VIEWS: 236 PAGES: 68 CATEGORY: Other POSTED ON: 2/19/2010 Public Domain
Introduction to Nonlinear Optics MOHAMMAD IMRAN AZIZ Assistant Professor PHYSICS DEPARTMENT SHIBLI NATIONAL COLLEGE, AZAMGARH (India). aziz_muhd33@yahoo.co.in How to make a laser in three easy steps … • Pick a medium that has the potential for optical gain – i.e., an amplifying medium. • Select a means of putting energy into that medium – i.e., an excitation system. • Construct an optical feedback system for stimulating further emission, i.e., an optical resonator. aziz_muhd33@yahoo.co.in Introduction Question: Is it possible to change the color of a monochromatic light? NLO sample input output Answer: Not without a laser light aziz_muhd33@yahoo.co.in Stimulated emission, The MASER and The LASER (1916) The concept of stimulated emission Albert Einstein (1928) Observation of negative absorption or stimulated emission near to resonant wavelengths, Rudolf Walther Ladenburg (1930) There is no need for a physical system to always be in thermal equilibrium, Artur L. Schawlow aziz_muhd33@yahoo.co.in aziz_muhd33@yahoo.co.in E2 E2 h h E1 E1 Absorption Spontaneous Emission E2 h h h E1 Stimulated Emission aziz_muhd33@yahoo.co.in Light (Microwave) Amplification by Stimulated Emission of Radiation LASER (MASER) aziz_muhd33@yahoo.co.in The Maser Two groups were working on Maser in 50s Alexander M. Prokhorov and Nikolai G. Bassov (Lebedev institute of Moscow) Charles H. Townes, James P. Gordon and Herbert J. Zeiger (Colombia University) aziz_muhd33@yahoo.co.in Left to right: Prokhorov, Townes and Basov at the Lebede institute (1964 Nobel prize in Physics for developing the “Maser-Laser principle”) aziz_muhd33@yahoo.co.in Townes (left) and Gordon (right) and the ammonia maser they had built at Colombia University aziz_muhd33@yahoo.co.in The LASER (1951) V. A. Fabrikant “A method for the application of electromagnetic radiation (ultraviolet, visible, infrared, and radio waves)” patented in Soviet Union. (1958) Townes and Arthur L. Schawlow, “Infrared and Optical Masers,” Physical Review (1958) Gordon Gould definition of “Laser” as “Light Amplification by Stimulated Emission of Radiation” (1960) Schawlow and Townes U. S. Patent No. 2,929,922 (1960) Theodore Maiman Invention of the first Ruby Laser (1960) Ali Javan The first He-Ne Laser aziz_muhd33@yahoo.co.in Maiman and the first ruby laser aziz_muhd33@yahoo.co.in Ali Javan and the first He-Ne Laser aziz_muhd33@yahoo.co.in aziz_muhd33@yahoo.co.in Properties of Laser Beam A laser beam Is intense Is Coherent Has a very low divergence Can be compressed in time up to few femto second aziz_muhd33@yahoo.co.in Applications of Laser (1960s) “A solution looking for a problem” (Present time) Medicine, Research, Supermarkets, Entertainment, Industry, Military, Communication, Art, Information technology, … aziz_muhd33@yahoo.co.in Start of Nonlinear Optics Nonlinear optics started by the discovery of Second Harmonic generation shortly after demonstration of the first laser. (Peter Franken et al 1961) aziz_muhd33@yahoo.co.in 2. The Essence of Nonlinear Optics When the intensity of the incident light Output to a material system increases the response of medium is no longer linear Input intensity aziz_muhd33@yahoo.co.in Response of an optical Medium The response of an optical medium to h the incident h electro magnetic h field is the induced dipole h moments inside the medium aziz_muhd33@yahoo.co.in Nonlinear Susceptibility Dipole moment per unit volume or polarization Pi Pi ij E j 0 The general form of polarization Pi Pi χ E j χ 0 (1) ij (2) ijk E j Ek χ E j Ek El (3) ijkl aziz_muhd33@yahoo.co.in Nonlinear Polarization Permanent Polarization First order P Ej 1 (1) polarization: i ij Second order Polarization Pi E j Ek 2 ( 2) ijk Third Order Polarization Pi E j Ek El 3 ( 3) ijkl aziz_muhd33@yahoo.co.in How does optical nonlinearity appear The strength of the electric field of the light e wave should be in the range of atomic fields h a0 N Eat e / a 2 0 a0 / me 2 2 7 Eat 2 10 esu aziz_muhd33@yahoo.co.in Nonlinear Optical Interactions The E-field of a laser beam ~ E (t ) Eeit C.C. 2nd order nonlinear polarization ~ ( 2) P (t ) 2 ( 2) EE* ( ( 2) E 2e 2it C.C.) 2 ( 2) aziz_muhd33@yahoo.co.in 2nd Order Nonlinearities The incident optical field ~ i1t i 2t E (t ) E1e E2e C.C. Nonlinear polarization contains the following terms P(21 ) E ( 2) 1 2 (SHG) P(2 2 ) ( 2 ) E2 2 (SHG) P(1 2 ) 2 E1 E2 ( 2) (SFG) P(1 2 ) 2 ( 2 ) E1 E2 * (DFG) P(0) 2 ( 2 ) ( E1 E1* E2 E2 ) (OR) * aziz_muhd33@yahoo.co.in Sum Frequency Generation 2 2 ( 2) 3 1 2 1 1 Application: 2 Tunable radiation in the 3 UV Spectral region. 1 aziz_muhd33@yahoo.co.in Difference Frequency Generation 2 2 ( 2) 3 1 2 1 1 Application: The low frequency photon, 2 amplifies in 2 1 the presence of high frequency beam . This 3 1 is known as parametric amplification. aziz_muhd33@yahoo.co.in Phase Matching ( 2) 2 •Since the optical (NLO) media are dispersive, The fundamental and the harmonic signals have different propagation speeds inside the media. •The harmonic signals generated at different points interfere destructively with each other. aziz_muhd33@yahoo.co.in SHG Experiments We can use a resonator to increase the efficiency of SHG. aziz_muhd33@yahoo.co.in aziz_muhd33@yahoo.co.in Third Order Nonlinearities When the general form of the incident electric field is in the following form, ~ i1t i 2t i3t E (t ) E1e E2e E3e The third order polarization will have 22 components which their frequency dependent are i ,3 i , (i j k ), (i j k ) (2 i j ), (2 i j ), i, j, k 1,2,3 aziz_muhd33@yahoo.co.in The Intensity Dependent Refractive Index The incident optical field ~ it E (t ) E ( )e C.C. Third order nonlinear polarization P ( ) 3 ( ) | E ( ) | E ( ) ( 3) ( 3) 2 aziz_muhd33@yahoo.co.in The total polarization can be written as P TOT ( ) E ( ) 3 ( ) | E ( ) | E ( ) (1) ( 3) 2 One can define an effective susceptibility eff 4 | E ( ) | (1) 2 ( 3) The refractive index can be defined as usual n 1 4eff 2 aziz_muhd33@yahoo.co.in By definition n n0 n2 I where n0c I | E ( ) | 2 2 12 2 ( 3) n2 2 n0 c aziz_muhd33@yahoo.co.in Typical values of nonlinear refractive index Mechanism n2 (cm2/W) 1111 ( 3) (esu) Response time (sec) Electronic Polarization 10-16 10-14 10-15 Molecular Orientation 10-14 10-12 10-12 Electrostriction 10-14 10-12 10-9 Saturated Atomic 10-10 10-8 10-8 Absorption Thermal effects 10-6 10-4 10-3 Photorefractive Effect large large Intensity dependent aziz_muhd33@yahoo.co.in Third order nonlinear susceptibility of some material Response Material 1111 time Air 1.2×10-17 CO2 1.9×10-12 2 Ps GaAs (bulk room 6.5×10-4 20 ns temperature) CdSxSe1-x doped 10-8 30 ps glass GaAs/GaAlAs (MQW) 0.04 20 ns Optical glass (1-100)×10-14 Very fast aziz_muhd33@yahoo.co.in Processes due to intensity dependent refractive index 1. Self focusing and self defocusing 2. Wave mixing 3. Degenerate four wave mixing and optical phase conjugation aziz_muhd33@yahoo.co.in Self focusing and self defocusing The laser beam has Gaussian intensity profile. It can induce a Gaussian refractive index profile inside the NLO sample. ( 3) aziz_muhd33@yahoo.co.in Wave mixing 2n0Sin( /2) aziz_muhd33@yahoo.co.in Optical Phase Conjugation Phase conjugation mirror PCM M s PCM M aziz_muhd33@yahoo.co.in Aberration correction by PCM Aberrating medium PCM s Aberrating medium PCM aziz_muhd33@yahoo.co.in What is the phase conjugation The signal wave ~ it Es ε s As e ˆ iks .r Es (r , t ) Es e C.C. The phase conjugated wave ~ * it Ec (r , t ) rEs e C.C. aziz_muhd33@yahoo.co.in Degenerate Four Wave Mixing A1 A2 ( 3) A3 A4 •All of the three incoming beams A1, A2 and A3 should be originated from a coherent source. •The fourth beam A4, will have the same Phase, Polarization, and Path as A3. •It is possible that the intensity of A4 be more than that of A3 aziz_muhd33@yahoo.co.in Mathematical Basis The four interacting waves ~ i ( ki .r t ) Ei (r.t ) Ai (r )e C.C. The nonlinear polarization * i (( k1 k 2 k3 ).r t ) P NL 6 E1E2 E 6 A1 A2 A e ( 3) * 3 ( 3) 3 The same form as the phase conjugate of A3 aziz_muhd33@yahoo.co.in Origin of Nonlinearities in Optics The fast response of media to an electromagnetic wave in visible and near IR is caused by a displacement of electrons, both free ones in metals and bound ones in dielectrics. aziz_muhd33@yahoo.co.in Origin of Nonlinearities in Optics The fast response of media to an electromagnetic wave in visible and near IR is caused by a displacement of electrons, both free ones in metals and bound ones in dielectrics. aziz_muhd33@yahoo.c o.in 1. Free electrons The motion of electron in the field of a light wave: E(t ) E0 exp i(t kr ) (1 H (t ) H0 exp i(t kr ) is described by an equation: e 1 2 d r E V H (2) dt 2 m c Becaus V E , the vector product H is proportional E 2 . e V to The solution of (2) can be found in a form: r E EE EEE ... (1) ( 2) ( 3) (3) wher (1) is ( 2 ) , ( 3) ... are nonlinear polarisabilities. e linear The induced electrical dipole moment isd er (4) , equal to aziz_muhd33@yahoo.c o.in 2. Bound electrons For the case of bound electron the equation has the following 2 form: e 2 r r F E (t ) r (5) NL m where the FNL takes into account real anharmonisity of term the oscillator:FNL ar r br r r ... Considerin FNL as a small term the solution of (5) can be presented as: r E EE EEE ... (1) ( 2) ( 3) (3) aziz_muhd33@yahoo.c o.in 3. Macroscopic characteristics To describe the media response for the electromagnetic field one must calculate a polarization vector P , which is a dipole moment of a unit volume. P Nd Ner (6 Where N is the concentration of electrons. ) If a nonlinear dependenced of E on takes place the d P vectors and can be presented in the form: d d ( E) d L d NL E EE EEE ... (1) ( 2) ( 3) (7 ) P P( E) PL PNL E EE EEE ... (1) ( 2) ( 3) (8) where (1) , (1) are tensors of 2 rank, ( 2 ) (2) , are tensors of 3 rank ( n ) so and ( non.2) are nonlinear susceptibilities aziz_muhd33@yahoo.c o.in 4. Local field factor In a microscopic model of nonlinearity (we presented two (n) such models) it is important to describe correctly microscopic (n) and macroscopic values. For crystals of cubic symmetry: n 2 ( ) 2 (1) ( ) N (1) (9) 3 where the term in brackets is so-called Lorentz factor (local field factor). For nonlinear susceptibility in particular for quadratic nonlinearity: n 2 (1 2 ) 2 n 2 (1 ) 2 ( 2 ) (1 2 ) N ( 2 ) (1 2 ) 3 3 n 2 (2 ) 2 (10) 3 aziz_muhd33@yahoo.c o.in 5. How high is the nonlinearity If the response of the media is caused by electrons in (n) nonresonant case for the following ratio is valid: ( n 1) 1 (11) (n) EA where E A is an interatomic field. For hydrogen E A 10 V cm . 9 One can see from this that appreciable nonlinear effects can be observed at relatively high light intensities, which are the features of pulse lasers. The nonlinear optics experiments became real after innovation of Q-switched laser with pulse duration of 10-8 s and intensities of 1010-1011 W/cm2. Now femtosecond lasers became available, which generate pulses with duration of 6-30·10-15 s at the intensity up to 1017-1020 W/cm2. In this case the electric field in the light wave exceeds the value of EA. It opens completely new branch of optics: physics of superstrong fields. aziz_muhd33@yahoo.c o.in Besides the above electronic nature of nonlinear response a strong nonlinearity can be caused by an anharmonisity of atomic oscillation in molecules, orientation of polar molecules in an electric field, heating of medium. The slower is a mechanism responsible for nonlinearity the stronger is the nonlinearity. Let us present the values of characteristic time constants and the values of ( n ) for different mechanism of nonlinear polarization. nonresonant resonant orientation in Mechanism electronic electronic liquid crystals Time constant, s 10-14 10-7-10-8 1-10-1 (2), esu 10-9 10-6-10-8 (3), esu 10-14-10-15 10-10 10-1-10-2 aziz_muhd33@yahoo.co.in III. Optical Harmonic Generation The high intensity light wave induces the nonlinear polarization in a medium. The wave of polarization is a source for new electromagnetic waves. aziz_muhd33@yahoo.c o.in 1. Second-harmonic generation First of all we should notice that the tensor ( 2 ) , for centrosymmetric media is equal to zero. ( 2) PNL EE ( 2) (12) The operation of symmetry transforms the terms from (12) in the following way: ( 2) ( 2) E E (13) P P ( 2) ( 2) Then PNL EE (E)(E) PNL ( 2) ( 2) , that can not take place under nonzero ( 2 ) . The same is valid for all even order ( n ) , n 2m . aziz_muhd33@yahoo.c o.in For a simplicity we assume that the medium is isotropic. Then the polarization: P P (1) P ( 2 ) P (3) ... (1) E ( 2) E 2 (3) E 3 ... (14) The incident waves propagating in z-direction can be presented as: E1 E10 cos(1t k1 z ) E2 E20 cos(2t k 2 z ) (15) PNL ) ( 2) [ E10 cos(1t k1 z ) E20 cos(2t k 2 z )]2 (2 ( 2) [ E10 cos2 (1t k1 z ) E20 cos2 (2t k 2 z ) 2 2 2 E10 E20 cos(1t k1 z ) cos(2t k 2 z )] ( 2) {0.5E10 [1 cos 2(1t k1 z )] 0.5E20 [1 cos 2(2t k 2 z )] 2 2 E10 E20 cos[( 1 2 )t (k1 k 2 ) z ] E10 E20 cos[( 1 2 )t (k1 k 2 ) z ]} (16) aziz_muhd33@yahoo.c o.in A spectrum of polarization waves contains new frequencies: 2 , 2 , , , 0 . 1 2 2 1 2 1 E1 , E2 0 ω1 ω2 ω P 0 ω2-ω 1 2ω1 ω2+ω1 2ω2 ω aziz_muhd33@yahoo.c o.in 2. Third-harmonic generation If the medium possesses cubic nonlinearity, under the action of two monochromatic waves1 2 and P ( 3) the polarization would contain the components with frequencies: 31 , 32 , 21 2 , 22 1 . aziz_muhd33@yahoo.c o.in IV. Wave Nonlinear Optics As the optical harmonic generation takes place both induced waves of polarization and free running electromagnetic waves of harmonics are propagating in the medium. If the dimensions of the medium are much larger than pumping wavelength the phase matching determines the efficiency of the energy transfer from the pumping wave to harmonics. Let us consider the phase matching conditions for the case of second harmonic generation. aziz_muhd33@yahoo.c o.in 1. Maxwell equations The propagation of the light in the medium is described by Maxwell equations: wher 1 B e rot E B H c t rot H 4 1 D D E 4P (18) j (17) (1) NL c c t E 4P 4P div B 0 For optical range div D 4 1, 0, j 0 (19) Combining first and second equations from (17) one may obtain so-called wave equation: 1 E 4 D 2 2 E 2 2 (20) c t c t 2 aziz_muhd33@yahoo.c o.in Inserting (18) into (20) we are getting: 1 E 4 P 2 2 (1) 4 P 2 NL E 2 2 (21) c t c t 2 c t 2 The nonlinear polarization term in the right hand side of (21) plays a role of a source of electromagnetic waves 2. Phase mismatch For quadratic media( ( 2) 0)and relatively low nonlinearity the plane wave solution of (21) for the intensity of the second harmonic looks like: 2 (n n2 ) z sin [ ( 2 ) I ]2 c I 2 (22) (n n2 ) 2 2 n2 c aziz_muhd33@yahoo.c o.in 2 Phase k 2k k2 (n n2 ) mismatch c max I2/I2 1 Δkz/2 -2π -π 0 π 2π aziz_muhd33@yahoo.c o.in For the case of the exact phase matching the energy of the pumping wave can be completely transferred into second harmonic I I2 I 0 L 2L z aziz_muhd33@yahoo.c o.in 3. Phase matching How the condition k 2k k 0 or n n 0 can be 2 2 realized? In an isotropic medium with normal dispersion 2 > n n and k never equals to zero Directions of phase matching But in birefringent uniaxial crystal there are two beams ordinary and extraordinary. For o so-called negative crystal no>ne. nω If pumping wave is ordinary one and second harmonic is e extraordinary one the material nω dispersion ( n2 > n ) can be n 2ω o compensate for the difference in refractive indices for o and e e beams: n n2 o e n 2ω aziz_muhd33@yahoo.c o.in For the process of third-harmonic generation the condition of phase matching looks the same: k 0 As it was mentioned already ( 2 ) and ( 3) values for the fast nonresonant electronic polarization do not much differ for many materials and the only way to enhance the efficiency of nonlinear energy transformation is to phase match the interacting waves. aziz_muhd33@yahoo.c o.in V. Other Nonlinear Effects 1. Modulation of a refractive index Cubic nonlinearity causes not only wave generation with new frequency but also appearance of a wave of nonlinear polarization with the frequency of pumping wave: 1 1 1 1 (23) PNL (1 ) (1; 1 , 1 ,1 ) E (1 ) E (1 ) ( 3) 2 As a result of such selfaction a nonlinear refractive index n2I appears at the frequency :1 n n0 n2 I n2 (3) (1 ; 1 , 1 ,1 ) (24) For the fast nonresonant nonlinearity n2 is relatively small: n2~10-13 cm2/kW. For slower mechanisms of the nonlinearity n2 can be much larger in particular for liquid crystals: n2~0.1 cm2/kW. aziz_muhd33@yahoo.c o.in 2. Selffocusing If the intensity of a laser beam is high enough instead of diffraction an opposite effect of selffocusing takes place. Phase velocity depends on the intensity through nonlinear refractive index: Vph=c/n0+n2I (25) If n2 > 0 the phase velocity at the axis of the beam is lower and nonlinear medium is working as a lens. Z aziz_muhd33@yahoo.c Phase front o.in VI. Nonlinear Optical Diagnostics Nonlinear susceptibilities ( 2 ) and ( 3) are tensors and they inherit the symmetry properties of the crystalline medium. It means that nonlinear optical effects are structure sensitive. It can be employed to study different structure transformations. A lot of such experiments were done. I will mention just one related with laser induced melting of semiconductors. R melting Ge, Si point RL R(t) RS laser pulse t aziz_muhd33@yahoo.c o.in 1. Nonlinear optical diagnostics of phase transitions Semiconductor in liquid Idea of stateMetal in liquid experiment beam Powerful laser R state melting 2 3 AB 6 5 which melts the surface point NL (Ruby, Nd:YAG , Eximer) RS nonlinear Pr ob NL reflection in NL gH n RL L c tio e- le L Ne ef RL Pr be arr linear ob am ne las ing N L i ar ) L reflection er be a d: YA nl ine (2ω RS m No ctio n (ω G le ) re f laser pulse Semiconductor t aziz_muhd33@yahoo.c o.in VIII. Conclusions 1.Nonlinear optics is an attractive and fast developing part of modern optics. 2.Nonlinear effects are structure sensitive in their nature. It can be used for time- resolved monitoring of structural transformation (up to femtosecond time resolution). 3.Artificial photonic media on the base of porous semiconductors open new exciting possibilities for the control of nonlinear aziz_muhd33@yahoo.c o.inoptical processes.