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Ottica non lineare_Saleh-Teich_Fundamentals of Photonics

Ottica non lineare_Saleh-Teich_Fundamentals of Photonics -...

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Unformatted text preview: SCIELIU 1%ch Fuudamquav .96. PI’LOIDWIQI . U... g; NONLINEAR OPTICS .EIJ NONLINEAR OPTICAL MEDIA B. The ElectrO-Optic Effect C. Three-Wave Mixing D. Phase Matching and Tuning Curves E. Quasi- Phase Matching $1.3 THIRD-ORDER NONLINEAR OPTICS C. Cross-Phase Modulation (XPM) D. Four-Wave Mixing (FWM) E. Optical Phase Conjugation (0P0) THEORY A. Second-Harmonic Generation (SHG) B. Optical Frequency Conversion (OFC) THEORY A. Four-Wave Mixing (FWM) '3: ._. ”-9- -'I.!' | J C. Optical Phase Conjugation (OPC) 2:16 ANISOTROPIO NONLINEAR MEDIA .7 DISPERSIVE NONLINEAR MEDIA _Vk'olaas Bloembergen (born I920) has carried out pio- «raring studies in nonlinear Optics since the early [9605. He . Ihnred the I98l Nobel Prize with Arthur Schawlow. 31.2 SECOND-ORDER NONLINEAR OPTICS 1:. A. Second-Harmonic Generation (SHG) and Rectification :4 SECOND ORDER NONLINEAR OPTICS: COUPLED-WAVE CHAPTER 21 875 879 894 .'- -'__ A. Third- Harmonic Generation (T HG) and Optical Kerr Effect ' ”' B. Self-Phase Modulation (3PM), Self-Focusing. and Spatial Solitons 905 _s _i C. Optical Parametric Amplification (CPA) and Oscillation (0P0) 1:15 THIRD-ORDER NONLINEAR OPTICS: COUPLED-WAVE 917 B. Three-Wave Mixing and Third-Harmonic Generation (THG) 924 927 $73 Throughout the long history of optics, and indeed until relatively recent] thought that all optical media were linear. The consequences of this assum far-reaching: I The optical properties of materials, such as refractive index and absorp ficient are independent of light intensity = _ II The principle of superposition, a fundamental tenet of classical optics, is '1 __ ble. - " I The frequency of light is never altered by its passage through a medium I Two beams of light in the same region of a medium have no effect on eaé so that light cannot be used to control light. The operation of the first laser in [960 enabled us to examine the behavior optical materials at higher intensities than previously possible. Experiments c- in the post- laser era clearly demonstrate that optical media do in fact exhibit no behavior as exemplified by the following observations: I The refractive index, and consequently the speed of light in a nonlinear medium, does depend on light intensity. I The principle of superposition is violated in a nonlinear optical medium... I The frequency of light is altered as it passes through a nonlinear Optical [11 the light can change from red to blue, for example. I Photons do interact within the confines of a nonlinear optical medium so th- can indeed be used to control light. _'. The field of nonlinear optics offers a host of fascinating phenomena, many of ". are also eminently useful. Nonlinear optical behavior IS not observed when light travels in free spa “nonlinearity” resides 1n the medium through which the light travels, rather than‘T' light itself. The interaction of light with light IS therefore mediated by the no l'lj' - medium: the presence of an optical field modifies the properties of the medium.‘ in turn causes another optical field or even the original field itself to be modifi As discussed 1n Chapter 5 the properties of a dielectric medium through an optical electromagnetic wave propagates are described by the relation betwu-j: polarization- -density vector ?(r, t) and the electric- field vector 8(1', t) Indeed it ful to view fP(r, t) as the output of a system whose input is 8(1', t). The mathe relation between the vector functions ?(r, t) and 8(r, t.) which IS governed b I__ characteristics of the medium, defines the system. The medium 15 said to be non. _-._ if this relation is nonlinear (see Sec. 5.2). "' This Chapter In Chapter 5, dielectric media were further classified with respect to their dispe. - ness, homogeneity, and' Isotropy (see Sec. 5.2). To focus on the principal effect ofl es is nondispersive, homogeneous, and isotropic. The vectors 9’ and 8 are consequ .. parallel at every position and time and may therefore be examined on a componenléi' component basis. - The theory of nonlinear optics and its applications is presented at two level simplified approach 13 provided' in Secs. 21.1—21.3 This 1s followed by a more data analysis of the same phenomena in Sec. 21 4 and Sec. 21.5. ' The propagation of light In media characterized by a second- order (quadratic) 11:: linear relation between SP and 8 IS described' 1n Sec 21.2 and Sec. 2| .4. Applicatlld': include the frequency doubling of a monochromatic wave (second-harmonic gen 11011), the mixing of two monochromatic waves to generate a third wave at their sum; - ' difference frequencies (frequency conversion), the use of two monochromatic wa- 374 \ 21.1 NONLINEAR OPTICAL MEDIA 875 it parametric-amplification device to create an oscillator (parametric oscillation). jive propagation in a medium with a third—order (cubic) relation between {P and 8 1‘3 discussed in Secs. 21.3 and 21.5. Applications include third-harmonic generation. : -phase modulation, selflfocnsing, four-wave mixing, and phase conjugation. The ._-. - Ivior of anisotropic and dispersive nonlinear optical media is briefly considered in :__'- - s. 21.6 and 21.7, respectively. 5;. _ nuencies may exchange energy with one another via the nonlinear property of the __' uium, but their total energy is conserved. This class of nonlinear phenomena are E- Iwn as parametric interactions. Several nonlinear phenomena involving nonpara- 'c interactions are described in other chapters of this book: I Laser interactions. The interaction of light with a medium at frequencies near " the resonances of an atomic or molecular transitions involves phenomena such as absorption. and stimulated and spontaneous emission, as described in Sec. 13.3. These interactions become nonlinear when the light is sufficiently intense so that the populations of the various energy levels are significantly altered. Nonlinear . optical effects are manifested in the saturation of laser amplifiers and saturable absorbers (Sec. 14.4). Multiphoton absorption. Intense light can induce the absorption of a collection of photons whose total energy matches that of an atomic transition. For k-photon absorption, the rate of absorption is proportional to 1’“, where I is the optical intensity. This nonlinear-optical phenomenon is described briefly in Sec. 13.53. Nonlinear scattering. Nonlinear inelastic scattering involves the interaction of light with the vibrational or acoustic modes of a medium. Examples include =' ' '. stimulated Raman and stimulated Brillouin scattering, as described in Secs. 13.5C ii: and 14.3D. .31 is also assumed throughout this chapter that the light is described by stationary __..f _-_ntinuous waves. Nonstationary nonlinear optical phenomena include: -L._- I Nonlinear optics ofpulsed light. The parametric interaction of optical pulses with g.“ a nonlinear medium is described in Sec. 22.5. I Optical solitons are light pulses that travel over exceptionally long distances through nonlinear dispersive media without changing their width or shape. This nonlinear phenomenon is the result of a balance between dispersion and nonlinear self—phase modulation, as described in Sec. 225B. The use of solitons in optical fiber communications systems is described in Sec. 24.213. {yet another nonlinear optical effect is optical bistability. This involves nonlinear opti- l‘l'll effects together with feedback. Applications in photonic switching are described in Joe. 23.4. 21.1 NONLINEAR OPTICAL MEDIA ' A lineardielectric medium is characterized by a linear relation between the polarization density and the electric field, (P = «“8. where so is the permittivity of free space and ~ 1 is the electric susceptibility of the medium (see Sec. 5.2A). A nonlinear dielectric medium, on the other hand, is characterized by a nonlinear relation between ‘J’ and 8 :txcc Sec. 5.23). as illustrated in Fig. 21.1-l. The nonlinearity may be of microscopic or macroscopic origin. The polarization 876 CHAPTER 21 NONLINEAR OPTICS ‘P E Figure 21 .1-1 The CP-ti for (a) a linear dielectri (a) Linear (b) Nonlinear and (b) a nonlinear medium density IP = Np is a product of the individual dipole moment p induced by the electric field E and the number density of dipole moments N. The nonlinear . may reside either in p or in N. r. The relation between p and 8 is linear when E is small, but becomes nonlinear. 8 acquires values comparable to interatomic electric fields, which are typically _ 103 Vlm. This may be understood in terms of a simple Lorentz model in dipole moment is p = —e:r:. where .r: is the displacement of a mass with charg which an electric force —e£ is applied (see Sec. 5.5C). If the restraining elast is proportional to the displacement (i.e., if Hooke’s law is satisfied). the equilt displacement :I: is proportional to 8. In that case {P is proportional to 8 and the is linear. However, if the restraining force is a nonlinear function of the displa the equilibrium displacement m and the polarization density 3P are nonlinear fu' of 8 and, consequently, the medium is nonlinear. The time dynamics of an anh oscillator model describing a dielectric medium with these features is discu Sec. 21.7. Another possible origin of a nonlinear response of an optical material to light". dependence of the number density N on the optical field. An example is provid a laser medium in which the number of atoms occupying the energy levels invol * g I the absorption and emission of light are dependent on the intensity of the light __ (see Sec. l4.4). '- Since externally applied optical electric fields are typically small in compariso ._ characteristic interatomic or crystalline fields, even when focused laser light is ' the noniinearity is usually weak. The relation between 9’ and 8 is then approxi linear for small 8, deviating only slightly from linearity as 8 increases (see Fig. 1). Under these circumstances, the function that relates {P to 8 can be expanded Taylor series about 8. = 0, : T:a.8+%a282+éa353+..., (2 and it suffices to use only a few terms. The coefficients a] , a2, and (13 are the first, ond, and third derivatives of (P with respect to 8, evaluated at 8 = 0. These coeffici are characteristic constants of the medium. The first term, which is linear, dom at small 8. Clearly. a1 = 60X, where X is the linear susceptibility, which is rela the dielectric constant and the refractive index of the material by n2 = 6/6,, = 1 _ -.-. [see (5.2-1 1)]. The second term represents a quadratic or second-order nonlinel _" the third term represents a third-order nonlinearity, and so on. ' ‘ [t is customary to write (21.1—l) in the formi {P : ms + 2d£2 + 4353’s” + - - .. (2i .i' i This nomenclature is used in a number of books. such as A. Yariv. Quantum Electronics. Wiley, 3rd ed. i' An alternative relation, {P .. “(XE + xmfll + v“”£“). is used in other books. cg. Y. R. Sheri, The Prim .i ._'; rngrmlinenr Optics. Wiley. I984. paperback ed. 2002. .. 21 I NONLINEAR OPTICAL MEDIA 877 _ “Ithhcre d - -r12 and X(“)— — (ng74 are coefficients describing the strength of the second- F" third-order nonlinear el',fects respectively -..- -Equation (2| 1 -2) provides the essential mathematical characterization of a nonlin- .g optical medium. Material dispersion inhomogeneity and anisotropy have not been en into account both for the sake of simplicity and to enable us to focus on the j- . ntial features of nonlinear optical behavior. Sections 21.6 and 2| 7 are devoted to _! 'sotropic and dispersive nonlinear media, respectively. II'I"-‘ ln centrosymmetric media which have' Inversion symmetry so that the properties of '2? medium are not altered by the transtormation r —I —r the iP—E function must have 3.. symmetry, so that the reversal of 8 results In the reversal of 1? without any other ge. The second-order nonlinear coefficient d must then vanish, and the lowest Icr nonlinearity is of third order. f.Typical values of the second- order nonlinear coefficient (1 for dielectric crystals, miconductors and organic materials used In photonics applications lie In the range _= 10—24-104l (C/V2' In MKS units). Typical values of the third- order nonlinear - I Ifficient I6") for glasses, crystals semiconductors, semiconductor-dopedg olasses, if; I organic materials of interest in photonics are in the vicinity of x( 3) — _10“ ”III —10 29 I/V3 In MKS units). Biased or asymmetric quantum wells offer large nonlinearities ithe mid and far infrared. 'i- Ir-‘I'__ =IERCISE 21.1-1 Ii sity of Light Required to Elicit Nonlinear Effects. 'II‘ II'! . 3-_ Determine the light intensity (in chm2) at which the ratio ot‘the second term to the first term in .:.__ (ELI-2) is 1% in an ADP(NH,;l-I2PO..) crystal for which n = 1.5 and d = 6.8 x 10"” CW2 iii-.3 at )to = l 06 um -:.iII- I} Determine the light intensity at which the third term in (2]. l- 2) Is 1% of the first term in carbon If: disulfidetCSg )for which II = 1.6. d: 0, and x“) = L4 x 10 ‘3 Cm/V‘ at A = 694 nm. "Itte In accordance with (5.-4 8) the light intensity is I = Kola/2n = (£2>/q, where n— = min II _Ithe impedance of the medium and 17“: (p,,/€.,)" /~2 ~37? Q' Is the impedance ot free space (see .IL' .5 .4). :1 : -{ : 1’s ""II\‘ 1..“ I-EIThe Nonlinear Wave Equation J'he propagation of light In a nonlinear medium' Is governed by the wave equation (5... 7- “.315), which was derived from Maxwell’s equations for an arbitrary homogeneous, I topic dielectric medium. The isotropy of the medium ensures that the vectors 3’ ; Lind E are always parallel so that they may be examined on a component- by- component Ihnsis, which provides :"r . . .- . l 638 02? -... 2 .,_ _ z __ - .-" V 8 (12’ 0152 no at? (21.1 3) _I It is convenient to write the polarization density in (2| . I -2) as a sum of linear (6.,v8) _ and nonlinear (1PM) parts, 'J’ = 6,,v8 + CPNL, (ZN-4) TNL=2d£2+1X(I‘)8I‘-I- (2l.l—5) {I Using (21.1-4), along with the relations (: = r:,,/n., ”I" = l + v, and c0 = l/(FUMU)I/2 provided in (5.2-I I) and (52-12). allows (21.1-3) to be written as 878 CHAPTER 21 NONLINEAR OPTICS Wave 11-— ' . . - - r- In Nonlinear [t is convenient to regard (2].l—6) as a wave equation in which the term 3 is as a source that radiates in a linear medium of refractive index it. Because 1P1.- - therefore 8) is a nonlinear function of E, (21.1-6) IS a nonlinear partial di .__; equation in 8. This 18 the basic equation that underlies the theory of nonlinearti Two approximate approaches to solving this nonlinear wave equation can be: upon. The first 13 an iterative approach known as the Born approximation. This 11-; imation underlies the simplified introduction to nonlinear optics presented in Seitin and 2l. 3. The second approach IS a coupled— wave theory 111 which the nonlin -i equation is used to derive linear coupled partial differential equations that go interacting waves. This is the basis of the more advanced study of wave interact . nonlinear media presented' in Sec 2| 4 and Sec. 21. 5. ' ' Scattering Theory of Nonlinear Optics: The Barn Approximation The radiation source 3 in (21.1-6) is a function of the field 8 that it, itself, radia emphasize this point we write 8 = 3(8) and illustrate the process by the simpler-1 diagram in Fig. 21.1-2. Suppose that an optical field 80 is incident on a non- medium confined to some volume as shown in the figure. This field creates a r11 '. source 8(80) that radiates an optical field 81. The corresponding radiation sourc ' ' radiates a field 82, and so on. This process suggests an iterative solution, the h of which is known as the first Born approximation. The second Born approxi _ carries the process an additional step, and so on. The first Born approxima l5 . Incident Radiatcd light 80 light 81 Radiation source 3(50) Figure 21.1-2 The first Born approximation. An incident optical field 8.; creates a source I which radiates an optical field 61. 8 8(8) Nonlinear medium adequate when the light intensity is sufficiently weak so that the nonlinearity is :1”- ln this approximation, light propagation through the nonlinear medium 15 regarded: _ scattering process in which the incident field IS scattered by the medium. The scat . light is determined from the incident light in two steps: I; I. The incident field 81, is used to determine the nonlinear polarization density il from which the radiation source 8(80) is determined. 2. The radiated (scattered) field 81 is determined from the radiation source adding the spherical waves associated with the different source points (as' in 1.=' ' theory of diffraction discussed in Sec. 4. 3). ' 21.2 SECOND-ORDER NONLINEAR OPTICS 879 The development presented in Sec. 21.2 and Sec. 21.3 are based on the first Born '. firesimation. The initial field 80 is assumed to contain one or several monochromatic -_ ms of different frequencies. The corresponding nonlinear polarization TPNL is then ' mined using (21.1-5) and the source function 5(80) is evaluated using (2| . l-7). _; e 8(80) is a nonlinear function, new frequencies are created. The source therefore Its an optical field 8. with frequencies not present in the original wave 80. This leads 'ly'numerous interesting phenomena that have been utilized to make useful nonlinear "” '- sdevices. Second-Harmonic Generation (SHG) and Rectification :wnsider the response of this nonlinear medium to a harmonic electric field of angular '=;'-_)r+twi1c w (wavelength A0 = 21rcO/w) and complex amplitude E(w). -' _ ‘ corresponding nonlinear polarization density (PM is obtained by substituting ' 1.2-2) into (2|.2-l), ;' _ 99mm «2 mm) + Re{PNL(2w) exp(j2wt)} (21.2-3) Igg’liihere I": PNLUJ) = dE(w)E*(w) (212-4) 5-; PNL(2w) = 013th). (212-5) .fl‘liis process is graphically illustrated in Fig. 21.2- I. ._ Becond—Harmoni'c Generation (SHG) ._"l‘he source 8(t} = —,u,,821PNL/0t2 corresponding to (2|.2-3) has a component at {frequency 2w with complex amplitude 3(2w) = 41t0w2 dE(w)E(w), which radiates .- an optical field at frequency 2w (wavelength A,,/ 2). Thus, the scattered optical field has ‘ a component at the second harmonic of the incident optical field. Since the amplitude _-. of the emitted second-hzmnonic light is proportional to 3(2w), its intensity 1(2w) is ' proportional to lS(2w)|2. which is proportional to the square of the intensity of the _ Incident wave 1(a)) = |E(w)12/'21} and to the square of the nonlinear coefficient d. 880 CHAPTER 21 NONLINEAR OPTICS Figure 21.2-1 A sinusoidal electric lield oi angular lrequcncy .11 in a second- order no- optical medium creates a polarization with a component at 21.: (second- harmonic) and a ste ' component Since the emissions are added coherently. the intensity of the second-harmonic _'-._' is proportional to the square of the length of the interaction volume L i The eIficiency of second- harmonic generation llsm; = I(2w}/I(w ) is the-- ' proportlonal to L210.) ). Since [(1.1 )= P/A wherePistheincidentpowerandAL”? cross- -sectional area of the interaction volume the SHG eIficiency Is oIten expres _. the Iorm .15. (21 - SHG EIIIeI__:_ where C'2 is a constant (units of W") proportional to d2 and 1.12. An expression In. _ will be provided In (2| 4- 36). . In accordance with (2] .2- 6) to maximize the SHG eIIiciency it is essential I'I the incident wave have the largest possible power P. This Is accomplished by 113,3 pulsed lasers for which the energy is confined in time to obtain large peak po _ Additionally to maximize the ratio L2/A the wave must be focused to the sm. - possible area A and provide the longest possible interaction length L. [f the dimenslei ot the nonlinear crystal are not limiting Iactors the maximum value of L for a gI; area A is limited by beam LliIfraction. For example a Gaussian beam focused beam width Il’u maintains a beam cross- -sectional area A - 11in over a depth: focus L— — 2:.) = 27rIrI/}f //\ [see (3. l -22)| so that the ratio L21A= 2L/A = 4.4]; The beam should then be locused to the largest spot size corresponding to the lit 1. depth of l...
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