam, complicating the phase-matching requirements. Conveniently, 4158:(necessary to obtain phase matching, see below), having a specific crystal symmetry, being transparent for both the impinging laser light and the frequency-doubled wavelength, and having high damage thresholds, which makes them resistant against the high-intensity laser light. 5551: 1632: 1285: 6375:

Similar highly conjugated chromophores with strong donor-acceptor characteristics are used due to their large difference in the dipole moments, and current efforts in extending their pi-conjugated systems to enhance their nonlinear optical properties are being made.

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The other method is quasi-phase-matching. In this method the frequencies involved are not constantly locked in phase with each other, instead the crystal axis is flipped at a regular interval Λ, typically 15 micrometres in length. Hence, these crystals are called

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of the nonlinear material is not changed by the interaction with the optical field. As a consequence of this, the process is "instantaneous". Energy and momentum are conserved in the optical field, making phase matching important and polarization-dependent.

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Gu, Bobo; Wu, Wenbo; Xu, Gaixia; Feng, Guangxue; Yin, Feng; Chong, Peter Han Joo; Qu, Junle; Yong, Ken-Tye; Liu, Bin (2017). "Precise Two-Photon Photodynamic Therapy using an Efficient Photosensitizer with Aggregation-Induced Emission Characteristics".

6181: 2144: 5713: 5433: 6126:{\displaystyle {\begin{aligned}\mathbf {P} _{\text{out}}&=-\hbar \mathbf {k} =-\mathbf {P} _{\text{in}}=\hbar \mathbf {k} ,\\{L_{\mathbf {z} }}_{\text{out}}&=-\hbar \ell =-{L_{\mathbf {z} }}_{\text{in}}=\hbar \ell .\end{aligned}}} 3596: 5626: 3348: 3823:

One undesirable effect of angle tuning is that the optical frequencies involved do not propagate collinearly with each other. This is due to the fact that the extraordinary wave propagating through a birefringent crystal possesses a

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acts as a driver/source of the electromagnetic waves. One of the consequences of this is a nonlinear interaction that results in energy being mixed or coupled between different frequencies, which is often called a "wave mixing".

969:{\displaystyle \nabla \times \nabla \times \mathbf {E} +{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =-{\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}},} 5299:

that is, the generated beam amplitude is the complex conjugate of the signal beam amplitude. Since the imaginary part of the amplitude contains the phase of the beam, this results in the reversal of phase property of the effect.

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of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >10 V/m and thus comparable to the atomic electric field of ~10 V/m) such as those provided by

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Yuan, Yufeng; Lin, Yining; Gu, Bobo; Panwar, Nishtha; Tjin, Swee Chuan; Song, Jun; Qu, Junle; Yong, Ken-Tye (2017). "Optical trapping-assisted SERS platform for chemical and biosensing applications: Design perspectives".

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Thaury, C.; Quéré, F.; Geindre, J.-P.; Levy, A.; Ceccotti, T.; Monot, P.; Bougeard, M.; Réau, F.; d’Oliveira, P.; Audebert, P.; Marjoribanks, R.; Martin, Ph (1 June 2007). "Plasma mirrors for ultrahigh-intensity optics".

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Basker, Dinesh K.; Brook, Michael A.; Saravanamuttu, Kalaichelvi (2015-09-03). "Spontaneous Emergence of Nonlinear Light Waves and Self-Inscribed Waveguide Microstructure during the Cationic Polymerization of Epoxides".

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means reversal of both polarization state and orbital angular momentum. Reversal of orbital angular momentum of optical vortex is due to the perfect match of helical phase profiles of the incident and reflected beams.

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is true in general, even for an isotropic medium. However, even when this term is not identically 0, it is often negligibly small and thus in practice is usually ignored, giving us the standard nonlinear wave equation:

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or two of which have a different refractive index than the other one(s). Uniaxial crystals, for example, have a single preferred axis, called the extraordinary (e) axis, while the other two are ordinary axes (o) (see

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Biria, Saeid; Malley, Philip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-03-03). "Tunable Nonlinear Optical Pattern Formation and Microstructure in Cross-Linking Acrylate Systems during Free-Radical Polymerization".

4109:. The laser must be linearly polarized, so that the electron returns to the vicinity of the parent ion. High-order harmonic generation has been observed in noble gas jets, cells, and gas-filled capillary waveguides. 1500:{\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} ={\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}.} 5294: 6363: 1008: 4137:

Practically, frequency doubling is carried out by placing a nonlinear medium in a laser beam. While there are many types of nonlinear media, the most common media are crystals. Commonly used crystals are BBO

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is nonzero, something that is generally true in any medium without any symmetry restrictions; in particular resonantly enhanced sum or difference frequency mixing in gasses is frequently used for extreme or

3852: = 2π/Λ (and hence momentum) to satisfy the phase-matching condition. Quasi-phase-matching can be expanded to chirped gratings to get more bandwidth and to shape an SHG pulse like it is done in a 3672: 2719:

as a function of wavelength (or increases as a function of frequency). This makes phase matching impossible in most frequency-mixing processes. For example, in SHG, there is no simultaneous solution to

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Burgess, Ian B.; Shimmell, Whitney E.; Saravanamuttu, Kalaichelvi (2007-04-01). "Spontaneous Pattern Formation Due to Modulation Instability of Incoherent White Light in a Photopolymerizable Medium".

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Biria, Saeid; Malley, Phillip P. A.; Kahan, Tara F.; Hosein, Ian D. (2016-11-15). "Optical Autocatalysis Establishes Novel Spatial Dynamics in Phase Separation of Polymer Blends during Photocuring".

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cancels this focal phase shift and often has a nearly self-canceling overall phase-matching condition, which relatively simplifies broad wavelength tuning compared to sum frequency generation. In

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Temperature tuning is used when the pump (laser) frequency polarization is orthogonal to the signal and idler frequency polarization. The birefringence in some crystals, in particular

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The most common way of producing optical phase conjugation is to use a four-wave mixing technique, though it is also possible to use processes such as stimulated Brillouin scattering.

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Green Fluorescent Protein (GFP) chromophore p-hydroxybenzylideneimidazolinone (HBDI) used in nonlinear bioimaging is an example of a pi-conjugated donor-acceptor (D-π-A) chromophore.

7286: 6257:{\displaystyle {\boldsymbol {\mu }}={\boldsymbol {\mu _{0}}}+\alpha \cdot {\boldsymbol {\mathrm {E} }}+{\frac {1}{2}}\beta :{\boldsymbol {\mathrm {E} }}{\boldsymbol {\mathrm {E} }}} 2752: 1132: 6277:

responses than those in single (𝜎) bonds. In these systems linear response scales with the length of the conjugated pi system, while nonlinear response scales even more rapidly.

4467: 4396: 4270: 2615:{\displaystyle {\begin{aligned}\mathbf {P} ^{\text{NL}}&=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t)\\&={\frac {\varepsilon _{0}}{4}}\chi ^{(2)}\left,\end{aligned}}} 5950: 4501:

Comparison of a phase-conjugate mirror with a conventional mirror. With the phase-conjugate mirror the image is not deformed when passing through an aberrating element twice.

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It is possible, using nonlinear optical processes, to exactly reverse the propagation direction and phase variation of a beam of light. The reversed beam is called a

5895: 3512: 165:(THG), generation of light with a tripled frequency (one-third the wavelength), three photons are destroyed, creating a single photon at three times the frequency. 8561: 8014:

Gu, Bobo; Zhao, Chujun; Baev, Alexander; Yong, Ken-Tye; Wen, Shuangchun; Prasad, Paras N. (2016). "Molecular nonlinear optics: recent advances and applications".

5559: 3175: 127:. However, some nonlinear effects were discovered before the development of the laser. The theoretical basis for many nonlinear processes was first described in 8331: 6768:

Lewis, Gilbert N.; Lipkin, David; Magel, Theodore T. (November 1941). "Reversible Photochemical Processes in Rigid Media. A Study of the Phosphorescent State".

5762: 4693: 3049: 6292:, are used as biomarkers for two-photon spectroscopy, in which the attenuation of incident light intensity as it passes through the sample is written as 1752: 729:{\displaystyle \mathbf {P} (t)=\varepsilon _{0}\left(\chi ^{(1)}\mathbf {E} (t)+\chi ^{(2)}\mathbf {E} ^{2}(t)+\chi ^{(3)}\mathbf {E} ^{3}(t)+\ldots \right),} 7128: 3853: 8650: 5963:. The photon reflected from phase conjugating-mirror (out) has opposite directions of linear and angular momenta with respect to incident photon (in): 226: 104:

for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at

6372:. The resulting signal adopts a Lorentzian lineshape with a cross-section proportional to the difference in dipole moments of ground and final states. 6733: 4651:{\displaystyle \Xi _{j}(\mathbf {x} ,t)={\frac {1}{2}}E_{j}(\mathbf {x} )e^{i\left(\omega _{j}t-\mathbf {k} \cdot \mathbf {x} \right)}+{\text{c.c.}},} 1870: 7573: 5556:

which is a linear momentum density of electromagnetic field. In the same way a phase-conjugated wave has an opposite angular momentum density vector

159:, generation of light with a doubled frequency (half the wavelength), two photons are destroyed, creating a single photon at two times the frequency. 6820:

Chen, Szu-yuan; Maksimchuk, Anatoly; Umstadter, Donald (17 December 1998). "Experimental observation of relativistic nonlinear Thomson scattering".

5008:{\displaystyle P_{\omega }={\frac {1}{2}}\chi ^{(3)}\varepsilon _{0}E_{1}E_{2}E_{3}^{*}e^{i(\omega t-\mathbf {k} \cdot \mathbf {x} )}+{\text{c.c.}}} 7102: 5095: 2800:

The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by

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One of the many applications of molecular nonlinear optics is the use in nonlinear bioimaging. These nonlinear materials, like multi-photon

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than the idler. For this reason, they are sometimes called IIA and IIB. The type numbers V–VIII are less common than I and II and variants.

6517: 1087:{\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} } 8139: 4093:

of light in air, in which the light travels without dispersion or divergence in a self-generated waveguide. At even high intensities the

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Bai, Zhenxu; Wang, Yulei; Lu, Zhiwei; Yuan, Hang; Jiang, Li; Tan, Tan; Liu, Zhaohong; Wang, Hongli; Cui, Can; Hasi, Wuliji (2016-10-01).

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is implemented via stimulated Brillouin scattering, four-wave mixing, three-wave mixing, static linear holograms and some other tools.

1742: + 1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization 6804: 8324: 7844: 4687:

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ, this produces a nonlinear polarization field:

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In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes:

8016: 186: 5546:{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t),} 2676: 1627:{\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =\mathbf {0} ,} 1516: 1280:{\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =\mathbf {0} .} 366: 180: 6566: 4828:

As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω

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Molecular nonlinear optics relate optical properties of bulk matter to their microscopic molecular properties. Just as the

6487: 2120:{\displaystyle \mathbf {E} (t)={\frac {1}{2}}E_{1}e^{-i\omega _{1}t}+{\frac {1}{2}}E_{2}e^{-i\omega _{2}t}+{\text{c.c.}},} 1178: 792: 202: 3119: 515: 357:(XPM), where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect. 8317: 6933:

Koga, J.; Esirkepov, T.Z.; Bulanov, S.V. (2005). "Nonlinear Thomson scattering in the strong radiation damping regime".

3861: 143:, polarization, phase or path of incident light. These nonlinear interactions give rise to a host of optical phenomena: 6280: 5307:

The frequency of the conjugate wave can be different from that of the signal wave. If the pump waves are of frequency ω

189:(OPA), amplification of a signal input in the presence of a higher-frequency pump wave, at the same time generating an 8614: 6471: 6390: 5839: 4401: 4398:. Because of conservation laws PC mirror experiences recoil: the vortex phonon (orange) with doubled linear momentum 4333: 4147: 498: 35: 5806: 4210: 1515:

The nonlinear wave equation is an inhomogeneous differential equation. The general solution comes from the study of

177:(SFG), generation of light with a frequency that is the sum of two other frequencies (SHG is a special case of this). 8445: 8418: 8268: 6613: 6557: 6385: 4101:

light flashes. The photon energies generated by this process can extend past the 800th harmonic order up to a few K

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Hernández-Acosta, M A; Soto-Ruvalcaba, L; Martínez-González, C L; Trejo-Valdez, M; Torres-Torres, C (2019-09-17).

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Reversal of wavefront means a perfect reversal of photons' linear momentum and angular momentum. The reversal of

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can be converted to visible light, with wavelengths of 532 nm (green) or 400 nm (violet) respectively.

3951: 2919:{\displaystyle E_{j}(\mathbf {x} ,t)=E_{j,0}e^{i(\mathbf {k} _{j}\cdot \mathbf {x} -\omega _{j}t)}+{\text{c.c.}}} 2757: 1720:{\displaystyle {\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}} 1300: 416: 171:(HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater). 162: 7593:

Soljacic, Marin (2000-01-01). "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media".

199:(OPO), generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input). 8551: 7250: 6875: 6369: 6146: 4106: 3054: 2672: 405: 208: 174: 168: 3840:

is highly temperature-dependent. The crystal temperature is controlled to achieve phase-matching conditions.

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owing to the nonlinear medium amplifying spatial and temporal noise. The effect is referred to as optical

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Zhang, Silu; Liu, Liwei; Ren, Sheng; Li, Zilin; Zhao, Yihua; Yang, Zhigang; Hu, Rui; Qu, Junle (2020).

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Pepper, David M.; Feinberg, Jack; Kukhtarev, Nicolai V. (October 1990). "The Photorefractive Effect".

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where N is the number of particles per unit volume, I is intensity of light, and δ is the two photon

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within the nonlinear medium, the oscillating second-order polarization radiates at angular frequency

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Robert Boyd plenary presentation: Quantum Nonlinear Optics: Nonlinear Optics Meets the Quantum World

5708:{\displaystyle \mathbf {L} _{\text{out}}(\mathbf {r} ,t)=-\mathbf {L} _{\text{in}}(\mathbf {r} ,t).} 5428:{\displaystyle \mathbf {S} _{\text{out}}(\mathbf {r} ,t)=-\mathbf {S} _{\text{in}}(\mathbf {r} ,t),} 30: 8574: 8433: 8428: 8413: 8388: 8365: 8341: 6265: 5783: 2712: 1520: 519: 411: 351:

Self-diffraction, splitting of beams in a multi-wave mixing process with potential energy transfer.

333: 5904: 5725: 4203: 3356: 2932: 8630: 8507: 8502: 8455: 8194: 7993: 7967: 7940: 7516: 7486: 7451: 7416: 7334: 7251:"Broadly tunable difference-frequency generation of VUV using two-photon resonances in H2 and Kr" 7033: 6855: 6829: 4059: 4023: 3990: 3957: 3915: 3882: 3482: 3378: 3092: 372: 246: 183:(DFG), generation of light with a frequency that is the difference between two other frequencies. 7547:

Okulov, A. Yu. (2008). "Optical and Sound Helical structures in a Mandelstam–Brillouin mirror".

7197:"Type-0 second order nonlinear interaction in monolithic waveguides of isotropic semiconductors" 7091:"Non-linear optical channels of the polarizability induction in a pair of interacting molecules" 6453: 6403: 4304: 4275: 4139: 1981: 8109: 4497: 8645: 8594: 8539: 8519: 8405: 8288: 8248: 8186: 8131: 8090: 7883: 7875: 7825: 7790: 7754: 7718: 7710: 7675: 7667: 7618: 7381: 7324: 7308: 7278: 7226: 7134: 7025: 6903: 6800: 6674: 6649: 6621: 6585: 6449: 6420: 6142: 4131: 3591:{\displaystyle {\vec {\mathbf {k} }}_{3}={\vec {\mathbf {k} }}_{1}+{\vec {\mathbf {k} }}_{2}.} 2708: 2688: 2131: 514:

Parametric and "instantaneous" (i.e. material must be lossless and dispersionless through the

211:, the special case of OPO or OPG when the signal and idler degenerate in one single frequency, 7183: 7168: 7124: 8584: 8524: 8450: 8280: 8222: 8178: 8121: 8080: 8072: 8033: 7985: 7932: 7867: 7817: 7782: 7746: 7702: 7657: 7610: 7556: 7526: 7478: 7443: 7408: 7373: 7316: 7270: 7216: 7071: 7017: 6978: 6942: 6895: 6847: 6777: 6711: 5621:{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t)} 4486: 3606: 3343:{\displaystyle P^{(2)}(\mathbf {x} ,t)\propto E_{1}^{n_{1}}E_{2}^{n_{2}}e^{i}+{\text{c.c.}}} 992: 780: 481: 402:

effect in which a wave with polarization vector perpendicular to the input one is generated.

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Jablan, Marinko; Buljan, Hrvoje; Manela, Ofer; Bartal, Guy; Segev, Mordechai (2007-04-16).

5880: 3872: 205:(OPG), like parametric oscillation but without a resonator, using a very high gain instead. 8676: 8604: 8514: 8460: 8360: 7580: 6573: 6561: 6549: 6536: 6475: 5345: 4791:{\displaystyle P_{\text{NL}}=\varepsilon _{0}\chi ^{(3)}(\Xi _{1}+\Xi _{2}+\Xi _{3})^{3},} 4151: 3845: 3837: 3825: 439: 124: 81: 7891: 7530: 7054:

Zakharov, V. E.; Ostrovsky, L. A. (2009-03-15). "Modulation instability: The beginning".

3605:. Typically, three-wave mixing is done in a birefringent crystalline material, where the 8272: 8174: 8068: 8029: 7981: 7928: 7863: 7653: 7606: 7574:

The Fascinating Behavior of Light in Photorefractive Media | Optics & Photonics News

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all four frequencies are mixing simultaneously, as opposed to sequential mixing via two

2667:, and 0. These three-wave mixing processes correspond to the nonlinear effects known as 1824:{\displaystyle \mathbf {P} ^{\text{NL}}=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t).} 8534: 8489: 8467: 8355: 8284: 8085: 8052: 7313:

CLEO '97., Summaries of Papers Presented at the Conference on Lasers and Electro-Optics

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and non-parametric effects. A parametric non-linearity is an interaction in which the

471: 461: 325: 69: 7482: 7447: 7412: 7354:"Efficient KDP frequency doubling SBS pulse compressed 532nm hundred picosecond laser" 8665: 8198: 7997: 7944: 7338: 7037: 6641: 6175: 4155: 4094: 523: 502: 445: 345: 297: 279: 113: 6859: 8569: 8247:(2nd ed.). Boca Raton London New York: CRC Press, Taylor & Francis Group. 4127: 4126:, or second-harmonic generation. With this technique, the 1064 nm output from 4102: 2793: 332:(spatial soliton) that does not change during propagation due to a balance between 329: 301: 287: 58: 2699: 7821: 4482:

within the material, which coherently radiates to form the phase-conjugate wave.

17: 7989: 7377: 7075: 6899: 6529: 6289: 3947: 2952: 1970:{\displaystyle \mathbf {E} (t)=E_{1}\cos(\omega _{1}t)+E_{2}\cos(\omega _{2}t),} 337: 311: 283: 240: 7614: 7021: 8589: 8579: 8226: 7560: 7434:

Pepper, David M. (January 1986). "Applications of Optical Phase Conjugation".

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of the medium, and the presence of such a term is generally referred to as an

527: 315: 84:, the vacuum itself is expected to become nonlinear. In nonlinear optics, the 8135: 8126: 7879: 7794: 7786: 7758: 7750: 7714: 7671: 7385: 7320: 7029: 5923:

is propagation axis. The angular momentum projection on propagation axis has

5195:{\displaystyle E_{4}={\frac {i\omega l}{2nc}}\chi ^{(3)}E_{1}E_{2}E_{3}^{*},} 8440: 8309: 7507:

Okulov, A. Yu. (2008). "Angular momentum of photons and phase conjugation".

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Kaiser, W.; Garrett, C. G. B. (1961). "Two-Photon Excitation in CaF2:Eu2+".

6178:, one can expand the induced dipole moment in powers of the electric field: 2716: 2704: 217:(SPDC), the amplification of the vacuum fluctuations in the low-gain regime. 140: 105: 8190: 8182: 8094: 7958:

Okulov, A Yu (2020). "Structured light entities, chaos and nonlocal maps".

7887: 7845:"Self-focusing and self-trapping of optical beams upon photopolymerization" 7829: 7722: 7679: 7622: 7399:

Shkunov, Vladimir; Zel'dovich, Boris (December 1985). "Phase Conjugation".

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Most common nonlinear crystals are negative uniaxial, which means that the

3472:{\displaystyle \|\mathbf {k} _{3}\|=\mathbf {n} (\omega _{3})\omega _{3}/c} 3024:{\displaystyle \|\mathbf {k} _{j}\|=\mathbf {n} (\omega _{j})\omega _{j}/c} 474:, the refractive index is affected by acoustic waves (ultrasound); used in 8037: 7871: 7662: 7637: 7274: 7221: 7196: 6141:

Optical fields transmitted through nonlinear Kerr media can also display

5064:), then the conjugate and signal beams propagate in opposite directions ( 2796:

materials avoid this problem by having two indices of refraction at once.

7490: 7455: 7420: 6834: 6781: 6554: 5089:, the electric field amplitude of the conjugate beam is approximated by 4272:

is reflected from perfect phase-conjugating mirror. Normal to mirror is

7936: 6493:

1550 nm: potassium titanyl phosphate (KTP), lithium niobate (LiNbO

799:

in an isotropic space, containing no free charge, it can be shown that

464:, the refractive index is affected by a static electric field; used in 8076: 8053:"Resonance Raman probes for organelle-specific labeling in live cells" 7915:

Okulov, A Yu (2000). "Spatial soliton laser: geometry and stability".

7706: 7195:

Abolghasem, Payam; Junbo Han; Bhavin J. Bijlani; Amr S. Helmy (2010).

6998:"Optical phase-change in plasmonic nanoparticles by a two-wave mixing" 6946: 8544: 7307:

Xhao, X.M.; Jones, R.J.; Strauss, C.E.M.; Funk, D.J.; Roberts, J.P.;

6982: 772: 485: 432: 50: 6911: 5959:

the interpretation of phase conjugation is much simpler compared to

4473:

One can interpret optical phase conjugation as being analogous to a

497:

Nonlinear effects fall into two qualitatively different categories,

7972: 7638:"Incoherent modulation instability in a nonlinear photonic lattice" 5289:{\displaystyle E_{4}(\mathbf {x} )\propto E_{3}^{*}(\mathbf {x} ),} 518:) nonlinear optical phenomena, in which the optical fields are not 431:, simultaneous absorption of two or more photons, transferring the 139:

Nonlinear optics explains nonlinear response of properties such as

8110:"Recent advances in nonlinear optics for bio-imaging applications" 7521: 6851: 6389: 6358:{\displaystyle {-dI \over dx}={N\delta I^{2} \over \hbar \omega }} 6279: 4496: 4202: 2698: 442:, near-simultaneous removal of many bound electrons by one photon. 227:

Nonlinear light-matter interaction with free electrons and plasmas

77: 54: 29: 5081:

Further, it can be shown that for a medium with refractive index

4801:

resulting in generation of waves with frequencies given by ω = ±ω

4154:. These crystals have the necessary properties of being strongly 4513:

For the four-wave mixing technique, we can describe four beams (

767:) are considered as scalar for simplicity. In general, χ is an ( 8313: 6734:"Schiff base metal complexes for second order nonlinear optics" 5438:("in" means incident field, "out" means reflected field) where 5078:). This results in the retroreflecting property of the effect. 5836:

and angular momentum, whose projection on propagation axis is

4191:

A device producing the phase-conjugation effect is known as a

3667:{\displaystyle \lambda _{p}\leq \lambda _{s}\leq \lambda _{i}} 3089:

is the index of refraction of the medium at angular frequency

7089:

Kouzov, A.P.; Egorova, N.I.; Chrysos, M.; Rachet, F. (2012).

5018:

This is the generating field for the phase-conjugate beam, Ξ

4122:

One of the most commonly used frequency-mixing processes is

3479:. Constructive interference, and therefore a high-intensity 751:-th-order nonlinearity. Note that the polarization density 123:, both shortly after the construction of the first laser by 5315:= ω, and the signal wave is higher in frequency such that ω 3116:. Thus, the second-order polarization at angular frequency 4813:

in addition to third-harmonic generation waves with ω = 3ω

4330:. Reflected photon (magenta) has opposite linear momentum 3860:(emulated by second-order processes) of the signal and an 2691:

uncertainty in the electric field initiates the process.

314:(and possibly higher-order nonlinearities) caused by the 286:(and possibly higher-order nonlinearities) caused by the 38:, viewed down b axis, used in second harmonic generation. 6876:"Observation of Nonlinear Effects in Compton Scattering" 6539:, with content on nonlinear optics, by Rüdiger Paschotta 4089:

as well. At high peak powers the Kerr effect can cause

1527:

solutions to the homogeneous part of the wave equation:

779:-dependent nature of the parametric interaction and the 5331:

Angular and linear momenta in optical phase conjugation

5050:, and so if the two pump beams are counterpropagating ( 96:

The first nonlinear optical effect to be predicted was

318:

creating a temporal variation in the refractive index.

6298: 6184: 5972: 5932: 5907: 5883: 5842: 5809: 5786: 5750: 5728: 5637: 5562: 5447: 5357: 5232: 5098: 4874: 4696: 4526: 4440: 4404: 4372: 4336: 4307: 4278: 4246: 4213: 4062: 4026: 3993: 3960: 3918: 3885: 3627: 3515: 3485: 3408: 3381: 3359: 3178: 3122: 3095: 3057: 3037: 2960: 2935: 2809: 2760: 2726: 2147: 1993: 1873: 1755: 1646: 1536: 1344: 1303: 1189: 1143: 1107: 1011: 808: 791:

Central to the study of electromagnetic waves is the

569: 375: 290:

creating a spatial variation in the refractive index.

249: 6166:

Connecting bulk properties to microscopic properties

5344:

a phase-conjugating mirror performs reversal of the

8623: 8560: 8488: 8404: 8376: 8348: 7049: 7047: 5319:= ω + Δω, then the conjugate wave is of frequency ω 3162:{\displaystyle \omega _{3}=\omega _{1}+\omega _{2}} 1002:Note that one can normally use the vector identity 6357: 6268:, β is the second hyperpolarizability, and so on. 6256: 6125: 5944: 5915: 5889: 5869: 5828: 5795: 5756: 5736: 5707: 5620: 5545: 5427: 5288: 5194: 5007: 4790: 4650: 4461: 4426: 4390: 4358: 4322: 4293: 4264: 4232: 4081: 4045: 4012: 3979: 3937: 3912:processes. It can be extended for processes where 3904: 3666: 3590: 3498: 3471: 3394: 3367: 3342: 3161: 3108: 3081: 3043: 3023: 2943: 2918: 2784: 2746: 2614: 2119: 1969: 1823: 1719: 1626: 1499: 1323: 1279: 1166: 1126: 1086: 968: 728: 394: 268: 7843:Kewitsch, Anthony S.; Yariv, Amnon (1996-01-01). 6380:Common second-harmonic-generating (SHG) materials 223:(OR), generation of quasi-static electric fields. 7169:1.2.10 Parametric versus Nonparametric Processes 6264:, where μ is the polarizability, α is the first 5870:{\displaystyle L_{\mathbf {z} }=\pm \hbar \ell } 4427:{\displaystyle \mathbf {P} =2\hbar \mathbf {k} } 4359:{\displaystyle \mathbf {P} =-\hbar \mathbf {k} } 3829:These methods are called temperature tuning and 363:(FWM), can also arise from other nonlinearities. 6727: 6725: 5829:{\displaystyle \mathbf {P} =\hbar \mathbf {k} } 4233:{\displaystyle \mathbf {P} =\hbar \mathbf {k} } 7542: 7540: 7244: 7242: 7240: 1842:) is made up of two components at frequencies 425:, interaction of photons with acoustic phonons 8325: 7502: 7500: 6543:An Intuitive Explanation of Phase Conjugation 3815:axis has a smaller refractive index than the 8: 7130:Encyclopedia of Laser Physics and Technology 7095:Nanosystems: Physics, Chemistry, Mathematics 6530:Encyclopedia of laser physics and technology 3424: 3409: 2976: 2961: 1167:{\displaystyle \nabla \cdot \mathbf {D} =0,} 5223:are plane (counterpropagating) waves, then 2785:{\displaystyle \mathbf {k} '=2\mathbf {k} } 1324:{\displaystyle \nabla \cdot \mathbf {E} =0} 8332: 8318: 8310: 6394:Dark-red gallium selenide in its bulk form 4207:Vortex photon (blue) with linear momentum 243:, intensity-dependent refractive index (a 8125: 8084: 7971: 7661: 7520: 7220: 6833: 6338: 6325: 6299: 6297: 6248: 6247: 6241: 6240: 6224: 6215: 6214: 6198: 6193: 6185: 6183: 6101: 6093: 6092: 6087: 6058: 6050: 6049: 6044: 6031: 6019: 6014: 6002: 5983: 5978: 5973: 5971: 5931: 5908: 5906: 5882: 5848: 5847: 5841: 5821: 5810: 5808: 5785: 5749: 5729: 5727: 5688: 5679: 5674: 5653: 5644: 5639: 5636: 5604: 5596: 5588: 5571: 5563: 5561: 5526: 5518: 5501: 5493: 5487: 5477: 5456: 5448: 5446: 5408: 5399: 5394: 5373: 5364: 5359: 5356: 5275: 5266: 5261: 5246: 5237: 5231: 5209:is the speed of light. If the pump beams 5183: 5178: 5168: 5158: 5142: 5112: 5103: 5097: 5000: 4987: 4979: 4963: 4953: 4948: 4938: 4928: 4918: 4902: 4888: 4879: 4873: 4779: 4769: 4756: 4743: 4724: 4714: 4701: 4695: 4640: 4625: 4617: 4605: 4592: 4580: 4571: 4557: 4540: 4531: 4525: 4439: 4419: 4405: 4403: 4371: 4351: 4337: 4335: 4309: 4308: 4306: 4280: 4279: 4277: 4245: 4225: 4214: 4212: 4170:beam, and thus the technique is known as 4067: 4061: 4031: 4025: 3998: 3992: 3965: 3959: 3923: 3917: 3890: 3884: 3658: 3645: 3632: 3626: 3579: 3568: 3566: 3565: 3555: 3544: 3542: 3541: 3531: 3520: 3518: 3517: 3514: 3490: 3484: 3461: 3455: 3442: 3430: 3418: 3413: 3407: 3386: 3380: 3360: 3358: 3335: 3318: 3306: 3294: 3289: 3279: 3274: 3263: 3251: 3246: 3241: 3229: 3224: 3219: 3198: 3183: 3177: 3153: 3140: 3127: 3121: 3100: 3094: 3082:{\displaystyle \mathbf {n} (\omega _{j})} 3070: 3058: 3056: 3036: 3013: 3007: 2994: 2982: 2970: 2965: 2959: 2936: 2934: 2911: 2894: 2882: 2873: 2868: 2860: 2844: 2823: 2814: 2808: 2777: 2762: 2759: 2725: 2595: 2586: 2571: 2561: 2543: 2533: 2502: 2489: 2475: 2465: 2458: 2453: 2446: 2422: 2409: 2395: 2385: 2375: 2354: 2340: 2330: 2323: 2318: 2303: 2289: 2279: 2272: 2267: 2249: 2234: 2228: 2203: 2198: 2185: 2175: 2158: 2153: 2148: 2146: 2134:. Plugging this into the expression for 2109: 2095: 2084: 2074: 2060: 2046: 2035: 2025: 2011: 1994: 1992: 1952: 1933: 1914: 1895: 1874: 1872: 1803: 1798: 1785: 1775: 1762: 1757: 1754: 1711: 1706: 1696: 1682: 1676: 1667: 1657: 1647: 1645: 1616: 1608: 1599: 1585: 1579: 1571: 1561: 1555: 1547: 1541: 1535: 1488: 1483: 1473: 1459: 1453: 1444: 1434: 1424: 1416: 1407: 1393: 1387: 1379: 1369: 1363: 1355: 1349: 1343: 1310: 1302: 1269: 1261: 1252: 1238: 1232: 1224: 1214: 1208: 1200: 1194: 1188: 1150: 1142: 1112: 1106: 1079: 1073: 1056: 1029: 1010: 957: 952: 942: 928: 922: 913: 903: 893: 882: 873: 859: 853: 845: 835: 829: 821: 807: 697: 692: 679: 657: 652: 639: 618: 606: 591: 570: 568: 380: 374: 254: 248: 7695:Journal of the American Chemical Society 7315:. Vol. 11. IEEE. pp. 377–378. 6770:Journal of the American Chemical Society 3871: 3616: 3051:is the velocity of light in vacuum, and 324:, an equilibrium solution for either an 8273:"Nonlinear Optics of Plasmonic Systems" 6605: 6555:SNLO - Nonlinear Optics Design Software 6346: 6195: 6186: 6110: 6074: 6028: 5999: 5936: 5861: 5818: 5787: 4676:are known as the two pump waves, with Ξ 4453: 4416: 4382: 4348: 4256: 4222: 1511:Nonlinearities as a wave-mixing process 8530:Atomic, molecular, and optical physics 4056:The Kerr effect can be described as a 3948:"vacuum" ultra-violet light generation 995:, which comes from the linear term in 484:, interaction of photons with optical 215:Spontaneous parametric down-conversion 8238: 8236: 8009: 8007: 2703:Most transparent materials, like the 1738:-th order nonlinearity will lead to ( 1127:{\displaystyle \rho _{\text{free}}=0} 787:Wave equation in a nonlinear material 783:(or lack) of the nonlinear material. 7: 7179: 7164: 6518:Category:Nonlinear optical materials 4684:being the generated conjugate wave. 4668:are the electric field amplitudes. Ξ 4517:= 1, 2, 3, 4) with electric fields: 4182:and is significantly different from 1519:and can be obtained by the use of a 310:(SPM), an effect due to the optical 7775:The Journal of Physical Chemistry C 7739:The Journal of Physical Chemistry C 6137:Nonlinear optical pattern formation 4865:, this gives a polarization field: 3601:The above equation is known as the 2625:which has frequency components at 2 316:temporal variation in the intensity 61:media, that is, media in which the 8285:10.1016/b978-0-12-811002-7.00023-0 6646:The Principles of Nonlinear Optics 4766: 4753: 4740: 4528: 4462:{\displaystyle L=\pm 2\hbar \ell } 3864:can be integrated monolithically. 1689: 1679: 1592: 1582: 1538: 1523:. Physically one gets the normal 1466: 1456: 1400: 1390: 1346: 1304: 1245: 1235: 1191: 1144: 1070: 1050: 1042: 1023: 1012: 935: 925: 866: 856: 815: 809: 288:spatial variation in the intensity 25: 7483:10.1038/scientificamerican1090-62 7448:10.1038/scientificamerican0186-74 7413:10.1038/scientificamerican1285-54 5342:classical Maxwell electrodynamics 4391:{\displaystyle L=\mp \hbar \ell } 4265:{\displaystyle L=\pm \hbar \ell } 2747:{\displaystyle \omega '=2\omega } 1294:does not imply that the identity 739:where the coefficients χ are the 131:'s monograph "Nonlinear Optics". 8017:Advances in Optics and Photonics 6673:(4th ed.). Academic Press. 6620:(3rd ed.). Academic Press. 6249: 6242: 6216: 6199: 6094: 6051: 6032: 6015: 6003: 5979: 5909: 5849: 5822: 5811: 5730: 5689: 5675: 5654: 5640: 5628:with respect to incident field: 5605: 5597: 5589: 5572: 5564: 5527: 5519: 5502: 5494: 5457: 5449: 5409: 5395: 5374: 5360: 5323:= ω − Δω. This is known as 5276: 5247: 4988: 4980: 4626: 4618: 4581: 4541: 4420: 4406: 4352: 4338: 4226: 4215: 3569: 3545: 3521: 3431: 3414: 3402:and a corresponding wave vector 3361: 3307: 3290: 3275: 3199: 3059: 2983: 2966: 2937: 2883: 2869: 2824: 2778: 2763: 2199: 2154: 1995: 1875: 1799: 1758: 1707: 1617: 1609: 1548: 1484: 1417: 1356: 1311: 1270: 1262: 1201: 1151: 1080: 1057: 1030: 953: 883: 822: 693: 653: 619: 571: 193:wave (can be considered as DFG). 187:Optical parametric amplification 53:that describes the behaviour of 8651:Timeline of physics discoveries 8142:from the original on 2021-01-21 7289:from the original on 2024-05-29 7249:Strauss, CEM; Funk, DJ (1991). 7147:from the original on 2011-08-22 7105:from the original on 2017-06-13 5945:{\displaystyle \pm \hbar \ell } 5803:also possesses linear momentum 5718:The above identities are valid 4130:or the 800 nm output from 2677:difference-frequency generation 1517:ordinary differential equations 423:Stimulated Brillouin scattering 367:Cross-polarized wave generation 282:, an effect due to the optical 181:Difference-frequency generation 8279:, Elsevier, pp. 569–582, 8215:Coordination Chemistry Reviews 7960:Chaos, Solitons & Fractals 7531:10.1088/0953-4075/41/10/101001 7509:J. Phys. B: At. Mol. Opt. Phys 7056:Physica D: Nonlinear Phenomena 6795:Bloembergen, Nicolaas (1965). 6732:Rigamonti, Luca (April 2010). 6151:nonlinear Schrödinger equation 5901:of photon, or winding number, 5766:ideal phase-conjugating mirror 5699: 5685: 5664: 5650: 5615: 5601: 5582: 5568: 5537: 5523: 5512: 5498: 5467: 5453: 5419: 5405: 5384: 5370: 5280: 5272: 5251: 5243: 5149: 5143: 5085:and a beam interaction length 4992: 4967: 4909: 4903: 4776: 4736: 4731: 4725: 4585: 4577: 4551: 4537: 4314: 4285: 4144:potassium dihydrogen phosphate 4107:high-order harmonic generation 4074: 4068: 4038: 4032: 4005: 3999: 3972: 3966: 3930: 3924: 3897: 3891: 3763:Type V (or type 0, or "zero") 3573: 3549: 3525: 3448: 3435: 3327: 3300: 3270: 3267: 3209: 3195: 3190: 3184: 3076: 3063: 3000: 2987: 2903: 2864: 2834: 2820: 2508: 2482: 2428: 2402: 2256: 2250: 2215: 2209: 2192: 2186: 2005: 1999: 1961: 1945: 1923: 1907: 1885: 1879: 1815: 1809: 1792: 1786: 709: 703: 686: 680: 669: 663: 646: 640: 629: 623: 613: 607: 581: 575: 387: 381: 261: 255: 197:Optical parametric oscillation 1: 6488:ammonium dihydrogen phosphate 5796:{\displaystyle \hbar \omega } 4475:real-time holographic process 3868:Higher-order frequency mixing 983:is the nonlinear part of the 203:Optical parametric generation 68:responds non-linearly to the 7822:10.1021/acsmacrolett.6b00659 6743:(3): 118–122. Archived from 6741:La Chimica & l'Industria 6398:Ordered by pump wavelength: 5916:{\displaystyle \mathbf {z} } 5737:{\displaystyle \mathbf {r} } 5022:. Its direction is given by 4680:being the signal wave, and Ξ 3862:optical parametric amplifier 3368:{\displaystyle \mathbf {x} } 2944:{\displaystyle \mathbf {x} } 1984:to convert to exponentials, 1177:to obtain the more familiar 8615:Quantum information science 7990:10.1016/j.chaos.2020.109638 7378:10.1016/j.ijleo.2016.07.021 7125:"Parametric Nonlinearities" 7123:Paschotta, Rüdiger (2008). 7076:10.1016/j.physd.2008.12.002 6900:10.1103/PhysRevLett.76.3116 6472:potassium titanyl phosphate 5722:, i.e. in each space point 4148:potassium titanyl phosphate 4082:{\displaystyle \chi ^{(3)}} 4046:{\displaystyle \chi ^{(2)}} 4013:{\displaystyle \chi ^{(3)}} 3980:{\displaystyle \chi ^{(3)}} 3952:difference frequency mixing 3938:{\displaystyle \chi ^{(3)}} 3905:{\displaystyle \chi ^{(2)}} 3499:{\displaystyle \omega _{3}} 3395:{\displaystyle \omega _{3}} 3109:{\displaystyle \omega _{j}} 1637:and the inhomogeneous term 1101:(assuming no free charges, 395:{\displaystyle \chi ^{(3)}} 269:{\displaystyle \chi ^{(3)}} 135:Nonlinear optical processes 8693: 8446:Classical electromagnetism 8243:McHale, Jeanne L. (2017). 8051:Kuzmin, Andrey N. (2016). 7615:10.1103/PhysRevLett.84.467 6386:Second-harmonic generation 6383: 6157:Molecular nonlinear optics 4509:Four-wave mixing technique 4323:{\displaystyle {\vec {z}}} 4294:{\displaystyle {\vec {n}}} 3791:Type VII (or IIA or IIIB) 3506:field, will occur only if 2669:second-harmonic generation 153:Second-harmonic generation 147:Frequency-mixing processes 110:second-harmonic generation 8227:10.1016/j.ccr.2017.03.013 7561:10.1134/S0021364008200046 6716:10.1103/PhysRevLett.7.229 6584:(4th ed.). Academi. 5961:classical electrodynamics 4492:Optical phase conjugation 4469:is excited within mirror. 4172:optical phase conjugation 4162:Optical phase conjugation 3777:Type VI (or IIB or IIIA) 3681: 3678: 417:Optical phase conjugation 234:Other nonlinear processes 163:Third-harmonic generation 8552:Condensed matter physics 8127:10.29026/oea.2020.200003 8114:Opto-Electronic Advances 7787:10.1021/acs.jpcc.5b11377 7751:10.1021/acs.jpcc.5b07117 7321:10.1109/CLEO.1997.603294 7022:10.1088/1402-4896/ab3ae9 6882:(Submitted manuscript). 6669:Agrawal, Govind (2006). 6370:absorption cross section 3603:phase-matching condition 2673:sum-frequency generation 2130:where "c.c." stands for 1290:For a nonlinear medium, 771: + 1)-th-rank 522:, can be described by a 516:Kramers–Kronig relations 476:acousto-optic modulators 466:electro-optic modulators 446:Chaos in optical systems 406:Modulational instability 209:Half-harmonic generation 175:Sum-frequency generation 169:High-harmonic generation 108:and the discovery of 7917:Optics and Spectroscopy 7595:Physical Review Letters 6696:Physical Review Letters 6438:monopotassium phosphate 6176:Taylor series expansion 5957:quantum electrodynamics 5780:the photon with energy 5778:quantum electrodynamics 759:) and electrical field 429:Multi-photon absorption 86:superposition principle 8636:Nobel Prize in Physics 8498:Relativistic mechanics 8245:Molecular spectroscopy 8183:10.1002/adma.201701076 6671:Nonlinear Fiber Optics 6648:. Wiley-Interscience. 6395: 6359: 6285: 6258: 6174:can be described as a 6147:modulation instability 6127: 5946: 5917: 5891: 5871: 5830: 5797: 5758: 5738: 5709: 5622: 5547: 5429: 5290: 5196: 5009: 4792: 4652: 4502: 4470: 4463: 4428: 4392: 4360: 4324: 4301:, propagation axis is 4295: 4266: 4234: 4193:phase-conjugate mirror 4083: 4047: 4014: 3981: 3939: 3906: 3876: 3668: 3592: 3500: 3473: 3396: 3369: 3344: 3163: 3110: 3083: 3045: 3025: 2945: 2920: 2797: 2786: 2748: 2616: 2121: 1971: 1825: 1721: 1628: 1501: 1325: 1281: 1168: 1128: 1088: 970: 775:representing both the 730: 535:electric dipole moment 396: 355:Cross-phase modulation 348:for spatial solitons). 328:(temporal soliton) or 270: 121:University of Michigan 39: 8641:Philosophy of physics 6393: 6360: 6283: 6272:Novel Nonlinear Media 6259: 6128: 5947: 5918: 5892: 5890:{\displaystyle \ell } 5872: 5831: 5798: 5759: 5739: 5710: 5623: 5548: 5430: 5291: 5197: 5010: 4793: 4653: 4500: 4464: 4434:and angular momentum 4429: 4393: 4366:and angular momentum 4361: 4325: 4296: 4267: 4240:and angular momentum 4235: 4206: 4084: 4048: 4015: 3982: 3940: 3907: 3875: 3858:self-phase modulation 3669: 3618:Phase-matching types 3593: 3501: 3474: 3397: 3370: 3345: 3164: 3111: 3084: 3046: 3026: 2946: 2921: 2787: 2749: 2702: 2681:optical rectification 2617: 2122: 1972: 1826: 1722: 1629: 1502: 1326: 1282: 1169: 1129: 1089: 971: 731: 435:to a single electron. 397: 342:self-phase modulation 308:Self-phase modulation 294:Kerr-lens modelocking 271: 221:Optical rectification 98:two-photon absorption 33: 8600:Mathematical physics 8038:10.1364/AOP.8.000328 7872:10.1364/OL.21.000024 7663:10.1364/OE.15.004623 7275:10.1364/ol.16.001192 7222:10.1364/OE.18.012681 6799:. World Scientific. 6513:Filament propagation 6296: 6182: 5970: 5930: 5905: 5881: 5840: 5807: 5784: 5748: 5726: 5635: 5560: 5445: 5355: 5230: 5096: 4872: 4694: 4524: 4438: 4402: 4370: 4334: 4305: 4276: 4244: 4211: 4060: 4024: 3991: 3958: 3916: 3883: 3879:The above holds for 3856:. SHG of a pump and 3831:quasi-phase-matching 3625: 3513: 3483: 3406: 3379: 3357: 3176: 3120: 3093: 3055: 3035: 2958: 2933: 2807: 2792:in these materials. 2758: 2724: 2145: 1991: 1871: 1753: 1644: 1534: 1525:electromagnetic wave 1342: 1301: 1187: 1141: 1105: 1009: 985:polarization density 806: 567: 531:polarization density 493:Parametric processes 373: 247: 102:Maria Goeppert Mayer 63:polarization density 8575:Atmospheric physics 8414:Classical mechanics 8342:branches of physics 8175:2017AdM....2901076G 8069:2016NatSR...628483K 8030:2016AdOP....8..328G 7982:2020CSF...13309638O 7929:2000OptSp..89..131O 7864:1996OptL...21...24K 7745:(35): 20606–20617. 7654:2007OExpr..15.4623J 7607:2000PhRvL..84..467S 7471:Scientific American 7436:Scientific American 7401:Scientific American 7370:2016Optik.127.9201B 7267:1991OptL...16.1192S 7213:2010OExpr..1812681A 7207:(12): 12681–12689. 7184:2.3. Phase Matching 7068:2009PhyD..238..540Z 7014:2019PhyS...94l5802H 6975:2007NatPh...3..424T 6892:1996PhRvL..76.3116B 6844:1998Natur.396..653C 6782:10.1021/ja01856a043 6708:1961PhRvL...7..229K 6580:Boyd, R.W. (2020). 6266:hyperpolarizability 5271: 5188: 4958: 3675: 3619: 3258: 3236: 2713:index of refraction 797:Maxwell's equations 412:Raman amplification 49:) is the branch of 8631:History of physics 8163:Advanced Materials 8057:Scientific Reports 7937:10.1134/BF03356001 7579:2015-04-02 at the 6935:Physics of Plasmas 6572:2016-03-17 at the 6560:2011-07-07 at the 6548:2007-11-08 at the 6535:2009-06-03 at the 6466:1319 nm: KNbO 6396: 6355: 6286: 6254: 6123: 6121: 5942: 5913: 5899:topological charge 5887: 5867: 5826: 5793: 5754: 5744:in a given moment 5734: 5705: 5618: 5543: 5425: 5325:frequency flipping 5286: 5257: 5192: 5174: 5005: 4944: 4788: 4648: 4503: 4471: 4459: 4424: 4388: 4356: 4320: 4291: 4262: 4230: 4180:wavefront reversal 4132:Ti:sapphire lasers 4124:frequency doubling 4118:Frequency doubling 4079: 4043: 4010: 3977: 3935: 3902: 3877: 3846:periodically poled 3735:Type III (or IIB) 3664: 3620: 3617: 3588: 3496: 3469: 3392: 3365: 3340: 3237: 3215: 3159: 3106: 3079: 3041: 3021: 2941: 2916: 2798: 2782: 2744: 2689:quantum-mechanical 2612: 2610: 2117: 1967: 1834:If we assume that 1821: 1717: 1624: 1497: 1321: 1277: 1164: 1124: 1084: 966: 726: 392: 300:as a mechanism to 296:(KLM), the use of 266: 157:frequency doubling 40: 8659: 8658: 8646:Physics education 8595:Materials science 8562:Interdisciplinary 8520:Quantum mechanics 8294:978-0-12-811002-7 8254:978-1-4665-8658-1 8077:10.1038/srep28483 7816:(11): 1237–1241. 7810:ACS Macro Letters 7707:10.1021/ja068967b 7701:(15): 4738–4746. 7364:(20): 9201–9205. 7330:978-0-7803-4125-8 7167:, pp. 13–15 7140:978-3-527-40828-3 6947:10.1063/1.2013067 6886:(17): 3116–3119. 6828:(6712): 653–655. 6776:(11): 3005–3018. 6680:978-0-12-369516-1 6655:978-0-471-43080-3 6627:978-0-12-369470-6 6591:978-0-12-811003-4 6508:Born–Infeld model 6450:lithium triborate 6430:980 nm: KNbO 6421:potassium niobate 6353: 6320: 6232: 6143:pattern formation 6104: 6061: 6022: 5986: 5757:{\displaystyle t} 5682: 5647: 5402: 5367: 5336:Classical picture 5136: 5003: 4896: 4704: 4643: 4565: 4317: 4288: 4105:. This is called 3809: 3808: 3805:Type VIII (or I) 3721:Type II (or IIA) 3576: 3552: 3528: 3353:At each position 3338: 3044:{\displaystyle c} 2914: 2709:normal dispersion 2707:shown here, have 2598: 2243: 2161: 2132:complex conjugate 2112: 2068: 2019: 1765: 1714: 1703: 1674: 1606: 1577: 1491: 1480: 1451: 1414: 1385: 1259: 1230: 1115: 960: 949: 920: 880: 851: 795:. Starting with 537:per unit volume) 526:expansion of the 453:Related processes 344:for temporal and 88:no longer holds. 27:Branch of physics 18:Nonlinear optical 16:(Redirected from 8684: 8672:Nonlinear optics 8585:Chemical physics 8525:Particle physics 8451:Classical optics 8334: 8327: 8320: 8311: 8304: 8303: 8302: 8301: 8277:Nonlinear Optics 8265: 8259: 8258: 8240: 8231: 8230: 8209: 8203: 8202: 8157: 8151: 8150: 8148: 8147: 8129: 8105: 8099: 8098: 8088: 8048: 8042: 8041: 8011: 8002: 8001: 7975: 7955: 7949: 7948: 7912: 7906: 7905: 7903: 7902: 7896: 7890:. Archived from 7849: 7840: 7834: 7833: 7805: 7799: 7798: 7781:(8): 4517–4528. 7769: 7763: 7762: 7733: 7727: 7726: 7690: 7684: 7683: 7665: 7633: 7627: 7626: 7590: 7584: 7571: 7565: 7564: 7544: 7535: 7534: 7524: 7504: 7495: 7494: 7466: 7460: 7459: 7431: 7425: 7424: 7396: 7390: 7389: 7349: 7343: 7342: 7304: 7298: 7297: 7295: 7294: 7246: 7235: 7234: 7224: 7192: 7186: 7177: 7171: 7162: 7156: 7155: 7153: 7152: 7120: 7114: 7113: 7111: 7110: 7086: 7080: 7079: 7051: 7042: 7041: 6993: 6987: 6986: 6983:10.1038/nphys595 6957: 6951: 6950: 6930: 6924: 6923: 6921: 6919: 6910:. Archived from 6880:Phys. Rev. Lett. 6870: 6864: 6863: 6837: 6817: 6811: 6810: 6797:Nonlinear Optics 6792: 6786: 6785: 6765: 6759: 6758: 6756: 6755: 6749: 6738: 6729: 6720: 6719: 6691: 6685: 6684: 6666: 6660: 6659: 6638: 6632: 6631: 6618:Nonlinear Optics 6610: 6595: 6582:Nonlinear optics 6461:gallium selenide 6364: 6362: 6361: 6356: 6354: 6352: 6344: 6343: 6342: 6326: 6321: 6319: 6311: 6300: 6263: 6261: 6260: 6255: 6253: 6252: 6246: 6245: 6233: 6225: 6220: 6219: 6204: 6203: 6202: 6189: 6132: 6130: 6129: 6124: 6122: 6106: 6105: 6102: 6100: 6099: 6098: 6097: 6063: 6062: 6059: 6057: 6056: 6055: 6054: 6035: 6024: 6023: 6020: 6018: 6006: 5988: 5987: 5984: 5982: 5951: 5949: 5948: 5943: 5922: 5920: 5919: 5914: 5912: 5896: 5894: 5893: 5888: 5876: 5874: 5873: 5868: 5854: 5853: 5852: 5835: 5833: 5832: 5827: 5825: 5814: 5802: 5800: 5799: 5794: 5763: 5761: 5760: 5755: 5743: 5741: 5740: 5735: 5733: 5714: 5712: 5711: 5706: 5692: 5684: 5683: 5680: 5678: 5657: 5649: 5648: 5645: 5643: 5627: 5625: 5624: 5619: 5608: 5600: 5592: 5575: 5567: 5552: 5550: 5549: 5544: 5530: 5522: 5505: 5497: 5492: 5491: 5482: 5481: 5460: 5452: 5434: 5432: 5431: 5426: 5412: 5404: 5403: 5400: 5398: 5377: 5369: 5368: 5365: 5363: 5295: 5293: 5292: 5287: 5279: 5270: 5265: 5250: 5242: 5241: 5201: 5199: 5198: 5193: 5187: 5182: 5173: 5172: 5163: 5162: 5153: 5152: 5137: 5135: 5124: 5113: 5108: 5107: 5014: 5012: 5011: 5006: 5004: 5001: 4996: 4995: 4991: 4983: 4957: 4952: 4943: 4942: 4933: 4932: 4923: 4922: 4913: 4912: 4897: 4889: 4884: 4883: 4797: 4795: 4794: 4789: 4784: 4783: 4774: 4773: 4761: 4760: 4748: 4747: 4735: 4734: 4719: 4718: 4706: 4705: 4702: 4657: 4655: 4654: 4649: 4644: 4641: 4636: 4635: 4634: 4630: 4629: 4621: 4610: 4609: 4584: 4576: 4575: 4566: 4558: 4544: 4536: 4535: 4487:angular momentum 4468: 4466: 4465: 4460: 4433: 4431: 4430: 4425: 4423: 4409: 4397: 4395: 4394: 4389: 4365: 4363: 4362: 4357: 4355: 4341: 4329: 4327: 4326: 4321: 4319: 4318: 4310: 4300: 4298: 4297: 4292: 4290: 4289: 4281: 4271: 4269: 4268: 4263: 4239: 4237: 4236: 4231: 4229: 4218: 4088: 4086: 4085: 4080: 4078: 4077: 4052: 4050: 4049: 4044: 4042: 4041: 4019: 4017: 4016: 4011: 4009: 4008: 3986: 3984: 3983: 3978: 3976: 3975: 3944: 3942: 3941: 3936: 3934: 3933: 3911: 3909: 3908: 3903: 3901: 3900: 3676: 3673: 3671: 3670: 3665: 3663: 3662: 3650: 3649: 3637: 3636: 3607:refractive index 3597: 3595: 3594: 3589: 3584: 3583: 3578: 3577: 3572: 3567: 3560: 3559: 3554: 3553: 3548: 3543: 3536: 3535: 3530: 3529: 3524: 3519: 3505: 3503: 3502: 3497: 3495: 3494: 3478: 3476: 3475: 3470: 3465: 3460: 3459: 3447: 3446: 3434: 3423: 3422: 3417: 3401: 3399: 3398: 3393: 3391: 3390: 3374: 3372: 3371: 3366: 3364: 3349: 3347: 3346: 3341: 3339: 3336: 3331: 3330: 3323: 3322: 3310: 3299: 3298: 3293: 3284: 3283: 3278: 3257: 3256: 3255: 3245: 3235: 3234: 3233: 3223: 3202: 3194: 3193: 3168: 3166: 3165: 3160: 3158: 3157: 3145: 3144: 3132: 3131: 3115: 3113: 3112: 3107: 3105: 3104: 3088: 3086: 3085: 3080: 3075: 3074: 3062: 3050: 3048: 3047: 3042: 3030: 3028: 3027: 3022: 3017: 3012: 3011: 2999: 2998: 2986: 2975: 2974: 2969: 2950: 2948: 2947: 2942: 2940: 2925: 2923: 2922: 2917: 2915: 2912: 2907: 2906: 2899: 2898: 2886: 2878: 2877: 2872: 2855: 2854: 2827: 2819: 2818: 2791: 2789: 2788: 2783: 2781: 2770: 2766: 2753: 2751: 2750: 2745: 2734: 2621: 2619: 2618: 2613: 2611: 2604: 2600: 2599: 2596: 2591: 2590: 2581: 2577: 2576: 2575: 2570: 2566: 2565: 2548: 2547: 2542: 2538: 2537: 2515: 2514: 2507: 2506: 2494: 2493: 2470: 2469: 2464: 2463: 2462: 2451: 2450: 2435: 2434: 2427: 2426: 2414: 2413: 2390: 2389: 2380: 2379: 2364: 2363: 2359: 2358: 2335: 2334: 2329: 2328: 2327: 2313: 2312: 2308: 2307: 2284: 2283: 2278: 2277: 2276: 2260: 2259: 2244: 2239: 2238: 2229: 2221: 2208: 2207: 2202: 2196: 2195: 2180: 2179: 2163: 2162: 2159: 2157: 2126: 2124: 2123: 2118: 2113: 2110: 2105: 2104: 2100: 2099: 2079: 2078: 2069: 2061: 2056: 2055: 2051: 2050: 2030: 2029: 2020: 2012: 1998: 1976: 1974: 1973: 1968: 1957: 1956: 1938: 1937: 1919: 1918: 1900: 1899: 1878: 1830: 1828: 1827: 1822: 1808: 1807: 1802: 1796: 1795: 1780: 1779: 1767: 1766: 1763: 1761: 1726: 1724: 1723: 1718: 1716: 1715: 1712: 1710: 1704: 1702: 1701: 1700: 1687: 1686: 1677: 1675: 1673: 1672: 1671: 1662: 1661: 1648: 1633: 1631: 1630: 1625: 1620: 1612: 1607: 1605: 1604: 1603: 1590: 1589: 1580: 1578: 1576: 1575: 1566: 1565: 1556: 1551: 1546: 1545: 1521:Green's function 1506: 1504: 1503: 1498: 1493: 1492: 1489: 1487: 1481: 1479: 1478: 1477: 1464: 1463: 1454: 1452: 1450: 1449: 1448: 1439: 1438: 1425: 1420: 1415: 1413: 1412: 1411: 1398: 1397: 1388: 1386: 1384: 1383: 1374: 1373: 1364: 1359: 1354: 1353: 1330: 1328: 1327: 1322: 1314: 1286: 1284: 1283: 1278: 1273: 1265: 1260: 1258: 1257: 1256: 1243: 1242: 1233: 1231: 1229: 1228: 1219: 1218: 1209: 1204: 1199: 1198: 1173: 1171: 1170: 1165: 1154: 1133: 1131: 1130: 1125: 1117: 1116: 1113: 1093: 1091: 1090: 1085: 1083: 1078: 1077: 1065: 1061: 1060: 1038: 1034: 1033: 993:refractive index 975: 973: 972: 967: 962: 961: 958: 956: 950: 948: 947: 946: 933: 932: 923: 921: 919: 918: 917: 908: 907: 894: 886: 881: 879: 878: 877: 864: 863: 854: 852: 850: 849: 840: 839: 830: 825: 745:susceptibilities 735: 733: 732: 727: 722: 718: 702: 701: 696: 690: 689: 662: 661: 656: 650: 649: 622: 617: 616: 596: 595: 574: 549:in terms of the 482:Raman scattering 401: 399: 398: 393: 391: 390: 361:Four-wave mixing 322:Optical solitons 275: 273: 272: 267: 265: 264: 43:Nonlinear optics 21: 8692: 8691: 8687: 8686: 8685: 8683: 8682: 8681: 8662: 8661: 8660: 8655: 8619: 8605:Medical physics 8556: 8515:Nuclear physics 8484: 8478:Non-equilibrium 8400: 8372: 8344: 8338: 8308: 8307: 8299: 8297: 8295: 8269:Boyd, Robert W. 8267: 8266: 8262: 8255: 8242: 8241: 8234: 8211: 8210: 8206: 8169:(28): 1701076. 8159: 8158: 8154: 8145: 8143: 8107: 8106: 8102: 8050: 8049: 8045: 8013: 8012: 8005: 7957: 7956: 7952: 7914: 7913: 7909: 7900: 7898: 7894: 7847: 7842: 7841: 7837: 7807: 7806: 7802: 7771: 7770: 7766: 7735: 7734: 7730: 7692: 7691: 7687: 7635: 7634: 7630: 7592: 7591: 7587: 7581:Wayback Machine 7572: 7568: 7546: 7545: 7538: 7506: 7505: 7498: 7468: 7467: 7463: 7433: 7432: 7428: 7398: 7397: 7393: 7351: 7350: 7346: 7331: 7306: 7305: 7301: 7292: 7290: 7248: 7247: 7238: 7194: 7193: 7189: 7178: 7174: 7163: 7159: 7150: 7148: 7141: 7122: 7121: 7117: 7108: 7106: 7088: 7087: 7083: 7053: 7052: 7045: 7002:Physica Scripta 6995: 6994: 6990: 6959: 6958: 6954: 6932: 6931: 6927: 6917: 6915: 6914:on 21 June 2019 6872: 6871: 6867: 6835:physics/9810036 6819: 6818: 6814: 6807: 6794: 6793: 6789: 6767: 6766: 6762: 6753: 6751: 6747: 6736: 6731: 6730: 6723: 6693: 6692: 6688: 6681: 6668: 6667: 6663: 6656: 6640: 6639: 6635: 6628: 6612: 6611: 6607: 6602: 6592: 6579: 6574:Wayback Machine 6562:Wayback Machine 6550:Wayback Machine 6537:Wayback Machine 6526: 6524:Further reading 6504: 6496: 6485: 6481: 6476:lithium niobate 6469: 6454:β-barium borate 6447: 6443: 6433: 6426: 6415: 6388: 6382: 6345: 6334: 6327: 6312: 6301: 6294: 6293: 6194: 6180: 6179: 6159: 6139: 6120: 6119: 6088: 6086: 6064: 6045: 6043: 6040: 6039: 6013: 5989: 5977: 5968: 5967: 5928: 5927: 5925:discrete values 5903: 5902: 5879: 5878: 5843: 5838: 5837: 5805: 5804: 5782: 5781: 5774: 5772:Quantum picture 5746: 5745: 5724: 5723: 5673: 5638: 5633: 5632: 5558: 5557: 5483: 5473: 5443: 5442: 5393: 5358: 5353: 5352: 5346:Poynting vector 5338: 5333: 5322: 5318: 5314: 5310: 5233: 5228: 5227: 5222: 5215: 5164: 5154: 5138: 5125: 5114: 5099: 5094: 5093: 5077: 5070: 5063: 5056: 5049: 5042: 5035: 5028: 5021: 4959: 4934: 4924: 4914: 4898: 4875: 4870: 4869: 4864: 4857: 4850: 4839: 4835: 4831: 4824: 4820: 4816: 4812: 4808: 4804: 4775: 4765: 4752: 4739: 4720: 4710: 4697: 4692: 4691: 4683: 4679: 4675: 4671: 4666: 4601: 4600: 4596: 4588: 4567: 4527: 4522: 4521: 4511: 4479:phase-conjugate 4436: 4435: 4400: 4399: 4368: 4367: 4332: 4331: 4303: 4302: 4274: 4273: 4242: 4241: 4209: 4208: 4201: 4185:retroreflection 4164: 4152:lithium niobate 4140:β-barium borate 4120: 4115: 4063: 4058: 4057: 4027: 4022: 4021: 3994: 3989: 3988: 3961: 3956: 3955: 3919: 3914: 3913: 3886: 3881: 3880: 3870: 3838:lithium niobate 3826:Poynting vector 3654: 3641: 3628: 3623: 3622: 3564: 3540: 3516: 3511: 3510: 3486: 3481: 3480: 3451: 3438: 3412: 3404: 3403: 3382: 3377: 3376: 3355: 3354: 3314: 3288: 3273: 3259: 3247: 3225: 3179: 3174: 3173: 3149: 3136: 3123: 3118: 3117: 3096: 3091: 3090: 3066: 3053: 3052: 3033: 3032: 3003: 2990: 2964: 2956: 2955: 2931: 2930: 2890: 2867: 2856: 2840: 2810: 2805: 2804: 2761: 2756: 2755: 2727: 2722: 2721: 2697: 2666: 2659: 2652: 2645: 2638: 2631: 2609: 2608: 2582: 2557: 2553: 2552: 2529: 2525: 2524: 2523: 2519: 2498: 2485: 2471: 2454: 2452: 2442: 2418: 2405: 2391: 2381: 2371: 2350: 2336: 2319: 2317: 2299: 2285: 2268: 2266: 2265: 2261: 2245: 2230: 2219: 2218: 2197: 2181: 2171: 2164: 2152: 2143: 2142: 2091: 2080: 2070: 2042: 2031: 2021: 1989: 1988: 1982:Euler's formula 1948: 1929: 1910: 1891: 1869: 1868: 1856:, we can write 1855: 1848: 1797: 1781: 1771: 1756: 1751: 1750: 1746:takes the form 1734:In general, an 1705: 1692: 1688: 1678: 1663: 1653: 1652: 1642: 1641: 1595: 1591: 1581: 1567: 1557: 1537: 1532: 1531: 1513: 1482: 1469: 1465: 1455: 1440: 1430: 1429: 1403: 1399: 1389: 1375: 1365: 1345: 1340: 1339: 1299: 1298: 1248: 1244: 1234: 1220: 1210: 1190: 1185: 1184: 1139: 1138: 1108: 1103: 1102: 1069: 1049: 1045: 1022: 1018: 1007: 1006: 951: 938: 934: 924: 909: 899: 898: 869: 865: 855: 841: 831: 804: 803: 789: 691: 675: 651: 635: 602: 601: 597: 587: 565: 564: 512: 495: 455: 440:photoionisation 376: 371: 370: 250: 245: 244: 236: 149: 137: 125:Theodore Maiman 94: 82:Schwinger limit 28: 23: 22: 15: 12: 11: 5: 8690: 8688: 8680: 8679: 8674: 8664: 8663: 8657: 8656: 8654: 8653: 8648: 8643: 8638: 8633: 8627: 8625: 8621: 8620: 8618: 8617: 8612: 8607: 8602: 8597: 8592: 8587: 8582: 8577: 8572: 8566: 8564: 8558: 8557: 8555: 8554: 8549: 8548: 8547: 8542: 8537: 8527: 8522: 8517: 8512: 8511: 8510: 8505: 8494: 8492: 8486: 8485: 8483: 8482: 8481: 8480: 8475: 8468:Thermodynamics 8465: 8464: 8463: 8458: 8448: 8443: 8438: 8437: 8436: 8431: 8426: 8421: 8410: 8408: 8402: 8401: 8399: 8398: 8397: 8396: 8386: 8380: 8378: 8374: 8373: 8371: 8370: 8369: 8368: 8358: 8352: 8350: 8346: 8345: 8339: 8337: 8336: 8329: 8322: 8314: 8306: 8305: 8293: 8260: 8253: 8232: 8204: 8152: 8120:(10): 200003. 8100: 8043: 8003: 7950: 7923:(1): 145–147. 7907: 7852:Optics Letters 7835: 7800: 7764: 7728: 7685: 7648:(8): 4623–33. 7642:Optics Express 7628: 7601:(3): 467–470. 7585: 7566: 7555:(8): 561–566. 7536: 7515:(10): 101001. 7496: 7461: 7426: 7391: 7344: 7329: 7299: 7261:(15): 1192–4. 7255:Optics Letters 7236: 7201:Optics Express 7187: 7172: 7157: 7139: 7115: 7081: 7062:(5): 540–548. 7043: 7008:(12): 125802. 6988: 6969:(6): 424–429. 6952: 6925: 6865: 6812: 6806:978-9810225995 6805: 6787: 6760: 6721: 6686: 6679: 6661: 6654: 6642:Shen, Yuen-Ron 6633: 6626: 6604: 6603: 6601: 6598: 6597: 6596: 6590: 6577: 6564: 6552: 6540: 6525: 6522: 6521: 6520: 6515: 6510: 6503: 6500: 6499: 6498: 6494: 6491: 6483: 6479: 6467: 6464: 6459:1300 nm: 6457: 6445: 6441: 6436:1064 nm: 6434: 6431: 6428: 6424: 6417: 6413: 6410:lithium iodate 6406: 6381: 6378: 6351: 6348: 6341: 6337: 6333: 6330: 6324: 6318: 6315: 6310: 6307: 6304: 6251: 6244: 6239: 6236: 6231: 6228: 6223: 6218: 6213: 6210: 6207: 6201: 6197: 6192: 6188: 6172:polarizability 6158: 6155: 6138: 6135: 6134: 6133: 6118: 6115: 6112: 6109: 6096: 6091: 6085: 6082: 6079: 6076: 6073: 6070: 6067: 6065: 6053: 6048: 6042: 6041: 6038: 6034: 6030: 6027: 6017: 6012: 6009: 6005: 6001: 5998: 5995: 5992: 5990: 5981: 5976: 5975: 5941: 5938: 5935: 5911: 5886: 5866: 5863: 5860: 5857: 5851: 5846: 5824: 5820: 5817: 5813: 5792: 5789: 5773: 5770: 5753: 5732: 5716: 5715: 5704: 5701: 5698: 5695: 5691: 5687: 5677: 5672: 5669: 5666: 5663: 5660: 5656: 5652: 5642: 5617: 5614: 5611: 5607: 5603: 5599: 5595: 5591: 5587: 5584: 5581: 5578: 5574: 5570: 5566: 5554: 5553: 5542: 5539: 5536: 5533: 5529: 5525: 5521: 5517: 5514: 5511: 5508: 5504: 5500: 5496: 5490: 5486: 5480: 5476: 5472: 5469: 5466: 5463: 5459: 5455: 5451: 5436: 5435: 5424: 5421: 5418: 5415: 5411: 5407: 5397: 5392: 5389: 5386: 5383: 5380: 5376: 5372: 5362: 5337: 5334: 5332: 5329: 5320: 5316: 5312: 5308: 5297: 5296: 5285: 5282: 5278: 5274: 5269: 5264: 5260: 5256: 5253: 5249: 5245: 5240: 5236: 5220: 5213: 5203: 5202: 5191: 5186: 5181: 5177: 5171: 5167: 5161: 5157: 5151: 5148: 5145: 5141: 5134: 5131: 5128: 5123: 5120: 5117: 5111: 5106: 5102: 5075: 5068: 5061: 5054: 5047: 5040: 5033: 5026: 5019: 5016: 5015: 4999: 4994: 4990: 4986: 4982: 4978: 4975: 4972: 4969: 4966: 4962: 4956: 4951: 4947: 4941: 4937: 4931: 4927: 4921: 4917: 4911: 4908: 4905: 4901: 4895: 4892: 4887: 4882: 4878: 4862: 4855: 4848: 4837: 4833: 4829: 4822: 4818: 4814: 4810: 4806: 4802: 4799: 4798: 4787: 4782: 4778: 4772: 4768: 4764: 4759: 4755: 4751: 4746: 4742: 4738: 4733: 4730: 4727: 4723: 4717: 4713: 4709: 4700: 4681: 4677: 4673: 4669: 4664: 4659: 4658: 4647: 4639: 4633: 4628: 4624: 4620: 4616: 4613: 4608: 4604: 4599: 4595: 4591: 4587: 4583: 4579: 4574: 4570: 4564: 4561: 4556: 4553: 4550: 4547: 4543: 4539: 4534: 4530: 4510: 4507: 4458: 4455: 4452: 4449: 4446: 4443: 4422: 4418: 4415: 4412: 4408: 4387: 4384: 4381: 4378: 4375: 4354: 4350: 4347: 4344: 4340: 4316: 4313: 4287: 4284: 4261: 4258: 4255: 4252: 4249: 4228: 4224: 4221: 4217: 4200: 4197: 4163: 4160: 4119: 4116: 4114: 4111: 4076: 4073: 4070: 4066: 4040: 4037: 4034: 4030: 4007: 4004: 4001: 3997: 3974: 3971: 3968: 3964: 3932: 3929: 3926: 3922: 3899: 3896: 3893: 3889: 3869: 3866: 3807: 3806: 3803: 3800: 3797: 3793: 3792: 3789: 3786: 3783: 3779: 3778: 3775: 3772: 3769: 3765: 3764: 3761: 3758: 3755: 3751: 3750: 3747: 3744: 3741: 3737: 3736: 3733: 3730: 3727: 3723: 3722: 3719: 3716: 3713: 3709: 3708: 3705: 3702: 3699: 3695: 3694: 3691: 3688: 3684: 3683: 3680: 3679:Polarizations 3661: 3657: 3653: 3648: 3644: 3640: 3635: 3631: 3612:crystal optics 3599: 3598: 3587: 3582: 3575: 3571: 3563: 3558: 3551: 3547: 3539: 3534: 3527: 3523: 3493: 3489: 3468: 3464: 3458: 3454: 3450: 3445: 3441: 3437: 3433: 3429: 3426: 3421: 3416: 3411: 3389: 3385: 3363: 3351: 3350: 3334: 3329: 3326: 3321: 3317: 3313: 3309: 3305: 3302: 3297: 3292: 3287: 3282: 3277: 3272: 3269: 3266: 3262: 3254: 3250: 3244: 3240: 3232: 3228: 3222: 3218: 3214: 3211: 3208: 3205: 3201: 3197: 3192: 3189: 3186: 3182: 3156: 3152: 3148: 3143: 3139: 3135: 3130: 3126: 3103: 3099: 3078: 3073: 3069: 3065: 3061: 3040: 3020: 3016: 3010: 3006: 3002: 2997: 2993: 2989: 2985: 2981: 2978: 2973: 2968: 2963: 2939: 2927: 2926: 2910: 2905: 2902: 2897: 2893: 2889: 2885: 2881: 2876: 2871: 2866: 2863: 2859: 2853: 2850: 2847: 2843: 2839: 2836: 2833: 2830: 2826: 2822: 2817: 2813: 2780: 2776: 2773: 2769: 2765: 2743: 2740: 2737: 2733: 2730: 2696: 2695:Phase matching 2693: 2683:respectively. 2664: 2657: 2650: 2643: 2636: 2629: 2623: 2622: 2607: 2603: 2594: 2589: 2585: 2580: 2574: 2569: 2564: 2560: 2556: 2551: 2546: 2541: 2536: 2532: 2528: 2522: 2518: 2513: 2510: 2505: 2501: 2497: 2492: 2488: 2484: 2481: 2478: 2474: 2468: 2461: 2457: 2449: 2445: 2441: 2438: 2433: 2430: 2425: 2421: 2417: 2412: 2408: 2404: 2401: 2398: 2394: 2388: 2384: 2378: 2374: 2370: 2367: 2362: 2357: 2353: 2349: 2346: 2343: 2339: 2333: 2326: 2322: 2316: 2311: 2306: 2302: 2298: 2295: 2292: 2288: 2282: 2275: 2271: 2264: 2258: 2255: 2252: 2248: 2242: 2237: 2233: 2227: 2224: 2222: 2220: 2217: 2214: 2211: 2206: 2201: 2194: 2191: 2188: 2184: 2178: 2174: 2170: 2167: 2165: 2156: 2151: 2150: 2128: 2127: 2116: 2108: 2103: 2098: 2094: 2090: 2087: 2083: 2077: 2073: 2067: 2064: 2059: 2054: 2049: 2045: 2041: 2038: 2034: 2028: 2024: 2018: 2015: 2010: 2007: 2004: 2001: 1997: 1978: 1977: 1966: 1963: 1960: 1955: 1951: 1947: 1944: 1941: 1936: 1932: 1928: 1925: 1922: 1917: 1913: 1909: 1906: 1903: 1898: 1894: 1890: 1887: 1884: 1881: 1877: 1853: 1846: 1832: 1831: 1820: 1817: 1814: 1811: 1806: 1801: 1794: 1791: 1788: 1784: 1778: 1774: 1770: 1760: 1728: 1727: 1709: 1699: 1695: 1691: 1685: 1681: 1670: 1666: 1660: 1656: 1651: 1635: 1634: 1623: 1619: 1615: 1611: 1602: 1598: 1594: 1588: 1584: 1574: 1570: 1564: 1560: 1554: 1550: 1544: 1540: 1512: 1509: 1508: 1507: 1496: 1486: 1476: 1472: 1468: 1462: 1458: 1447: 1443: 1437: 1433: 1428: 1423: 1419: 1410: 1406: 1402: 1396: 1392: 1382: 1378: 1372: 1368: 1362: 1358: 1352: 1348: 1332: 1331: 1320: 1317: 1313: 1309: 1306: 1288: 1287: 1276: 1272: 1268: 1264: 1255: 1251: 1247: 1241: 1237: 1227: 1223: 1217: 1213: 1207: 1203: 1197: 1193: 1175: 1174: 1163: 1160: 1157: 1153: 1149: 1146: 1123: 1120: 1111: 1095: 1094: 1082: 1076: 1072: 1068: 1064: 1059: 1055: 1052: 1048: 1044: 1041: 1037: 1032: 1028: 1025: 1021: 1017: 1014: 977: 976: 965: 955: 945: 941: 937: 931: 927: 916: 912: 906: 902: 897: 892: 889: 885: 876: 872: 868: 862: 858: 848: 844: 838: 834: 828: 824: 820: 817: 814: 811: 788: 785: 737: 736: 725: 721: 717: 714: 711: 708: 705: 700: 695: 688: 685: 682: 678: 674: 671: 668: 665: 660: 655: 648: 645: 642: 638: 634: 631: 628: 625: 621: 615: 612: 609: 605: 600: 594: 590: 586: 583: 580: 577: 573: 551:electric field 511: 508: 494: 491: 490: 489: 479: 472:Acousto-optics 469: 462:Pockels effect 454: 451: 450: 449: 443: 436: 426: 420: 414: 409: 403: 389: 386: 383: 379: 364: 358: 352: 349: 319: 305: 291: 277: 263: 260: 257: 253: 235: 232: 231: 230: 224: 218: 212: 206: 200: 194: 184: 178: 172: 166: 160: 148: 145: 136: 133: 93: 90: 70:electric field 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8689: 8678: 8675: 8673: 8670: 8669: 8667: 8652: 8649: 8647: 8644: 8642: 8639: 8637: 8634: 8632: 8629: 8628: 8626: 8622: 8616: 8613: 8611: 8610:Ocean physics 8608: 8606: 8603: 8601: 8598: 8596: 8593: 8591: 8588: 8586: 8583: 8581: 8578: 8576: 8573: 8571: 8568: 8567: 8565: 8563: 8559: 8553: 8550: 8546: 8545:Modern optics 8543: 8541: 8538: 8536: 8533: 8532: 8531: 8528: 8526: 8523: 8521: 8518: 8516: 8513: 8509: 8506: 8504: 8501: 8500: 8499: 8496: 8495: 8493: 8491: 8487: 8479: 8476: 8474: 8471: 8470: 8469: 8466: 8462: 8459: 8457: 8454: 8453: 8452: 8449: 8447: 8444: 8442: 8439: 8435: 8432: 8430: 8427: 8425: 8422: 8420: 8417: 8416: 8415: 8412: 8411: 8409: 8407: 8403: 8395: 8394:Computational 8392: 8391: 8390: 8387: 8385: 8382: 8381: 8379: 8375: 8367: 8364: 8363: 8362: 8359: 8357: 8354: 8353: 8351: 8347: 8343: 8335: 8330: 8328: 8323: 8321: 8316: 8315: 8312: 8296: 8290: 8286: 8282: 8278: 8274: 8270: 8264: 8261: 8256: 8250: 8246: 8239: 8237: 8233: 8228: 8224: 8220: 8216: 8208: 8205: 8200: 8196: 8192: 8188: 8184: 8180: 8176: 8172: 8168: 8164: 8156: 8153: 8141: 8137: 8133: 8128: 8123: 8119: 8115: 8111: 8104: 8101: 8096: 8092: 8087: 8082: 8078: 8074: 8070: 8066: 8062: 8058: 8054: 8047: 8044: 8039: 8035: 8031: 8027: 8023: 8019: 8018: 8010: 8008: 8004: 7999: 7995: 7991: 7987: 7983: 7979: 7974: 7969: 7966:(4): 109638. 7965: 7961: 7954: 7951: 7946: 7942: 7938: 7934: 7930: 7926: 7922: 7918: 7911: 7908: 7897:on 2020-04-20 7893: 7889: 7885: 7881: 7877: 7873: 7869: 7865: 7861: 7857: 7853: 7846: 7839: 7836: 7831: 7827: 7823: 7819: 7815: 7811: 7804: 7801: 7796: 7792: 7788: 7784: 7780: 7776: 7768: 7765: 7760: 7756: 7752: 7748: 7744: 7740: 7732: 7729: 7724: 7720: 7716: 7712: 7708: 7704: 7700: 7696: 7689: 7686: 7681: 7677: 7673: 7669: 7664: 7659: 7655: 7651: 7647: 7643: 7639: 7632: 7629: 7624: 7620: 7616: 7612: 7608: 7604: 7600: 7596: 7589: 7586: 7582: 7578: 7575: 7570: 7567: 7562: 7558: 7554: 7550: 7543: 7541: 7537: 7532: 7528: 7523: 7518: 7514: 7510: 7503: 7501: 7497: 7492: 7488: 7484: 7480: 7476: 7472: 7465: 7462: 7457: 7453: 7449: 7445: 7441: 7437: 7430: 7427: 7422: 7418: 7414: 7410: 7406: 7402: 7395: 7392: 7387: 7383: 7379: 7375: 7371: 7367: 7363: 7359: 7355: 7348: 7345: 7340: 7336: 7332: 7326: 7322: 7318: 7314: 7310: 7303: 7300: 7288: 7284: 7280: 7276: 7272: 7268: 7264: 7260: 7256: 7252: 7245: 7243: 7241: 7237: 7232: 7228: 7223: 7218: 7214: 7210: 7206: 7202: 7198: 7191: 7188: 7185: 7181: 7176: 7173: 7170: 7166: 7161: 7158: 7146: 7142: 7136: 7132: 7131: 7126: 7119: 7116: 7104: 7100: 7096: 7092: 7085: 7082: 7077: 7073: 7069: 7065: 7061: 7057: 7050: 7048: 7044: 7039: 7035: 7031: 7027: 7023: 7019: 7015: 7011: 7007: 7003: 6999: 6992: 6989: 6984: 6980: 6976: 6972: 6968: 6964: 6956: 6953: 6948: 6944: 6940: 6936: 6929: 6926: 6913: 6909: 6905: 6901: 6897: 6893: 6889: 6885: 6881: 6877: 6869: 6866: 6861: 6857: 6853: 6852:10.1038/25303 6849: 6845: 6841: 6836: 6831: 6827: 6823: 6816: 6813: 6808: 6802: 6798: 6791: 6788: 6783: 6779: 6775: 6771: 6764: 6761: 6750:on 2016-01-01 6746: 6742: 6735: 6728: 6726: 6722: 6717: 6713: 6709: 6705: 6701: 6697: 6690: 6687: 6682: 6676: 6672: 6665: 6662: 6657: 6651: 6647: 6643: 6637: 6634: 6629: 6623: 6619: 6615: 6609: 6606: 6599: 6593: 6587: 6583: 6578: 6576:SPIE Newsroom 6575: 6571: 6568: 6565: 6563: 6559: 6556: 6553: 6551: 6547: 6544: 6541: 6538: 6534: 6531: 6528: 6527: 6523: 6519: 6516: 6514: 6511: 6509: 6506: 6505: 6501: 6492: 6489: 6477: 6473: 6465: 6462: 6458: 6455: 6451: 6439: 6435: 6429: 6422: 6419:860 nm: 6418: 6411: 6408:806 nm: 6407: 6405: 6402:800 nm: 6401: 6400: 6399: 6392: 6387: 6379: 6377: 6373: 6371: 6366: 6349: 6339: 6335: 6331: 6328: 6322: 6316: 6313: 6308: 6305: 6302: 6291: 6282: 6278: 6274: 6273: 6269: 6267: 6237: 6234: 6229: 6226: 6221: 6211: 6208: 6205: 6190: 6177: 6173: 6168: 6167: 6163: 6156: 6154: 6152: 6148: 6144: 6136: 6116: 6113: 6107: 6089: 6083: 6080: 6077: 6071: 6068: 6066: 6046: 6036: 6025: 6010: 6007: 5996: 5993: 5991: 5966: 5965: 5964: 5962: 5958: 5953: 5939: 5933: 5926: 5900: 5884: 5864: 5858: 5855: 5844: 5815: 5790: 5779: 5771: 5769: 5767: 5751: 5721: 5702: 5696: 5693: 5670: 5667: 5661: 5658: 5631: 5630: 5629: 5612: 5609: 5593: 5585: 5579: 5576: 5540: 5534: 5531: 5515: 5509: 5506: 5488: 5484: 5478: 5474: 5470: 5464: 5461: 5441: 5440: 5439: 5422: 5416: 5413: 5390: 5387: 5381: 5378: 5351: 5350: 5349: 5347: 5343: 5335: 5330: 5328: 5326: 5305: 5301: 5283: 5267: 5262: 5258: 5254: 5238: 5234: 5226: 5225: 5224: 5219: 5212: 5208: 5189: 5184: 5179: 5175: 5169: 5165: 5159: 5155: 5146: 5139: 5132: 5129: 5126: 5121: 5118: 5115: 5109: 5104: 5100: 5092: 5091: 5090: 5088: 5084: 5079: 5074: 5067: 5060: 5053: 5046: 5039: 5032: 5025: 4997: 4984: 4976: 4973: 4970: 4964: 4960: 4954: 4949: 4945: 4939: 4935: 4929: 4925: 4919: 4915: 4906: 4899: 4893: 4890: 4885: 4880: 4876: 4868: 4867: 4866: 4861: 4854: 4847: 4843: 4826: 4785: 4780: 4770: 4762: 4757: 4749: 4744: 4728: 4721: 4715: 4711: 4707: 4698: 4690: 4689: 4688: 4685: 4667: 4645: 4637: 4631: 4622: 4614: 4611: 4606: 4602: 4597: 4593: 4589: 4572: 4568: 4562: 4559: 4554: 4548: 4545: 4532: 4520: 4519: 4518: 4516: 4508: 4506: 4499: 4495: 4493: 4488: 4483: 4480: 4476: 4456: 4450: 4447: 4444: 4441: 4413: 4410: 4385: 4379: 4376: 4373: 4345: 4342: 4311: 4282: 4259: 4253: 4250: 4247: 4219: 4205: 4198: 4196: 4194: 4189: 4187: 4186: 4181: 4177: 4176:time reversal 4174:(also called 4173: 4169: 4161: 4159: 4157: 4153: 4149: 4145: 4141: 4135: 4133: 4129: 4128:Nd:YAG lasers 4125: 4117: 4112: 4110: 4108: 4104: 4100: 4096: 4095:Taylor series 4092: 4091:filamentation 4071: 4064: 4054: 4035: 4028: 4002: 3995: 3969: 3962: 3953: 3949: 3927: 3920: 3894: 3887: 3874: 3867: 3865: 3863: 3859: 3855: 3851: 3847: 3841: 3839: 3834: 3832: 3827: 3821: 3818: 3814: 3804: 3801: 3798: 3795: 3794: 3790: 3787: 3784: 3781: 3780: 3776: 3773: 3770: 3767: 3766: 3762: 3759: 3756: 3753: 3752: 3748: 3745: 3742: 3739: 3738: 3734: 3731: 3728: 3725: 3724: 3720: 3717: 3714: 3711: 3710: 3706: 3703: 3700: 3697: 3696: 3692: 3689: 3686: 3685: 3677: 3659: 3655: 3651: 3646: 3642: 3638: 3633: 3629: 3615: 3613: 3608: 3604: 3585: 3580: 3561: 3556: 3537: 3532: 3509: 3508: 3507: 3491: 3487: 3466: 3462: 3456: 3452: 3443: 3439: 3427: 3419: 3387: 3383: 3332: 3324: 3319: 3315: 3311: 3303: 3295: 3285: 3280: 3264: 3260: 3252: 3248: 3242: 3238: 3230: 3226: 3220: 3216: 3212: 3206: 3203: 3187: 3180: 3172: 3171: 3170: 3154: 3150: 3146: 3141: 3137: 3133: 3128: 3124: 3101: 3097: 3071: 3067: 3038: 3018: 3014: 3008: 3004: 2995: 2991: 2979: 2971: 2954: 2908: 2900: 2895: 2891: 2887: 2879: 2874: 2861: 2857: 2851: 2848: 2845: 2841: 2837: 2831: 2828: 2815: 2811: 2803: 2802: 2801: 2795: 2774: 2771: 2767: 2741: 2738: 2735: 2731: 2728: 2718: 2717:monotonically 2714: 2710: 2706: 2701: 2694: 2692: 2690: 2684: 2682: 2678: 2674: 2670: 2663: 2660: − 2656: 2649: 2646: + 2642: 2635: 2628: 2605: 2601: 2592: 2587: 2583: 2578: 2572: 2567: 2562: 2558: 2554: 2549: 2544: 2539: 2534: 2530: 2526: 2520: 2516: 2511: 2503: 2499: 2495: 2490: 2486: 2479: 2476: 2472: 2466: 2459: 2455: 2447: 2443: 2439: 2436: 2431: 2423: 2419: 2415: 2410: 2406: 2399: 2396: 2392: 2386: 2382: 2376: 2372: 2368: 2365: 2360: 2355: 2351: 2347: 2344: 2341: 2337: 2331: 2324: 2320: 2314: 2309: 2304: 2300: 2296: 2293: 2290: 2286: 2280: 2273: 2269: 2262: 2253: 2246: 2240: 2235: 2231: 2225: 2223: 2212: 2204: 2189: 2182: 2176: 2172: 2168: 2166: 2141: 2140: 2139: 2137: 2133: 2114: 2106: 2101: 2096: 2092: 2088: 2085: 2081: 2075: 2071: 2065: 2062: 2057: 2052: 2047: 2043: 2039: 2036: 2032: 2026: 2022: 2016: 2013: 2008: 2002: 1987: 1986: 1985: 1983: 1964: 1958: 1953: 1949: 1942: 1939: 1934: 1930: 1926: 1920: 1915: 1911: 1904: 1901: 1896: 1892: 1888: 1882: 1867: 1866: 1865: 1863: 1859: 1852: 1845: 1841: 1837: 1818: 1812: 1804: 1789: 1782: 1776: 1772: 1768: 1749: 1748: 1747: 1745: 1741: 1737: 1732: 1697: 1693: 1683: 1668: 1664: 1658: 1654: 1649: 1640: 1639: 1638: 1621: 1613: 1600: 1596: 1586: 1572: 1568: 1562: 1558: 1552: 1542: 1530: 1529: 1528: 1526: 1522: 1518: 1510: 1494: 1474: 1470: 1460: 1445: 1441: 1435: 1431: 1426: 1421: 1408: 1404: 1394: 1380: 1376: 1370: 1366: 1360: 1350: 1338: 1337: 1336: 1318: 1315: 1307: 1297: 1296: 1295: 1293: 1274: 1266: 1253: 1249: 1239: 1225: 1221: 1215: 1211: 1205: 1195: 1183: 1182: 1181: 1180: 1179:wave equation 1161: 1158: 1155: 1147: 1137: 1136: 1135: 1121: 1118: 1109: 1100: 1074: 1066: 1062: 1053: 1046: 1039: 1035: 1026: 1019: 1015: 1005: 1004: 1003: 1000: 998: 994: 990: 986: 982: 963: 943: 939: 929: 914: 910: 904: 900: 895: 890: 887: 874: 870: 860: 846: 842: 836: 832: 826: 818: 812: 802: 801: 800: 798: 794: 793:wave equation 786: 784: 782: 778: 774: 770: 766: 762: 758: 754: 750: 746: 742: 723: 719: 715: 712: 706: 698: 683: 676: 672: 666: 658: 643: 636: 632: 626: 610: 603: 598: 592: 588: 584: 578: 563: 562: 561: 559: 555: 552: 548: 544: 540: 536: 532: 529: 525: 524:Taylor series 521: 517: 509: 507: 504: 503:quantum state 500: 492: 487: 483: 480: 477: 473: 470: 467: 463: 460: 459: 458: 452: 447: 444: 441: 437: 434: 430: 427: 424: 421: 418: 415: 413: 410: 407: 404: 384: 377: 368: 365: 362: 359: 356: 353: 350: 347: 346:self-focusing 343: 339: 335: 331: 327: 326:optical pulse 323: 320: 317: 313: 309: 306: 303: 299: 298:self-focusing 295: 292: 289: 285: 281: 280:Self-focusing 278: 258: 251: 242: 238: 237: 233: 228: 225: 222: 219: 216: 213: 210: 207: 204: 201: 198: 195: 192: 188: 185: 182: 179: 176: 173: 170: 167: 164: 161: 158: 154: 151: 150: 146: 144: 142: 134: 132: 130: 126: 122: 118: 115: 114:Peter Franken 111: 107: 103: 99: 91: 89: 87: 83: 79: 74: 71: 67: 64: 60: 56: 52: 48: 44: 37: 34:Structure of 32: 19: 8570:Astrophysics 8384:Experimental 8298:, retrieved 8276: 8263: 8244: 8218: 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Retrieved 8117: 8113: 8103: 8060: 8056: 8046: 8021: 8015: 7963: 7959: 7953: 7920: 7916: 7910: 7899:. Retrieved 7892:the original 7855: 7851: 7838: 7813: 7809: 7803: 7778: 7774: 7767: 7742: 7738: 7731: 7698: 7694: 7688: 7645: 7641: 7631: 7598: 7594: 7588: 7569: 7552: 7548: 7512: 7508: 7477:(4): 62–75. 7474: 7470: 7464: 7442:(1): 74–83. 7439: 7435: 7429: 7407:(6): 54–59. 7404: 7400: 7394: 7361: 7357: 7347: 7312: 7309:Taylor, A.J. 7302: 7291:. Retrieved 7258: 7254: 7204: 7200: 7190: 7175: 7160: 7149:. Retrieved 7129: 7118: 7107:. Retrieved 7098: 7094: 7084: 7059: 7055: 7005: 7001: 6991: 6966: 6962: 6955: 6938: 6934: 6928: 6916:. Retrieved 6912:the original 6883: 6879: 6868: 6825: 6821: 6815: 6796: 6790: 6773: 6769: 6763: 6752:. Retrieved 6745:the original 6740: 6699: 6695: 6689: 6670: 6664: 6645: 6636: 6617: 6614:Boyd, Robert 6608: 6581: 6470:, BBO, KDP, 6397: 6374: 6367: 6290:chromophores 6287: 6275: 6271: 6270: 6169: 6165: 6164: 6160: 6140: 5960: 5956: 5954: 5924: 5898: 5777: 5775: 5765: 5719: 5717: 5555: 5437: 5341: 5339: 5324: 5306: 5302: 5298: 5217: 5210: 5206: 5204: 5086: 5082: 5080: 5072: 5065: 5058: 5051: 5044: 5037: 5030: 5023: 5017: 4859: 4852: 4845: 4841: 4827: 4800: 4686: 4662: 4660: 4514: 4512: 4504: 4484: 4478: 4472: 4192: 4190: 4183: 4179: 4175: 4171: 4167: 4165: 4156:birefringent 4136: 4123: 4121: 4113:Example uses 4055: 3878: 3849: 3842: 3835: 3822: 3816: 3812: 3810: 3602: 3600: 3352: 2929:at position 2928: 2799: 2794:Birefringent 2685: 2661: 2654: 2647: 2640: 2633: 2626: 2624: 2135: 2129: 1979: 1861: 1857: 1850: 1843: 1839: 1835: 1833: 1743: 1739: 1735: 1733: 1729: 1636: 1514: 1333: 1289: 1176: 1096: 1001: 996: 988: 980: 978: 790: 777:polarization 768: 764: 760: 756: 752: 748: 740: 738: 557: 553: 546: 542: 538: 513: 496: 456: 330:spatial mode 190: 156: 138: 116: 95: 80:. Above the 72: 65: 46: 42: 41: 8473:Statistical 8389:Theoretical 8366:Engineering 7858:(1): 24–6. 6918:6 September 4053:processes. 2953:wave vector 2951:, with the 1292:Gauss's law 1099:Gauss's law 338:Kerr effect 312:Kerr effect 284:Kerr effect 241:Kerr effect 129:Bloembergen 36:KTP crystal 8666:Categories 8590:Geophysics 8580:Biophysics 8424:Analytical 8377:Approaches 8300:2023-11-27 8146:2023-11-27 8024:(2): 328. 7973:1901.09274 7901:2019-08-26 7293:2015-06-23 7151:2011-08-16 7109:2015-10-31 6754:2015-10-21 6702:(6): 229. 6600:References 6452:(LBO) and 6384:See also: 4199:Principles 4099:attosecond 2715:decreases 1980:and using 781:symmetries 743:-th-order 545:) at time 528:dielectric 499:parametric 334:dispersion 155:(SHG), or 8540:Molecular 8441:Acoustics 8434:Continuum 8429:Celestial 8419:Newtonian 8406:Classical 8349:Divisions 8199:205279732 8136:2096-4579 8063:: 28483. 7998:118828500 7945:122790937 7880:1539-4794 7795:1932-7447 7759:1932-7447 7715:0002-7863 7672:1094-4087 7549:JETP Lett 7522:0801.2675 7386:0030-4026 7339:120016673 7180:Boyd 2008 7165:Boyd 2008 7133:. Wiley. 7101:(2): 55. 7038:202145209 7030:0031-8949 6350:ω 6347:ℏ 6332:δ 6303:− 6235:β 6212:⋅ 6209:α 6196:μ 6187:μ 6114:ℓ 6111:ℏ 6084:− 6078:ℓ 6075:ℏ 6072:− 6029:ℏ 6011:− 6000:ℏ 5997:− 5940:ℓ 5937:ℏ 5934:± 5885:ℓ 5865:ℓ 5862:ℏ 5859:± 5819:ℏ 5791:ω 5788:ℏ 5671:− 5594:× 5516:× 5475:ϵ 5391:− 5268:∗ 5255:∝ 5185:∗ 5140:χ 5119:ω 4985:⋅ 4977:− 4971:ω 4955:∗ 4916:ε 4900:χ 4881:ω 4767:Ξ 4754:Ξ 4741:Ξ 4722:χ 4712:ε 4623:⋅ 4615:− 4603:ω 4529:Ξ 4457:ℓ 4454:ℏ 4448:± 4417:ℏ 4386:ℓ 4383:ℏ 4380:∓ 4349:ℏ 4346:− 4315:→ 4286:→ 4260:ℓ 4257:ℏ 4254:± 4223:ℏ 4168:conjugate 4065:χ 4029:χ 3996:χ 3963:χ 3921:χ 3888:χ 3656:λ 3652:≤ 3643:λ 3639:≤ 3630:λ 3574:→ 3550:→ 3526:→ 3488:ω 3453:ω 3440:ω 3425:‖ 3410:‖ 3384:ω 3316:ω 3312:− 3304:⋅ 3213:∝ 3151:ω 3138:ω 3125:ω 3098:ω 3068:ω 3005:ω 2992:ω 2977:‖ 2962:‖ 2892:ω 2888:− 2880:⋅ 2742:ω 2729:ω 2705:BK7 glass 2500:ω 2496:− 2487:ω 2477:− 2467:∗ 2420:ω 2407:ω 2397:− 2352:ω 2342:− 2301:ω 2291:− 2247:χ 2232:ε 2183:χ 2173:ε 2093:ω 2086:− 2044:ω 2037:− 1950:ω 1943:⁡ 1912:ω 1905:⁡ 1783:χ 1773:ε 1690:∂ 1680:∂ 1655:ε 1593:∂ 1583:∂ 1553:− 1539:∇ 1467:∂ 1457:∂ 1432:ε 1401:∂ 1391:∂ 1361:− 1347:∇ 1308:⋅ 1305:∇ 1246:∂ 1236:∂ 1206:− 1192:∇ 1148:⋅ 1145:∇ 1110:ρ 1071:∇ 1067:− 1054:⋅ 1051:∇ 1043:∇ 1027:× 1024:∇ 1016:× 1013:∇ 936:∂ 926:∂ 901:ε 891:− 867:∂ 857:∂ 819:× 816:∇ 813:× 810:∇ 716:… 677:χ 637:χ 604:χ 589:ε 520:too large 438:Multiple 378:χ 369:(XPW), a 302:mode-lock 252:χ 141:frequency 106:Bell Labs 59:nonlinear 8271:(2020), 8191:28556297 8140:Archived 8095:27339882 7888:19865292 7830:35614732 7723:17378567 7680:19532708 7623:11015940 7577:Archived 7491:24997062 7456:24975872 7421:24967871 7311:(1997). 7287:Archived 7283:19776917 7231:20588396 7145:Archived 7103:Archived 6963:Nat Phys 6908:10060879 6860:16080209 6644:(2002). 6616:(2008). 6570:Archived 6558:Archived 6546:Archived 6533:Archived 6502:See also 6448:, KDP), 5877:, where 4146:), KTP ( 4142:), KDP ( 3749:Type IV 3031:, where 2768:′ 2732:′ 336:and the 276:effect). 239:Optical 8624:Related 8508:General 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4586:) 4582:x 4578:( 4573:j 4569:E 4563:2 4560:1 4555:= 4552:) 4549:t 4546:, 4542:x 4538:( 4533:j 4515:j 4451:2 4445:= 4442:L 4421:k 4414:2 4411:= 4407:P 4377:= 4374:L 4353:k 4343:= 4339:P 4312:z 4283:n 4251:= 4248:L 4227:k 4220:= 4216:P 4138:( 4075:) 4072:3 4069:( 4039:) 4036:2 4033:( 4006:) 4003:3 4000:( 3973:) 3970:3 3967:( 3931:) 3928:3 3925:( 3898:) 3895:2 3892:( 3850:k 3817:o 3813:e 3660:i 3647:s 3634:p 3621:( 3586:. 3581:2 3570:k 3562:+ 3557:1 3546:k 3538:= 3533:3 3522:k 3492:3 3467:c 3463:/ 3457:3 3449:) 3444:3 3436:( 3432:n 3428:= 3420:3 3415:k 3388:3 3362:x 3333:+ 3328:] 3325:t 3320:3 3308:x 3301:) 3296:2 3291:k 3286:+ 3281:1 3276:k 3271:( 3268:[ 3265:i 3261:e 3253:2 3249:n 3243:2 3239:E 3231:1 3227:n 3221:1 3217:E 3210:) 3207:t 3204:, 3200:x 3196:( 3191:) 3188:2 3185:( 3181:P 3155:2 3147:+ 3142:1 3134:= 3129:3 3102:j 3077:) 3072:j 3064:( 3060:n 3039:c 3019:c 3015:/ 3009:j 3001:) 2996:j 2988:( 2984:n 2980:= 2972:j 2967:k 2938:x 2909:+ 2904:) 2901:t 2896:j 2884:x 2875:j 2870:k 2865:( 2862:i 2858:e 2852:0 2849:, 2846:j 2842:E 2838:= 2835:) 2832:t 2829:, 2825:x 2821:( 2816:j 2812:E 2779:k 2775:2 2772:= 2764:k 2739:2 2736:= 2665:2 2662:ω 2658:1 2655:ω 2651:2 2648:ω 2644:1 2641:ω 2637:2 2634:ω 2630:1 2627:ω 2606:, 2602:] 2593:+ 2588:0 2584:e 2579:) 2573:2 2568:| 2563:2 2559:E 2555:| 2550:+ 2545:2 2540:| 2535:1 2531:E 2527:| 2521:( 2517:+ 2512:t 2509:) 2504:2 2491:1 2483:( 2480:i 2473:e 2460:2 2456:E 2448:1 2444:E 2440:2 2437:+ 2432:t 2429:) 2424:2 2416:+ 2411:1 2403:( 2400:i 2393:e 2387:2 2383:E 2377:1 2373:E 2369:2 2366:+ 2361:t 2356:2 2348:2 2345:i 2338:e 2332:2 2325:2 2321:E 2315:+ 2310:t 2305:1 2297:2 2294:i 2287:e 2281:2 2274:1 2270:E 2263:[ 2257:) 2254:2 2251:( 2241:4 2236:0 2226:= 2216:) 2213:t 2210:( 2205:2 2200:E 2193:) 2190:2 2187:( 2177:0 2169:= 2155:P 2136:P 2115:, 2107:+ 2102:t 2097:2 2089:i 2082:e 2076:2 2072:E 2066:2 2063:1 2058:+ 2053:t 2048:1 2040:i 2033:e 2027:1 2023:E 2017:2 2014:1 2009:= 2006:) 2003:t 2000:( 1996:E 1965:, 1962:) 1959:t 1954:2 1946:( 1935:2 1931:E 1927:+ 1924:) 1921:t 1916:1 1908:( 1897:1 1893:E 1889:= 1886:) 1883:t 1880:( 1876:E 1862:t 1860:( 1858:E 1854:2 1851:ω 1847:1 1844:ω 1840:t 1838:( 1836:E 1819:. 1816:) 1813:t 1810:( 1805:2 1800:E 1793:) 1790:2 1787:( 1777:0 1769:= 1759:P 1744:P 1740:n 1736:n 1708:P 1698:2 1694:t 1684:2 1669:2 1665:c 1659:0 1650:1 1622:, 1618:0 1614:= 1610:E 1601:2 1597:t 1587:2 1573:2 1569:c 1563:2 1559:n 1549:E 1543:2 1495:. 1485:P 1475:2 1471:t 1461:2 1446:2 1442:c 1436:0 1427:1 1422:= 1418:E 1409:2 1405:t 1395:2 1381:2 1377:c 1371:2 1367:n 1357:E 1351:2 1319:0 1316:= 1312:E 1275:. 1271:0 1267:= 1263:E 1254:2 1250:t 1240:2 1226:2 1222:c 1216:2 1212:n 1202:E 1196:2 1162:, 1159:0 1156:= 1152:D 1122:0 1119:= 1081:V 1075:2 1063:) 1058:V 1047:( 1040:= 1036:) 1031:V 1020:( 997:P 989:n 981:P 964:, 954:P 944:2 940:t 930:2 915:2 911:c 905:0 896:1 888:= 884:E 875:2 871:t 861:2 847:2 843:c 837:2 833:n 827:+ 823:E 769:n 765:t 763:( 761:E 757:t 755:( 753:P 749:n 741:n 724:, 720:) 713:+ 710:) 707:t 704:( 699:3 694:E 687:) 684:3 681:( 673:+ 670:) 667:t 664:( 659:2 654:E 647:) 644:2 641:( 633:+ 630:) 627:t 624:( 620:E 614:) 611:1 608:( 599:( 593:0 585:= 582:) 579:t 576:( 572:P 558:t 556:( 554:E 547:t 543:t 541:( 539:P 533:( 488:. 478:. 468:. 448:. 419:. 408:. 388:) 385:3 382:( 262:) 259:3 256:( 229:. 73:E 66:P 45:( 20:)

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