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# Light Matter Interaction OSE 5312

University of Central Florida

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Fundamentals of Optical Science OSE 5312 Fall 2003 Tuesday October 14 2003 Moving from the microscopic to the macroscopic world Continued Another averaging issue we face when moving from microscopic to macroscopic optical properties is that each atom or molecule may exhibit a slightly different resonance frequency The resonances are usually distributed according to a simple statistical law This gives rise to a type of broadening called inhomogeneous broadening that may or may not dominate the lifetime broadening due to damping Statistical Inhomogeneous broadening In many cases the atoms or molecules do not experience the same environment In some cases they may be in some random host matrix eg a glass where the surrounding electric field is slightly different and somewhat random for each atom causing the resonance frequency of each atom or molecule to be slightly different In the case of a dilute gas the atoms or molecules are moving at substantial velocities and their resonance frequencies are Doppler shifted In either case since the shift in the resonance frequency of each atommolecule is random the probability of having frequency shift with respect to the mean unshifted frequency is usually described by a Gaussian distribution Since each molecule has a different resonant frequency broadening of an absorption line due to this type of effect is called Inhomogeneous broadening We will treat the case of Doppler broadening but the description applies to all sorts of inhomogeneous broadening For an atom moving at velocity component v along the direction of propagation the observed resonance frequency is shifted from stationary 030 to 0301 vc Now in a gas the velocity distribution is Maxwellian ie the probability fvdv of having velocity in the range v vdv is l v2 fvdv J exp u Zjdv where u V ZkT M M is the mass of the molecules and T the temperature Given that the imaginary part of the polarizability is given in the resonance approximation by e2 r2 200m 00 02F22 then for a molecule moving at velocity v the imaginary part of the polarizability 0c 0 becomes 62 FZ 200 m 00 wovc w2 1V22 39 aquota aquotav Hence integrating over all molecules and velocities to get the susceptibility we find Ne2 T r2 fvdv 2 60 mo moa0 a0vc a2 lG22 iF2L2 l exp v2u2dv 2w0 mo L1vcl aa02l c2002 And in the weak susceptibility approximation the absorption coefficient 0c0 is given by 0c0 0cx 0 Hence the above integral gives the general inhomogeneous lineshape This is known as a Voight lineshape We can look at the extreme cases of strongly homogeneous and strongly inhomogeneous The integrand is just a product of a Gaussian function and a Lorentzian If the inhomogeneous width dominates ie u gtgt Fc200 then the Lorentzian behaves like a delta function and the integral reduces to a Gaussian giving a Gaussian absorption lineshape 2 2 2 a 1 050 1 expl w0 20 21 Zwo u 7 L u wo J 95quotw If the homogeneous width dominates u ltlt FcZw 0 then the Gaussian acts like a delta function and the absorption lineshape reverts back to the original Lorentzian form m z air n 20 00 a2 ll22 We will look at this in more detail when we study lasers Fundamentals of Optical Science OSE 5312 Spring 2006 2003 Physical meaning of the Kramers Kronig relations The above treatment gives us some sort of physical insight into the Kramers Kronig relations in the sense that they are really just a frequency domain expression of causality But it is reasonable to ask why causality should necessarily lead to a relationship between refraction and absorption The answer to this may be found from the following illustration which was first published by Toll in 1956 a We consider the case where a short optical pulse is incident on a material Such a pulse is sketched in part a of the figure on the next page This input J S Toll Phys Rev 104 1956 page 1760 pulse is zero for times t lt 0 Being a short pulse it has a finite bandwidth and may be considered as being a superposition of many Fourier components each of which extend from 00 lt t lt 00 Since the pulse is zero for t lt 0 and for times well after the pulse has ended then the Fourier components of the pulses must sum exactly to zero in these time ranges Now suppose the pulse is incident on an idealized material that exhibits strong absorption over a very narrow spectral range much narrower than the spectrum of the input pulse so that it completely absorbs just one of these Fourier components The absorbed component is shown in b It would therefore make sense that the output pulse should just be the input waveform with the one Fourier component subtracted from it But the subtraction of a single Fourier component from the input pulse field is shown in part c of the figure Clearly it no longer obeys causality as there is now a finite field for times t lt 0 This is not possible yet we know it is possible to have a very narrowband absorber Physically then to have a narrowband absorber that obeys causality then the phases of other Fourier components in the pulse must be shifted in just such a way that they will once again sum to zero for t lt 0 Optical Properties of Materials OSE 5312 Fall 2003 Thursday September 18 2003 Interaction of light With nuclear atomic motions We have studied the Lorentz oscillator model for electronic motion Actually this is relative motion of the electron amp nucleus but the as the nucleus is so heavy we only need consider the electron motion Now we want to consider the relative motions of nuclei in a molecule or solid The nucleus is about 3 or 4 orders of magnitude 103 104 heavier than the electron depending on the nucleus and the interatomic force is similar to or less than the electron nucleus force Hence using a VK m we can expect that resonance frequencies for nuclear vibrations should be a factor of about 50 1000 times less than electronic resonances Hence it is infrared radiation rather than visible light that interacts strongly with nuclear vibrations Rotations of molecules interact with even longer wavelengths quotfar infraredquot As we will see there are many degrees of freedom that can be associated with nuclear vibrations and rotations but not all of them can interact with electromagnetic radiation Degrees of freedom We first consider all degrees of freedom for molecules We assume only one electron will interact with the light Hence for N atoms there are 2N particles N electrons and N nuclei each particle has 3 degrees of freedom 3 coordinates A set of coordinates Fa where ct l 6N is needed to describe a system of N atoms Example Nitrogen molecule N2 2N x 3 12 coordinate axes describe all possible motions However there are not a convenient set of axes not quotnormalquot coordinates A more convenient way to describe motions Translation USC center Of mass 3 coordinates Rotation X rotation is about X or z aXis 2 coordinates angles Vibration l coordinate separation Electronic motions Still 6 coordinates Xyz for each relative to molecule Note that this still gives g coordinates But these are more convenient as they are independent of each other ie they are quotnormal coordinatesquot or quotnormal modesquot Center of mass motion is not interesting for optics but the other coordinates are The vibration and rotation normal modes of the molecule determine the absorption and refraction of the molecule in addition to electronic cloud motions already described by the Lorentz model Born Oppenheimer Approximation assumes that the electrons respond so quickly compared to the nuclear motions that we can consider the electrons as following the nuclear configuration quotinstantaneouslyquot as the molecules vibrate or rotate Usually we can also assume that only one electron per molecule is of importance to the lightmatter interaction In this case the number of degrees of freedom becomes 3N 3 degrees of freedom per molecule Examples 39 Diatomic molecule 3 translational 2 rotational l Vibrational and 3 electronic 9 3N 3 Triatomic General Rule for number of degrees of freedom for Natom molecule N 2 2 TOTAL 3N 3 3 electronic 3 translational Linear chain Other 2 rotational 3 rotational 3N 5 Vibrational 3N 6 Vibrational Typical spectral ranges for transitions Electronic near infrared about 1 um to uV about 100 nm inner core electrons go up to xray energies Vibrational near infrared about 1 um to long wave infrared about 50 um Rotational Far infrared about 100 500 um Dipole active Modes If a mode is to directly interact with radiation its dipole moment must change along the direction of the radiation field as the mode amplitude changes This is best seen by example Clearly a molecule containing only one type of atom e g 02 N2 etc can have no dipole moment no matter how strong the Vibration Carbon Monoxide CO has an oxygen atom that is slightly negatively charged while the carbon is slightly positively charged Hence it changes its dipole moment as it is stretched so that any component of radiation polarized along the molecular axis does couple to the Vibrational mode Hence the Vibrational mode of CO is quotdipole activequot while for 02 it is quotdipole inactivequot Since CO has a permanent dipole moment applied radiation may exert a torque on the molecule so that the rotational modes are also dipole active Another useful example is C02 Since this has N 3 the number of modes is 3N3 12 Of these not all are dipole active Vibrational 4 modes C02 is linear so there are 3N 5 vibrational modes These are H Carbon 0 Oxygen Rotational 2 modes Linear molecule so same two modes as for diatomic molecule shown earlier Ram an active Modes Modes that are not dipoleactive can also interact with electromagnetic radiation indirectly For example vibrational and rotational modes in H2 are not dipole active but if an electric field is applied the electrons are move in response to the field thus polarizing the entire molecule This induced dipole moment will be modulated by a Vibrational or rotational mode so a dipole active mode has effectively been induced through the electronic polarization This leads to Raman Scattering and so this type of dipole inactive mode is sometimes termed Raman Active The effect of the induced dipole is that the electromagnetic wave can interact with the rotational or vibrational modes This can result in the rotational or vibrational state of the molecule being changed by the electromagnetic radiation This is analyzed classically in H opf amp Stegeman chapter 3 but we will not discuss the analysis in this class Usually Raman scattering is described quantum mechanically as shown in the following diagram Stokes AntiStokes Scattering Scattering An incident wave frequency 031 is incident on the molecule polarizing the molecule through a dipoleactive mode e g electronic polarization There is no absorption as 031 is usually far from the resonance of the dipoleactive mode The dashed line is not a real state but referred to as a virtual state dashed line which is just to say that the molecule is being driven at the incident frequency 031 without absorption When in the virtual state the molecule can interact with the electric field and may gain or lose a quantum of vibrational or rotational energy from or to the EM field Hence the scattered light will be emitted with either a smaller photon energy 71035 71031 M which is referred to as Stokes scattering or with a larger photon energy 71035 71031 M which is referred to as Anti Stokes scattering We will discuss later what a quantum of vibrational or rotational energy is when we look at quantum mechanical variations from classical models Classical Oscillator models for vibrational modes First we will look at Vibration in the diatomic molecule and then at the collective modes that form Vibrations phonons in solids Vibrational Modes in a Diatomic Molecule Assume atoms with identical masses for now BX PX Define general displacement Dt X1 01 X 2 02 Equations of motions are le IltX1 X2 mX2 KX1 X2 Now we look for solutions Dt that depend only on a single frequency ie X1 and X2 have the same time dependence This will give us the fundamental modes of oscillation Dt 2 x11 00500111 x21 00500112 where XliCOS0it X1t and XQiCOS0it X2t i 12 is the mode index There can be more than one in general We then substitute into the equations of motion mx1i al2cosalt x11 cosalt szl cosa2t mx2i a2cosait Kx1icosat szl cosa2t H which reduces to an eigenvalue equation K mcol2 K K K mwl2 for a solution the determinant 0 so that 2 2 2 K mca K 0 or K ma2 iK 139 Therefore we have two possible eigenvalues 031 0 and 2K m Labeling mode index i l and 2 respectively we have 031 0 and 032 2K m There are the normal mode frequencies of the system For i l 031 0 so cos031t 1 giving X1t X11 Hence x1921 xzin But since m 0 then 0 KX11 KX21 so that X11 X21 so that 131 x11f1 22 ie both atoms move exactly the same way So a mode with 0 0 is just linear translation For 032 xZKm we have a 2 K A 2K A Dt 2 x12 cos tX1 x22 cos tX2 m m le m 2Kmx12 cos m IZK 2K 2 Kgtc12 cos t Kx22 cos t m m gt 2Kx12 Kx12 Kx22 x12 2 x22 In this case Hence the displacement is DU 2 x12 cosEIX1 X2 m Normal Vibrational mode corresponding to stretching frequency 2K m For a diatomic molecule with different atomic masses M and m we dfind that the resonant Vibrational frequency is given by mo 1Ku l l where u is the reduced mass of the molecule l i 7 Note that this is u m M consistant with the above case of M m where u m2 Vibrational Modes in a Crystalline Lattice In a crystal the indiVidual molecular Vibrational modes are replaced by collectiVe modes of the lattice Crystals are much easier to deal with analytically than noncrystalline solids but much of what we do here for crystals applies to noncrystalline solids also We start with the simplest case of a ldimensional monatomic lattice 4 a gt m m I K m l K m l K l n O a O a O 0 5 L ga X571 X a n1 here a unit cell length a contains only one atom mass m We consider only nextneighbor interactions giVing the following equations of motion KXa Xa71KXal XamXa K2Xa Xail Xa1mXa We again look for solutions where all atoms oscillate at the same frequency ie all Ts have the same time dependence e In this case maZXaK2XaXailXal We will see that this difference equation has travelingwave solutions of the form X iak aeiik a ail x e where k3 is the wavevector of the 3th mode We plug this trial solution into equation 1 mng eiak a Kx ialk a eiailk a zeta u k sin a 2 a 2ampl cosk a m 2K 2K w VW11 cosUc a 1 k w 2 5 sin 5a m 2 K m DB 753 0 753 Note the maximum frequency a collective mode can have is 2 x K m This occurs at k i Tca This wavenumber corresponds to a Vibration wavelength of 2a For wavenumbers of magnitude gt Tca the physical configuration also has a wavenumber that is less than Tca so there is no new physical situation outside the range 7a lt k lt Tca This range is known as the first Brillouin Zone Boundary conditions and number of degrees of freedom In a 1D lattice each atom has 1 degree of freedom so we eXpect that there should be a total of N of these collective modes ie 3 l N where N is the number of atoms in the crystal Usually periodic boundary conditions are applied to the solutions so that ik aa ik aNa ik Na e e gt 1 e 3 k Na227r 32012 27x 27x k 7 7 Na L so there are N possible values of k evenly spaced at 27139 47139 67139 2N7 k03 3 3 3quot 3 L L L L 100 L 5 2a 2 3 This discrete nature of k is not important for the optical properties However all this is for a monatomic lattice for which the vibrational modes cannot interact with radiation To look at the interaction of light with lattice modes we must next examine the lattice modes of a diatomic lattice Diatomic 1D lattice A a MI K m I K M ml 5 5 39 39 I V H E 39 P s l I us l VS us VSH Masses M and m characterized by displacement amplitudes us and VS respectively Note that a unit cell contains two atoms one of each kind Assuming next neighbor interactions the equations of motion are Miis KVS1 Vs 2us mys Kus us1 2V3 39 Again we look for travelling wave solutions where us VS have the same oscillation frequency and wayenumber but different amplitudes Hence we anticipate solutions of the form iska 7m VS V06 6 iska fig 3 Us uoe e Plugging into the equations of motions we obtain Ma2u0 KV01 ei ZKuo mwzv0 Kuo 1 97 ZKV0 Solution found by setting determinant equal to zero 2K Mw2 Kle 0 K1e 2K mw2 mMaJ4 2KmMa2 2K2l coska 0 gt a2 zikm MJr mM2 4mMl coska mM Rather than seek a general solution we solve in the limits of ka ltlt 1 and ka 2 in For ka 2 in we have lcoska E 2 so that the solution simplifies to a2 mMi1MM2 4mM ltmMim M Hence a 7 or a 2 forkasirn M For ka ltlt 1 coska E l k2a22 and m M ZmM ka2 mM m M 2 2 wz K szMll1 mM ka2 mM mM2 2 hence we have two solutions for this range of ka also 012 22KM2Kiijamp mM m M u where Ll is the reduced mass of the unit cell i l umM 2 kza2 K K Or a E gt a 2 ka 2 mM 2mM k Fundamentals of Optical Science OSE 5312 Fall 2003 Thursday October 23 2003 Sellmeier equations Sellmeier equations are essentially empirical fits to the actual refractive index of a material using the result for x for Lorentz oscillators as basis functions Hence Selhneier equations are of the form 2 a 2 y n l Z 2 2 j 01 These equations are very useful ways of providing data on the refractive index of materials vs wavelength without the need for extensive tables They are usually valid only in high transparency spectral regions far from resonances where X is real Often but not always Sellmeier equation for a material may contain a pole ie a resonance at low frequency 0 0 ltlt 0 and one at very high frequency 030 gtgt 0 Hence one could write 2 2 2 a a wo n2a1 ZOO E 2 I 2 72 000 wOJ a a 2 2 where 1 a poo 000 is more commonly written as 800 In practice these equations are usually expressed in terms of wavelength 2 b 12 n2a 1AZ d z asz d22 139 01 1 j 1 C so that coefficients a bj cj and 1 may completely describe the refractive index vs wavelength for a material There are usually only one or two values of j ie only one or two resonances in the Sellmeier equation for a given material This may depend on the material and on the level of accuracy required To get an idea of the wide variety of Selhneier equations some Sellmeier equations are given in the following table from the OSA handbook of Optics Px PROPERTIES OF CRYSTALS AND GLASSES KBLE 22 Room tempurature Dispersion Formulas for Crystals 3361 Mmerial Dispersion formula wavelength A in mm Range pun Ag3A553 Z4 7 0001ng2 um n 7483 n A 1342 quot 100111z is n 6346 A2 7 009 n2 1 00993942 n2 2 52505 112 0070 37 206250812 0946146542 43007854 A1 7 010390542 12 024386912 A2 70857232 2 21685712 2175312 no 7 36280 A Z 01003 A2 950 1527 2 21692 2 7 quotE 439017 12 01310 A2 7 950 0001 5012 172 1 2205742 1837712 A2 01879 A2 1600 139702 1928212 2 quota 52912 A2 02845 12 1500 nu 2184 n 220006S9 pun no2104 n21151318 14m ng 46453 6084012 190012 2 1 20792 W 12 27622 2 w 1378612 386142 7 31399 A2 041715 1 15032 1617312 413911 A2 017462 113 15032 14313492 065054713A2 53414021 13 30729 Hi 1 quot2 1 15039752 05506914142 0592737912 7 A 2137542 458212 1270102562 127113868 00184 2 2 n 27405 m 001551 712 1 quot3 23730 A2001 7 00044112 00156 0643356A 050676212 8261 12400577892 Az 0109682 124463864 quotL1 418711 0 12702231 4004 2 3 02112 21 1922744Z 1242097 quotquot 8 76107908 A2971312 r1371 29593 2 quot31 10033902 W733 127 1047972 8 7 007260311 AZ 01193242Z 12 7 130232512 1 007402882 A2 01215529i 12 7 200722482 063 460 039405 0497067 054 211 049 12 0737135 056 21 122 50 02755 Oil23 022 106 027 103 04 07 044 711 169 110 170 170 171 172 173 174 175 176 177 178 For fused silica a good fit can be obtained by using a 3pole Sellmeier equation with poles at approximately 99 um 116 nm and 68 nm The contributions of the individual poles can be seen below 2 2 2 069616632 04079426 087747947 HO 2 2 121 7 2 2 m 2 2 2 x 7 00684043 x 7 01162414 x 7 9896161 Reference Handbook of Optics OSA r10 1 V1 Hm mm 1305 alid 021 371 um I I 391 3 I I I 2 h 111 quot 2 mm 0 7l 111 81 39 l 1 W 2 II I I 00 01 01 I1 i0 10c 001 01 1 10 100 39 A 2 111 145 Other ways of representing the dispersion of optical materials Schott glass form power series 112 a0 61122 c1212 c1314 61416 adquot Hertzberger form combines power series and Sellmeier forms n2Ai C D l2 E14 1213 23 292 Abbe number The Abbe number Vd is a very commonly used singlenumber measure of the dispersion of glasses It is defined by n 1 Vd d nf no where nd is the refractive index at 7 5876 nm the sodium quotdquotline and nf and n0 are the refractive indices at 7 4861 nm and 6563 nm respectively Typical values are in the range 20 60 A low Abbe number indicates high chromatic dispersion Some examples are given in the following table Material Refractive index Abbe number Crown Glass 152 58 Polycarbonate 159 31 BK7 152 62 CaF2 1433 94 Fundamentals of Optical Science OSE 5312 Fall 2003 Tuesday December 02 2003 Nonlinear Propagation Equations Now that we have seen how it is possible to generate nonlinear polarizations we must address how these polarizations cause new waves to grow or how they affect existing waves Recall that we found that nonlinear polarizations for discrete frequency inputs also have discrete frequency components where the frequencies generated are at the sums amp differences of the spectral components of the input Eg 131 w 60 NW1 w2w1w2Eltw1Eltw2 OI P2w3 eo252w3w1w2Ew1Ew2 where 033 01 032 The time and z dependence of the polarization at 033 is assuming all waves are collinear and propagating in the zdirection Pagjtz 60 12Q3wla02Ea1eik1Z mltEa2eik22mzt cc 260 12Q3a1a2Ea1Ea2eiklk2zm3t Note that while this polarization wave has frequency 033 its propagation constant k1 k2 depends on the refractive indices at 031 and 032 This is because the phase of the polarization Pagfkhz depends locally on the phases of the electric fields at 031 and 032 We anticipate that a wave E3 at frequency 033 will be generated by this polarization so we will take a look at the wave equation for E3 2 2 2 6E3i6E306Pm3 622 02 6t2 6t2 62 0 g 1251 r z 135 r 2 where PCS I z 60 11 03 Ea3 eik3zm3t 00 and k3 na 0 Hence 0 62E3 1 10 62E3 azPaEfrZ 522 c2 622 0 622 Now we will anticipate that the amplitude of the generated field E0133 may change with z ie E3 t z A3 zeiquot3z 3 06 We can plug this into the wave equation but first we can take the derivatives 6E3 i ik3A3 eiltk3zm3t cc Oz 2 Oz 62E 6A 621 61 3 2 ik3 3ik3A3j 3 z39k3 a3 610632 6030 00 622 6Z 622 Z 621 61 a 23 2ik3 6 32143 610632 6030 cc 2 a25 3 1 2A ik3Z ca3t Also at 2 3a3 3Ze cc 2 2 aszs 2 2 i03 E0 6 0301 2 at2 2 gtlt Ea1Ea 2 eik1k2zw3t cc Now we apply the slowly varying envelope approximation SVEA1 5213 622 6A3 E ltlt 2k3 which is valid provided the changes in A3z are small over length scales on the order of 7 Under the SVEA the second derivative is eliminated from the propagation equation aA 1 1 2l39k3 a3k32A3ezk3z w3t ccz 2w3w32A3 Zezk3z mt 66 Z c w u0 e0 22w3w1w2Ew1Ew2eik1k2z 3t ac Noting that k32 1 1 a 032 this simplifies to c A 3 e39kBZ who 60 262 301202E01Ea2elk1k2z 6 21k 3 62 1 Sometimes called the slowly varying amplitude and phase approximation SVAPA which recognizes that the amplitude may be complex and the phase may change due to nonlinearities as well as the magnitude or 51 a C o 2 39Akz 3 213 60 Z 03Q12E01E02el 62 2na3 where Ak k1 k2 k3 is the phase mismatch We note that 132 03 60 262 030102E01E02 so that in general terms we may write 1 0 3 WO P2w3eiAkz Oz 2na3 or 6A a 3 l Q3 elAkZ 62 2na3 60 c where 132 can represent any second order process ie P203 30103 2 P203 320 1 etc The phase mismatch results from the waves at the three different frequencies propagating with different phase velocities and hence getting out of phase with each other which adversely affects the efficiency of conversion from one frequency to another Perhaps it is easiest to understand the phase mismatch by looking at the secondharmonic generation process SecondHarmonic Generation For second harmonic generation the nonlinear polarization is defined as P22a 13 60 12 20 a aE2 0 so we have 6A 2w l z39Acz 62 2n2a 60 c 0 22gt2wwwgtAEe 20 Here NC 21 km quot250 It is common to use the deff coefficient defined by deff X22 so that 6A wd 2 2 le AlzezAkz 62 r1200 2 iAkZ e where K2 is the figure of merit for second harmonic generation Provided we can assume that the fundamental field A1 is constant along 2 this differential equation is easily solved 2 eiAcz 1 A A2lt0gtlt2A1 Ak 1420 iK2A12eiAkz2Z e m z2 eiAkz2 ZiAkZZ A 0iK AgentQM 2 2 1 AkZZ We are usually interested in the case where A20 0 so we see that AkzZ A A2 tAkzZ Sln 22 m2 126 AkzZ 2 2 1 1 NOW Ia inwc E0 A1 and 12a 3 2wceo A2 nwc eo znzlezlzz2 2 2 M2 gt 12wz2 1wsmc 7 2 2 2 2a de z 2 2 Akz nwnmc GO 2 So that the second harmonic output irradiance is proportional to the square of the input fundamental irradiance The sinc2 dependence is plotted below 1 I203 o Akz2 Note that for Ak 0 the phasematched case 120 oc 22 But for Akz i275 the second harmonic output is zero This is the usual expression we see for the second harmonic irradiance Usually z L the thickness of the nonlinear material which is not variable The phase mismatch is often tunable though as we will see below and it is often quite easy to generate the above curve experimentally However it is usually more insightful to look at 1202 In the above analysis we multiplied the top and bottom by z in the expression for A2 in order to get the sinc function If we do not do this we find A Z AZeiAkzZ 2 2 Ala2 1 2n lK 2 I2 Akz I 2 2a 2 a 2 gt 2wz nc GO Ak22 s1n E 2 1 We see this plotted below In all cases for very small 2 the second harmonic grows as 22 as sinx x for x ltltl But if the indices of refraction at 0 and 20 are unequal the SHG produced near the entrance which propagates at cn2m and the polarization that generates new SHG which propagates at cnm eventually get out of phase By the time AkzZ 752 the new SHG is already 752 out of phase with the old SHG that was produced at the front surface Hence at this point the new and old SHG start to cancel and further propagation just reduces the SHG field Unless we can make Ak 0 there is no advantage in haVing a material longer than 2 TcAk This distance is known as the coherence length 10 1 l l C M 4quotw n2w I 10 Ak01 3 e 3 Ak05 4 2 AK1 0o 1 2 3 Techniques of Phase matching In any normal material n20 3 nm There are three main ways of overcoming this Anomalous Dispersion Phase Matching It may occasionally be possible to match the indices if 0 and 20 are on either sides of a resonance as sketched below This is highly unusual and not a generally applicable method of phase matching an 0 20 V nw Bire ingent phase matching It is often possible to find propagation direction in a birefringent crystal where the ordinary wave at 0 has the same index of refraction as the extraordinary wave at 20 Or viceversa This is illustrated in the figure below This requires a Xe tensor element that gives a second order polarization polarized perpendicular to the fundamental field but nevertheless there are many materials for which this is so and this is the most common means of achieving phase matched Xe processes z optic axis I l I lgt I I Figure 88 Normal index surfaces for the ordinary and extraordinary rays in a negative n2 lt n uniaxial crystal If n lt n2 the condition nf 6 ng is satis ed at 0 0quot The eccentricities shown are vastly exaggerated In this case the exact phase matching can be tuned by adjusting the angle as illustrated in the experimental data below 44 L0 X 81 X 10 0 Experimental 39 TEMOOII 08 P 148 X lO39awatt 2 1 23 cm 06 sun 180 0m Km 3 2 c l3 9 90 04 02 r I I 7 l I w t 01 0 0l 02 0 9m Figure 89 Variation of the second harmonic power P20 with the angular departure 6 6m from the phasematching angle After Reference 11 Quasi PhaseMatching QPM It is possible to obtain an large SHG or other efficient Xe processes by periodically modulating the X9 along the propagation direction as shown below PZmauXIElt3LIll1 Distance zLL d Figure 821 The evolution of the secondeharmonic phasor E ZWz in a a nonphaseematched case b quasi phaseimatched and c bulk birefringent phaseqnalching Ak 0 d The secondharmonic power eld Em in a crystal for cases a b c above as well as curve 7139 d for quasi phaseimatched operation using the third Fourier coefhment m 3 LC E 7 Pan 1 is reproduced from Reference 34 Here the crystal is reversed every coherence length ie for each successive layer X0 gt me Hence instead of having a reduction of Em after 2 10 we reverse the phase of the quotnewquot SHG so that it continues to add to the SHG generated in previous layers This adds tremendous exibility as we are no longer restricted to the polarizations and directions imposed by birefringent phase matching However this is a difficult materials fabrication problem QPM has been known since the early 1960 s but it is only in the last few years that QPM materials have become commercially available Other 762 processes There is insufficient time to go over the theory for other processes but it is quite straightforward to apply the same techniques to sum frequency generation SFG and difference frequency generation DFG Phase matched DFG is particularly interesting This involves a beam at high frequency 0 3 mixing with another lower frequency 031 to give a difference frequency beam at 032 033 031 2 X 032 033 39031 It turns out that assuming do depletion at 033 the beam at 031 grows as cosh2Pl and the beam at 032 grows as sinh2Pl This corresponds to near exponential gain The gain can be so high that with only an input at 0 3 the other waves can grow out of noise The particular values of 031 and 032 will depend on phase matching This process is known as Optical Parametric Generation and Amplification OPG amp OPA Just like amplification by stimulated emission this can be put into a laser cavity to build up a selfconsistent mode This is known as an optical Parametric Oscillator OPO Nonlinear Refraction and Absorption We now wish to look at the degenerate thirdorder susceptibility 9630 0 o03 The SVEA propagation equation for a monochromatic field at frequency 0 in a medium with susceptibility X3o39030303 will look like dA a 3 2 z aa aa A A dz 41101 I I 39 3 3 3 Setting X w7w7a7a X 195 wehave 031 0 2 3 3 1 1 A A dz 4nc gr II I 39 Now the slowly varying field amplitude may contain a phase component as we write Az Azei z so the propagation equation becomes d a Alew EzzE IIl3e zf3lAl3equot dA d a 3 3 iA i 3 A 314 dz I I dz 4nc gr I I Z I I Therefore we have two propagation equations one for amplitude and one for phase Mai 3 3 dz 4ncZi IAI 3 2 dz 4nczr IAI This 53 leads to nonlinear refraction while 53 leads to nonlinear absorption Nonlinear absorption Multiplying the equation for nonlinear absorption by A dIAI lt3 4 IAI dz 4nczi IAI dlAlz 2 dz 2 we can wr1te Since I IA d a 3 2 fl 1 dZ 2n 0 60 gt 6 I2 Where 3 is the twophoton absorption coefficient This has solution 12 I0 l IOZ Nonlinear re action 21 lt3gtA2 z a Zlt3gtA22 The equation dz 4m Zr I I has solution 4 10 r I I Hence the propagating wave may be written as EQ A a h w AZ sz 119414sz at 4110 using k nowc k0n0 where no is the linear index of refraction we can write this as 2 z9M Ezt Az cos kz n0 4 wt quot0 Hence the total refractive index is nLM2 4110 Of n1 n0 n21 where E0 C 3 quot2 2 4 So the refractive index can be irradiance dependent For Gaussian laser beams this can lead to quot SelfFocussingquot IQZgtO Lad In the case of a negative n2 we have quotselfdefocussingquot r lt0 gt1 Optical Properties of Materials OSE 5312 Summer 2002 Monday May 20 2002 Some comments on the applicability of the Lorentz model to real materials a Insulators The Lorentz model works surpri singly well provided we remember that real materials correspond to a collection of Lorentz oscillators with different frequencies The outer or valence electrons predominantly determine the characteristics of the optical properties a solid In an ionically bonded material e g alkalihalides such as KCl the valence electrons are quite strongly localized at the negative ion for KCl this would be the Cl atom and hence the optical spectrum contains some atomiclike features with many resonances As the valence electrons are tightly bound the resonance frequency is high so that these materials may have a transparency range that extends far into the uv This can be seen in the re ectance spectrum for KCl shown below taken from Wooten Ch 3 For these types of materials the external eld and the local eld can be quite different and it is not trivial to calculate the local eld For this reason the Lorentz model does not give quantitatively accurate results for ionic materials 40 20 o Jlll Ref Ieclonce o L o 2 4 6 3 l0 l2 l4 IS 18 20 22 24 hvleV Fig 36 The spectral dependence of the re ectance of KC The region of transparency extends to about 7 eV Above 7 eV there are a number of sharp peaks related to narrow energy bands and excitons From H R Philipp and H Ehrenreich Phys Rev 131 2016 1963 ii Semiconductors Semiconductors are covalently bonded materials where the electrons are evenly shared between neighboring atoms Some insulators are covalently bonded too This means that the electrons are smeared out into broader bands and that their resonance frequencies are lower than for ionically bonded materials Usually these materials can be described by a single energy gap and single broad absorption band above the energy gap The example of Silicon is shown below 80 R o l5 E eV Fig 37 The spectral dependence of the re ectance and dielectric functions of Si Regions 1 IL 111 and IV correspond to the regions with the same designation shown in Figs 31 33 and 34 H R Philipp and H Ehrenreich Phys Rev 129 1550 1963 Estimation of 800 for Si Noting that the re ectance of Si rises sharply at about 3 eV we may take this as an estimate for coo Hence coo z3 X 16 X103919h 453 X 1015 rads a 2 a 2 2 P sothat E r0 1 sothatif a 0 a 2 If a 0 we can determine cop we can estimate 80 Now a P 1lNe2 E 0 m and since each Si atom has 4 valence electrons N 4NSi z 4X 2 1028 m3 This gives an estimate of cop z 16 X 1016 rads corresponding to about 105 eV and hence 80 z 14 This is compared to a measured value for 80 of 12 so the approximations are reasonable Note that Si appears as a grayish re ector throughout the visible spectrum 17 32 eV Now Er0 1 iii Metals Drude theory of optical properties of metals We can eXtend the Lorentz model to metals in which case since the electrons are unbound or quotfreequot they eXperience zero restoring force and hence the resonance frequency coo Km is also zero This is known as the Drude model The equation of motion is then 2 gt MW m1 w eEt M in which has solution39 a 6 50 rt 0 maJZHTa and hence 9a is given by 2 I w P I while where once again the plasma frequency is defined by of NeZegm Hence l 2 1 n 2 Fa 2wwpa2lquot2 1wwpw2lz OI v 2 u 2 Fw 3rW1wp wzrz sw ap w2r2 Now in a metal the damping term F is just the electron collision rate which is just the inverse of the mean electron collision time T ie F 1391 Hence 2 2 2 or e39wI P quot1 P A 1wzr2 ra1a212 The collision rate can be quite rapid tens of femtoseconds But for optical frequencies eg for 9 500 nm co 21cc9 38x1015 rads 0172 gtgt 1 Under this approximation we nd a a 0 2T Er0zl 2 Er 3 3 0 0 T 0 This approximation may break down in the farinfrared spectral region where damping may be significant Note that damping is absolutely necessary to have an imaginary part of xco or 8a Recalling that the absorption coefficient is given by occo ZkOK coarcocnco Now in the long wavelength limit 02 ltlt mp2 nco 1 so that HP r12 239 2 clp where hp is the wavelength corresponding to the plasma frequency The 93 10 mequotwe do dependence of x is commonly seen in the long wavelength limit It is useful to look at some plots of 8a n03 0a and Rco These are plotted on the next page for cop 10 and for P z 0 or F 05 In the limit of no damping the n 0 and R 1 for 0 lt co lt cop Above cop K is zero and the re ectance drops as n rises from zero to unity Note that even for 8r 0 K and hence x is not zero Introducing some damping causes R to be lt l and the re ectance drop at cap is less severe The behavior of if n and K is consistent with what we now expect for a Lorentz oscillator with coo 0 Clearly the sharp edge in the re ectance seen at the plasma frequency can be expected to be the predominant spectral feature in the optical properties of metals P 00005 F 05 c igni cance of 1 he eXpressions have been written for 8r rather than for X as they more 1 clearly reveal something significant about the 3lasma frequency in this form Notice that at co cap the real part of the dieleL tric constant becomes zero Hence n03p 0 which means the phase velocity 00 A more rational way to describe this is that the wavelength 9 27ccnco gt oo as co gt 03p This means that all the electrons are oscillating in phase throughout the propagation length of the material cosoapt v metal Note that as all the electrons are moving together there is no charge separation polarization and hence no restoring force or sustained oscillation after the eld is removed Plasma oscillations The above figure shows an entirely transverse field compared to the surfaces of the material Should there be a component of the field perpendicular to the surface there can be a net surface charge as a result of the applied field q 31313 W gtlt O A The attractive restoring force between the surface charges can result in a free oscillation For no net charge qf 0 then D 0 soarE But E 0 so then 8r amp ier 0 Hence 850 Now P charge X displacement volume Thus NeA x39L AL Ne3x amp The restoring force is given by eE Ne25x eE Eo 82x Ne2 x 82 2 N2 Bx 65x20 2 a me0 which is equal to the acceleration m which is the equation 0 of motion Hence the resonance frequency for the plasma oscillation is given by 0h Ne2 P me0 Modi cations 0fDrude theory to account for properties of real metals The Drude model implies that the only the plasma frequency should dictate the appearance of metals This works for many metals see the example of Zinc Fig 312 in Wooten But is does not explain why copper is red gold is yellow and silver is colorless In fact the appearance of these metals is characterized by an edge in the re ectance spectrum similar to that predicted by the Drude model but the problem is that all three metals have the same number of valence electrons Also the calculated plasma frequency for all three should lie at about 9 eV well outside the Visible region so the plasma frequency cannot in itself account for the colors of Cu and Au All three have filled dshells Copper has the electronic configuration Ar3d1 4s1 Silver KI 4d1 5 1 and Gold Xe4f145d106sl These metals are known as the Noble Metals The delectron bands lie below the Fermi energy of the conduction band Transitions from the dband to the empty states above the Fermi level can be occur over a fairly narrow band of energies around 711 0 E F E d which can be modeled as additional Lorentz oscillator The combined effects of the freeelectrons Drude model and the bound delectrons Lorentz model in uence the re ectance properties of the metal E A Conduction band EF fFermienergy Ed Dband k e Hence E Efree Sbomd Where 8f is described by the Drude model coo 0 and 8b is described by the Lorentz model coo EF Edh Example Silver The re ectance spectrum of silver shows a string drop at about 4 eV well below the eXpected plasma frequency The re ectance also rises again for frequencies just above 4 eV See Wooten Fig 315 shown below 100 r39 I I l T Reflectance S I l I I IL I J I J J J 1 ILl 102 I J L l4 2 4 6 8 10 12 Photon energy ev It turns out that this behavior is because Silver has a dband resonance at M x 4 eV This can be determined from experimental data by fitting the Drude model to the low frequency data as shown in Wooten Fig 318 Shown below The difference between dielectric functions from Drude model and from experiment gives the dielectric function due to the dband resonance abound written as 5803 in Wooten The effect is to pull the Sr 0 frequency in from 9 eV to about 39 eV Wooten Fig 318 Real part of dielectric constant for Silver This shift in op means that there is a shift in the free plasma oscillation in silver due to the delectrons This can be explained by noting that the highly polarizable delectrons will reduce the electric eld that provides the restoring force involved in these oscillations illustrated on page 6 A reduced restoring force gives a reduced oscillation frequency See Wooten fig 320 for an illustration of how the delectrons do this Copper The case of copper is almost identical to that of silver except that the dband resonance is at about 2 eV Now since e ee becomes very large and negative at low frequencies it turns out that e bound due to the delectrons is not sufficient to pull the net 8 through zero Hence 8 becomes small at about 2 eV but there is no true plasma frequency there However the effect of this is sufficient to cause R to start to drop at 2 eV but the reduction is gradual throughout the visible This gives copper its characteristic redorange appearance Fig 322 from Wooten Dielectric function of Copper g I q D gt o m 7 L 1 5 4 0 c J x0l J L 1 Vx I 390 Z 4 6 l3 l2 14 EM Re ectance of Copper From Wooten Fig 321 Optical properties of LiNbO3 anisotropy Anisotropic media optical properties depend on direction of k and polarization I 39 39 39 quotquotquot IOEO 3 39039 I I I II I I r E f r i A 039l M 390 II I 5 Io I I E I 5 all i 5 I I i l I I l l l I I0 I l0quot l l I E 39 I t I I i l i IO Z l IO39Z l o a I 5 x l 393 39 I a 1 2 I c I x log II 39E 3906 i 39 l I 3939 l I 39 39 I I0 2 l0 45 I E I 5 I l IO5 A IHHI I UNI ml I hull 0 5 JIImII III II uI IOl IOo IO 02 I03 Io Io I01 I02 Io3 WAVELENGTH Lm WAVELENGTH lam Fig 7 a Log log plot of n and Ico versus wavelength in micrometers for lithium niobale b Log log plot of ne and kc versus wavelength in micrometers for lithium niobate quot Fundamentals of Optical Science Spring 2006 Materials practice set slide 1 Optical properties of amorphous SiO2 I IIIIIII I Illllll lillllll l f lo IiononlikeI 3 Siozglass E a SiO bonds polar character I O negatively charged Two clear phonon resonances one main electronic transition refractive index n I I I l I I I I I l I l I I I I I I I I I I I I l see Fox page 37 IIIIIII iiinml III 1 ml I I IIIIIII I llHllI I IIIIIIII 10 1 I IIIIIIII Illllllll ll 10 2 2E15 Hz k150nm E8eV IIIIIII i IIIIIIII ll Extinction coef cient llIII IlllllllI IlIIII HIIIII lllIIIILI IIIiIIIII II A 9 4 Mr llllllu IIIIlIIlI Illll HI I IlIIIIlI l IIIJHII I IIIIIIII l 1012 103 1014 1015 1016 10I7 Frequency Hz Fundamentals of Optical Science Spring 2006 Materials practice set slide 2 Isolated absorbers in a transparent host material mi Nd3 ions in glass Distinct absorption lines within host glass transmissive region Describe P as superposition Ptot Phost P Percent Jilxarpiion l dopant 2 mp Bot 80 Zhost 002 a2 iFa l i i a 4000 5000 6000 7000 8000 9000 l0000 Wavelength A I Fig 29 Absorption versus wavelength of Ndzglass Material ED2 thickness 63 mm Wth Xhost aSSumed Constant Around an absorption line the dielectric function can be described by hqg39emtand 2 5a 11 mp or mp r OS g a 800 ht wg wz lFa wg wz iFa where 800 is known as the high frequency dielectric constant 2 At zero frequency this gives a static dielectric constant an 2 800 2 Fundamentals of Optical Science Spring 2006 Materials practice set 0 slide 3 Drude description of the dielectric function of silver Drude model predicts monotonous decrease of ar for decreasing frequency 2 P sr39ltwgt1 2 with sr crossing zero at up Measurements show a peak in sr as for bound electrons caused by delectrons ie bound electrons in a d orbit at 4 eV below the Fermi energy E Conduction band Fermi energy D band Fundamentals of Optical Science Spring 2006 Materials practice set slide 4 reflectivity Comparison of Drude model and Ag Silver 8 and R Drude model 2E R33 5 3 393 Irma o I D In El quotquot we 1 D I I III I El 2 4 6 3 ID 3 I I 100 r IvIccr 2 g m 39 E 390 Ifcj 332 E 1 I If L Ha El 395 1D 15 0 22 L 4 i 39 g 39 13 m Photon energy eV Drop not monotonous due to d electrons slide 5 Drop in reflectivity around 8 0 Fundamentals of Optical Science Spring 2006 Materials practice set Comparison of silver and copper reflectivity Cu sand R C u a I IL 3 m 1 o0 r quotI r C u E c quotf i o 1 C v c h I h I quotH quot E A 9 H x 1 0 2 r l I II t x 2 4 6 113 l gt m Photon energy ev E Levl Cu effect of d electrons insufficient to reach sr Reduced reflectance at short blue wavelengths gt reddish color slide 6 Fundamentals of Optical Science Spring 2006 Materials practice set Metal n and K vs energy Response of gold Au Low frequency n lt 1 and large K gt signature of negative 2 apparently we re above resonance but absorption max at co0 metal nlt1 also results in large reflection metallic appearance at low freq 14 12 10 index of refraction Energy eV Fundamentals of Optical Science Spring 2006 Materials practice set slide 7 Metal dielectric function vs wavelength Response of gold Au Low frequency long wavelength negative 2 due to free carriers 20 I I I I 20 dielectric function 40 3960 I I I l I I I I I 02 04 06 08 10 12 14 wavelength um Fundamentals of Optical Science Spring 2006 Materials practice set slide 8 Insulator dielectric refractive index vs energy Response of titania TiOz Low frequency finite index and low absorption could be POLAR liquid or insulator See absorption band at finite frequency polar liquid or insulator See normal dispersion ie increasing index NOT a polar liquid index of refraction 39 I 39 I 1 2 3 Energy eV 4 01 Fundamentals of Optical DCIence bpring zuuo IVIaterIaIs practice set slide 9 Fundamentals of Optical Science Spring 2006 Materials practice set slide 10 waveleth m Energy em refl ectan ce 0 0 0 0 o N A m I 20 dielectric function AmwAm ooooooo xnwAm II II 10 re 39active index refl ectan ce 0 o o dielectric function re 39active index AmwAm 0 o N A m I I I 20 ix x A m o o o o o o o o I I I I l Example 1 Example 2 Ag I I 8 l I I I 7 n 7 I n K K I 6 6 I I I 5 39 5 395 5 E E II 4 II 4 I I I x s 3 3 I 3 39 e 39 39 2 2 I 2 2 x 1 quot 1 0 I l I I I 0 l 10 10 I 39 0 A quot 0 g 10 a g 10 E 1 E a 20 quot39quot52 a 20 2 2 3 3930 3 30 9 9 39 40 39 40 50 50 60 I I I I I I I I 60 I I I I I I 10 1 0 R 08 08 3 06 8 E E 06 o o In I s 04 s L L 04 02 02 0390 I I I I I I I I 0390 I I I I I I 00 02 04 06 08 10 12 14 16 1 2 3 4 5 6 waveleth 0m Energy em Fundamentals of Optical Science Spring 2006 Materials practice set slide 11 Optical Properties of Materials OSE 5312 Summer 2002 Friday July 19 2002 Nonlinear Propagation Equations Now that we have seen how it is possible to generate nonlinear polarizations we must address how these polarizations cause new waves to grow or how they affect existing waves Recall that we found that nonlinear polarizations for discrete frequency inputs also have discrete frequency components where the frequencies generated are at the sums amp differences of the spectral components of the input Eg 131 w 60 NW1 w2w1w2Eltw1Eltw2 OI P2w3 eo252w3w1w2Ew1Ew2 where 033 01 032 The time and z dependence of the polarization at 033 is assuming all waves are collinear and propagating in the zdirection Pagjtz 60 12Q3wla02Ea1eik1Z mltEa2eik22mzt cc 260 12Q3a1a2Ea1Ea2eiklk2zm3t Note that while this polarization wave has frequency 033 its propagation constant k1 k2 depends on the refractive indices at 031 and 032 This is because the phase of the polarization Pagfkhz depends locally on the phases of the electric fields at 031 and 032 We anticipate that a wave E3 at frequency 033 will be generated by this polarization so we will take a look at the wave equation for E3 52153 i 2153 lt3sz 0 622 c2 62 62 52 1 2 P603 ZJZP U3 ZZZ where Pa t z 60 11 03 Ea3 eik3zm3t 00 and k3 Hence 62E3 1 10 62E3 62PIZ 522 c2 622 0 622 Now we will anticipate that the amplitude of the generated field E0133 may change with z ie 1 ik z a t E3 I z 3A3 26 3 3 00 We can plug this into the wave equation but first we can take the derivatives 6E 3A 3 3 z39k3A3 jet quot30 00 Oz Oz 62E 6A 621 61 3 2i z39k3 3z39k3A3 3 ik3 3 Jaw 30 cc 622 2 62 622 32 621 61 i 3 2ik3 3 32143 Jaw 030 00 2 622 62 5253 1 2A ik3Z a3t Also V 303 3Ze cc And 2 lt2 azp quot l 2 2 3 atz 3w3 E0 6 0133012012 gtlt Ea1Ea2eik1k2zw3t cc Now we apply the slowly varying envelope approximation SVEA1 5213 622 6A3 E ltlt 2k3 which is valid provided the changes in A3z are small over length scales on the order of 7 Under the SVEA the second derivative is eliminated from the propagation equation 5A 1 1 2ik3 6 3 k32A3 e k3z 30 cc Z 2a3a32A3 Ze k3z 30 cc Z w u0 e0 Z2w3w1w2Ew1Ew2eik1k2z 3t no 1 1 a Noting that 1632 this simplifies to C A 3 e39kBZ who 60 z2gtltw3w1w2gtEltw1gtEltw2gte39k1 2 6 21k 3 62 1 Sometimes called the slowly varying amplitude and phases approximation SVAPA which recognizes that the amplitude may be complex and the phase may change due to nonlinearities as well as the magnitude OI a we 2 39Ak E0 Z 0301202E01E02el Z 62 2na3 where Ak k1 k2 k3 is the phase mismatch We note that 132 03 60 262 030102E01E02 so that in general terms we may write l 30 P2a3eiAkz Oz 2na3 or a 3 l 1132a3 eiAkz 62 2na3 60 c where 132 can represent any second order process ie P203 30103 2 P203 320 1 etc The phase mismatch results from the waves at the three different frequencies propagating with different phase velocities and hence getting out of phase with each other which adversely affects the efficiency of conversion from one frequency to another Perhaps it is easiest to understand the phase mismatch by looking at the secondharmonic generation process SecondHarmonic Generation For second harmonic generation we have 2w z ie 2 2000 Azemkz 62 2n2a60 02 0 I 1 20 Here NC 21 km quot250 It is common to use the deff coefficient defined by deff X22 so that 6142 l adeff AIZez39AkZ 62 r1200 2 iAkZ e where K2 is the figure of merit for second harmonic generation Provided we can assume that the fundamental field A1 is constant along 2 this differential equation is easily solved eiAz 1 i 1422 2 1420 iK2A12 39Akz2 iiAkzZ e e A20 1K2A12e AkZ22 ZlAkZZ Z sinAkZ 2 A 0 iK A222 N z2 2 2 l We are usually interested in the case where A20 0 so we see that A Z ZiK AzzeiAkZZM 2 2 1 AkzZ ZIK IZZZ AkZ 3 1250 Z zz IjschE j a Go 2 So that the second harmonic output irradiance is proportional to the square of the input fundamental irradiance The sinc2 dependence is plotted below I203 o Akz2 Note that for Ak 0 the phasematched case 120 oc 22 But for Akz i275 the second harmonic output is zero This is the usual expression we see for the second harmonic irradiance Usually z L the thickness of the nonlinear material which is not variable The phase mismatch is often tunable though as we will see below and it is often quite easy to generate the above curve experimentally However it is usually more insightful to look at 1202 In the above analysis we multiplied the top and bottom by z in the expression for A2 in order to get the sinc function If we do not do this we find sin AkzZ A2Z lK2A12 Ak2 Ak2 Z K 2 I2 Akz 3 113222 wsng j nwc GO AkZ 2 We see this plotted below In all cases for very small 2 the second harmonic grows as 22 as sinx x for x ltltl But if the indices of refraction at 0 and 20 are unequal the SHG produced near the entrance which propagates at Cl lzm and the polarization that generates new SHG which propagates at cnm eventually get out of phase By the time AkzZ 752 the new SHG is already 752 out of phase with the old SHG that was produced at the front surface Hence at this point the new and old SHG start to cancel and further propagation just reduces the SHG field Unless we can make Ak 0 there is no advantage in having a material longer than 2 TcAk This distance is known as the coherence length 10 7r xi 10 Z Z M 4quotm n2a 1o 8 6 g Ak05 4 2 Ak1 0o 1 2 3 Example For a phase matched 1 cm thick crystal with deff l pmV and refractive indeX nm n20 15 find the irradiance of the second harmonic for a 10 MWcm2 irradiance fundamental input at a wavelength of l um Solution lm1011Wm2 L 001 m d 1012 mV 0 188x1015 rads 2 2 2 2 2 2K2 L 2a d L 12wltL I 12 a 12wz 3 If 587x109Wm2 nwce0 116002 60 0587MWcm2 Hence the conversion efficiency is 0587 10 587 High conversion ef ciency case It must be stressed that the above equations are valid only for the case of low conversion efficiency so that little energy is transferred from the fundamental field to the SH field This allowed us to assume that A1 is constant along 2 Should A2 become large this cannot be assumed and we must consider the depletion of the fundamental This is described by the nonlinear polarization term P2a 60 122a a2a aE2aEa so we have to solve coupled propagation equations 6A 2 2 39 a 60 ZZZaaaAlzelAkZ 62 2n20 60 0 6A 1 iL 2 60 gm aZa aA2A1e AkZ 62 2170 60 c Now if we have large conversion probably Ak 0 and hence nm ngm It turns out that the two susceptibilities are equal so that K2 K1 K say The fields can be considered real so we have 6A2 2 m4 62 6A1 iKA A 62 2 1 In addition energy is conserved so that A12 2 A22 2 constant 2 14120 We can solve to get 12012 1m lt0tanh2ltA1lt0za 1a 0 tanh2Fz where F KA10 Since L 120 100 we also find an expression for the depletion of the fundamental irradiance 1 z 1 0 sinh2 r2 Techniques of Phase matching In any normal material n20 3 nm There are three main ways of overcoming this Anomalous Dispersion Phase Matching It may occasionally be possible to match the indices if 0 and 20 are on either sides of a resonance as sketched below This is highly unusual and not a generally applicable method of phase matching an 0 20 V nw Bire ingent phase matching It is often possible to find propagation direction in a birefringent crystal where the ordinary wave at 0 has the same index of refraction as the extraordinary wave at 20 Or viceversa This is illustrated in the figure below This requires a Xe tensor element that gives a second order polarization polarized perpendicular to the fundamental field but nevertheless there are many materials for which this is so and this is the most common means of achieving phase matched Xe processes z optic axis I l I lgt I I Figure 88 Normal index surfaces for the ordinary and extraordinary rays in a negative n2 lt n uniaxial crystal If n lt n2 the condition nf 6 ng is satis ed at 0 0quot The eccentricities shown are vastly exaggerated In this case the exact phase matching can be tuned by adjusting the angle as illustrated in the experimental data below 44 L0 X 81 X 10 0 Experimental 39 TEMOOII 08 P 148 X lO39awatt 2 1 23 cm 06 sun 180 0m Km 3 2 c l3 9 90 04 02 r I I 7 l I w t 01 0 0l 02 0 9m Figure 89 Variation of the second harmonic power P20 with the angular departure 6 6m from the phasematching angle After Reference 11 10 Quasi PhaseMatching QPM It is possible to obtain an large SHG or other efficient Xe processes by periodically modulating the X9 along the propagation direction as shown below PZmauXIElt3LIll1 Distance zLL d Figure 821 The evolution of the secondeharmonic phasor E ZWz in a a nonphaseematched case b quasi phaseimatched and c bulk birefringent phaseqnalching Ak 0 d The secondharmonic power eld Em in a crystal for cases a b c above as well as curve 7139 d for quasi phaseimatched operation using the third Fourier coefhment m 3 LC E 7 Pan 1 is reproduced from Reference 34 Here the crystal is reversed every coherence length ie for each successive layer X0 gt me Hence instead of having a reduction of Em after 2 10 we reverse the phase of the quotnewquot SHG so that it continues to add to the SHG generated in previous layers This adds tremendous exibility as we are no longer restricted to the polarizations and directions imposed by birefringent phase matching However this is a difficult materials fabrication problem QPM has been known since the early 1960 s but it is only in the last few years that QPM materials have become commercially available Other 762 processes There is insufficient time to go over the theory for other processes but it is quite straightforward to apply the same techniques to sum frequency generation SFG and difference frequency generation DFG Phase matched DFG is particularly interesting This involves a beam at high frequency 0 3 mixing with another lower frequency 031 to give a difference frequency beam at 032 033 031 2 X 032 033 39031 It turns out that assuming do depletion at 033 the beam at 031 grows as cosh2Pl and the beam at 032 grows as sinh2Pl This corresponds to near exponential gain The gain can be so high that with only an input at 0 3 the other waves can grow out of noise The particular values of 031 and 032 will depend on phase matching This process is known as Optical Parametric Generation and Amplification OPG amp OPA Just like amplification by stimulated emission this can be put into a laser cavity to build up a selfconsistent mode This is known as an optical Parametric Oscillator OPO Nonlinear Refraction and Absorption We now wish to look at the degenerate thirdorder susceptibility 9630 0 o03 The SVEA propagation equation for a monochromatic field at frequency 0 in a medium with susceptibility X3o39030303 will look like dA a 3 2 z aa aa A A dz 41101 I I 39 3 3 3 Setting X w7w7a7a X 195 wehave 031 0 2 3 3 1 1 A A dz 4nc gr II I 39 Now the slowly varying field amplitude may contain a phase component as we write Az Azei z so the propagation equation becomes d a Alew EzzE IIl3e zf3lAl3equot dA d a 3 3 iA i 3 A 314 dz I I dz 4nc gr I I Z I I Therefore we have two propagation equations one for amplitude and one for phase Mai 3 3 dz 4ncZi IAI 3 2 dz 4nczr IAI This 53 leads to nonlinear refraction while 53 leads to nonlinear absorption Nonlinear absorption Multiplying the equation for nonlinear absorption by A dIAI lt3 4 IAI dz 4nczi IAI dlAlz 2 dz 2 we can wr1te Since I IA d a 3 2 fl 1 dZ 2n 0 60 gt 6 I2 Where 3 is the twophoton absorption coefficient This has solution 12 I0 l IOZ Nonlinear re action 21 lt3gtA2 z a Zlt3gtA22 The equation dz 4m Zr I I has solution 4 10 r I I Hence the propagating wave may be written as EQ A a h w AZ sz 119414sz at 4110 using k nowc k0n0 where no is the linear index of refraction we can write this as 2 z9M Ezt Az cos kz n0 4 wt quot0 Hence the total refractive index is nLM2 4110 Of n1 n0 n21 where E0 C 3 quot2 2 4 So the refractive index can be irradiance dependent For Gaussian laser beams this can lead to quot SelfFocussingquot lQgtO In the case of a negative n2 we have quotselfdefocussingquot Optical Properties of Materials OSE 5312 Summer 2002 Monday June 24 2002 Effect of dipole radiation eld from Lorentz oscillators Another way of looking at the optical properties of materials is to consider the filed reradiated by the induced dipoles of the classical Lorentz oscillators This dipole radiation field will interfere with the incident field in such a way as to produce absorption or refraction Due to time constraints in this semester we will only have time to look very brie y at this subject Dipole radiation We make use of the vector potential A We know that V of 0 and since there is a general identity V 0 V X 17 0 which applies for any arbitrary vector field V This implies that there can exist a vector potential A such that B V X A Actually there are a set of possible A s We use the Coulomb gauge or transverse gauge where V 0 21 0 The wave equation for A in the Coulomb gauge is a 6221 VZA uo e atz Iuojl where JL is the transverse component of the current This has a general solution gt 77739 3 l J JLUJ C clr 2 C 7 739 which we can use to find the dipole radiation field We assume a point dipole placed at the origin I it f l9 137 p0 cosal57l30 j 2 g 2 wp0 sincot57i90 jl apo Sinat57j70 h Now 30 L is the component of the unit vector that is perpendicular to r oh I xfxz30 the magnitude of which is cos6 A polar plot of cos 6 so shown below Now we can substitute for JL in our solution for A to get 7 fgtltfgtltiv Sinlwl zcll A7gtt 4 20 pr 723960 c Now the reradiated or scattered field from the induced dipole Es is therefore given by COS a Z E I X I X 2 6 Es w 190 51 47 GO 02 39739 a k2 coswt L A A A E WW 47 60 r rJ a4 Spherical wave Dipole pattern Note that in for a dipole with no damping p0 is in phase with the incident field and hence Es is in phase with the incident field We will come back to this later Now we want to find the radiated power which means finding the Poynting vector which in turn requires us to find the magnetic field which we do as follows where we used the fact that f 19 in this geometry Now fgtlt79gtlt9gtlt 0 mp0 some A A B 7 rgtlt Sothat s 4 60 C r P0 55 an 39 39 39 F by the dipole the scattered eld Es Hence the total poyhtjhg vector is 1 SXio 1 s i 1gtlt 1 lgtlt s 55 X1 1 S X s 0 Smclderzz Sabxorbed chazzered New the scattered elds are wak cumpared tn the mement e d5 5 we can rat the scattered intensity as bemg neglxgxble cumpared tn the ma amt Hence the nance is pnmanly a cctcd b mcc bet th 3 lt messluwl Wnl thcpelarplethelew me Hupf at Stcgehen cheptccA m as m mmtcmm vatrm cascva around a sphere surmumhn x a were the mate micrsecu m in arm of h m nl you when he the ts xem Hum mile at cucle m mutt amt Doimve m mu A mu e a me cllcm m Dehxlnd meme at m lnlerlermze m Resmx at average mcc the effect erthc dlpule ls te impress a small shaduvf39 un the Lmnsmmed eldbehmd ll Ifwe mtcgetc the uulwardr uwmg cempencnt ens uvaall anglestheresulnsthelu39zl absurbed qua Fhl c 3117 am 6319 where the mth is uvaall suhd angle 0 Ths yields Pubs f70 E Imr Where A 0 e 50 39E EIW e 50 OEEIa m 03 a2 iaF m Da ImroEI WIEIIZ Hence eAOE 2 r0 IEII 2 m Dltwgt Now we may define a molecular absorption cross section cabs that relates the absorbed power to the incident irradiance by Pabs aabsltS1gtt Hence Oabs SI IA70 El Imr wEl 1 2 Since S 3710 Go E1 we obtain 2 Jabs 07lt 0 EI2gt F 6 mDa2 Where the averaging is now over all angles between the dipole coordinates and the applied field For a medium containing randomly oriented molecules ie isotropic we have already seen that ltj70 oEIzgt g 9 Since Pabs is an absorbed power per molecule we can find the net absorbed power per unit volume as N Pabs NoabsSI But the power absorbed per unit volume for a plane wave is just dSIdz Hence d5 7 N0abs 2 512 2 510e 510e 39NZ S1 2 0091 Hence the absorption coefficient is given by 050 2 cabs wN Scattered Power a 1 a a Recall that 5mm Es XBS where 0 u k2 u ES 2 po expik o 7 at 00 x x 0 47 60 r and ES 2 p0 expi o 7 at cc f X g This simplifies down to 4 2 lt scallgt k 0p0 cos2 6 7 2472 60 r2 4 IPOI2 2 A ZMV ZCOS 6 V The scattered power is then given by 27239 7239 2 A Pscatt Sln ltSscattgtt1 0 0 ZAP 2 12713960 c3 0 Hence scattering power is proportional to 034 or 7J4 Rayleigh scattering We can similarly define a scattering cross section Pscatt ascatt SI gtt Z a 0464 o scarf 2 2 67 60 04m2 Dal Refractive Index Unlike absorption we cannot calculate the refractive index from the field of a single dipole We must look at the field due to an ensemble of induced dipoles To calculate the refractive indeX real and imaginary we look at the field produced by a sheet of dipoles all induced by the same incident plane wave 4 K M 1xquot r l v l quot 81 f Figure 43 Illustration of sheet at dipoles 0f dcmhty 3 thickness dz and area A where we are ultimately interested in the limit A 4 an Summing over the NAdz dipoles in the sheet we find that the net radiated field is also plane wave described some distance behind the sheet by 6quotchngth 130gt cc ERI2 1 and since B ZXE ERIL2Ndzltzx130gt cc 260 c Note the cc which means that the radiated field ER is 90 out of phase with the oscillation phase of the dipole This implies that for a real p0 and a cosinusoidal incident field the reradiated field is sinusoidal Hence in the absence of loss the reradiated field is 90 out of phase with the applied field For a very weak reradiated field always the case this translates to a phase shift on the incident field rather than an amplitude change E1 ER J Resultant Lossless oscillator fleld 7 below resonance For the Lorentz oscillator model given above we find 2 A A2 EREIikNe dZltPO 2 60 m Dco where the averaging is over the molecular orientations Now the total field is Etot E ER A A2 EtotERdEIlkN lpo39El E dz dz 2 60 Da but 2 kn 3E1 dz 2 I Hence 2 A A 2 77niltl Ne DOE 260m Do And therefore A A2 2 Ne2 pOoE 6277 277l 60m Do which is the Lorentz oscillator model result including orientational averaging Clearly then the phase of the induced dipoles and hence the phase of the re radiated field dictates the optical properties of the material On resonance the oscillators are 752 out of phase with the driVing field The reradiated field is shifted by yet another 752 upon propagation so that ER is 7 out of phase with E1 E1 i L Resultant I Lossy oscillator 7 field at resonance Clearly we may extend the s arguments to gain If the phase of the oscillator is 452 then the radiated field will add to E1 Previous lectures 14 Found description of propagating plane waves J J EgU Hat En where the dispersion relation in a medium was given by electric response only if g H39X Cu Taking into account the magnetic response and remembering c E llLou0 k mi 0 I Xc m L l 4 le Or simply k2 IS 039 with permittivity IL and permeability u given by 58IXe and pyLlJgtm where Xm ltlt 1 or u z 0 in most materials and Lr1Xeco the dielectric function Fundamentals of Optical Science Spring 2006 Class 5 slide 1 Previous lectures 24 In vacuum plane waves propagate with a phase velocity V 393 Pogo h39l39 39d I I d39 th h I 39t39 v quot 33 WIeInSIeaowossmeIum epaseveOCIyIs Pk rs n with n the real refractive index In isotropic nonmagnetic materials wave propagation is governed by the complex electrical susceptibility X with Xu X39hu ixquotu we can write the dispersion relation k2 um 032 in terms of a complex wave vector k k i k and a complex refractive index n n i x U The complex wave vector is now given by k 3916 or srlk39D with k0 21cm the free space wave vector and 7v clf the free space wavelength Fundamentals of Optical Science Spring 2006 Class 5 slide 2 Previous lectures 34 The relation between the complex refractive index n and the susceptibility X is and r10 4 K00 ll J X39IU iXquot Taking the square of this and equating imaginary and real parts we found quot 04 who a iX39u 2amp O 6 Wu 6 With llL and llL the real and imaginary parts of the complex permittivity um Klw X39Yu I An complex wave vector ksrlk0 resulted in absorption o 2 In Id 2K Z In the case of dilute media small X amp X the relations between 1 and X became D II V X nu Hr X7 and Kw 7 1 Fundamentals of Optical Science Spring 2006 Class 5 slide 3 Previous lectures 44 KramersKronig relations link real susceptibility X to a spectrum of the imaginary susceptibility X 09 2 439 quot039 X W RIF 0 real refractive index n to an absorption spectrum oc 3 5 u39J Mu l 7 Who do 0 a phase shift upon reflection p to a reflectivity spectrum R a Jquot a cwum 90 1739 uquot 0quot A 0 Fundamentals of Optical Science Spring 2006 Class 5 slide 4 Today s lecture 1 Simple model of a polarizable atom the Lorentz model slides 619 single resonance frequency one absorption only expressions for Xb and consequently 110 ocb and Ro 2 Extended model multiple absorptions slides 2024 3 Modified Lorentz model absorbers in a dielectric host material slide 25 Fundamentals of Optical Science Spring 2006 Class 5 slide 5 Equation of motion 3 Illuminate atom E TW lt9 i l J Charge displacement E 1 E E Equation of motion ma Em Average electron position 17t All forces acting on the electrons E Felecmcal F ic on Fresmre Fundamentals of Optical Science Spring 2006 Class 5 slide 6 Individual forces on single atom Electrical force Fe 2 eE s a 2 K Restoring force Fr Kr ma0r With 00 m Assumes force linear in displacement always true for sufficiently small amplitude This linear dependence of restoring force on position is Hooke s Law am Friction loss Ff 2 mr a Assumed linearly related to to charge velocity with F units s a damping rate Contributions to F Radiation energy loss due to emission fast relaxation high F Scatteringcollisions eg excited ions colliding with other ions in gas phase oscillating electrons exciting vibrations in a material Fundamentals of Optical Science Spring 2006 Class 5 slide 7 Fourier transform of the equation of motion Equation of motion m a Ff Fr Fe 2 or m9mrmw 7t 6Et To get a relation for roa write rt and Et in terms of their Fourier components a iwl a J 10 FIG lawn w and Eu EU0e u 0 w Write out all the derivatives For example co 1 m t FY g ml 4quot 7 I39m coquot FadeWaco a ca Fundamentals slide 8 Complex amplitude vs frequency Then obtain the Fourier component at a single frequency 0 This results in ma2 139me mco0217a e a giving us the frequency dependent charge oscillation and thus polarization s e 5a r a m 002 602 in Note that again the loss term is responsible for a phase shift roo shows the features of the driven damped harmonic oscillator 1 Maximum amplitude when denominator minimum when at ammo resonance 2 For excitation at resonance ammo r03 is entirely imaginary gt at resonance 90 degree phase difference between E and r 3 At high frequencies 0 gtgt 030 the amplitude roa vanishes 4 At low frequencies 0 ltlt 030 a nite amplitude is obtained Fundamentals of Optical Science Spring 2006 Class 5 slide 9 Amplitude and phase of a driven oscillator man on a swing model Driven harmonic oscillator amplitude and phase depend on frequency gl Low TKEQMEHLV A1 SonAME m6quot Humtum Low frequency medium amplitude displacement in phase with F At resonance large amplitude displacement and F 90 out of phase High frequency vanishing amplitude displacement and F in antiphase Fundamentals of Optical Science Spring 2006 Class 5 slide 10 G Amplitude and phase of a driven oscillator pt 2 Behavior of a driven and damped harmonic oscillator can be summarized as follows 5 a low high frequency frequency limit limit AHPUTMDE PHASE LAG Jisrlwcch4 v5 Jrivin 392 FIRE away This type of response of bound charges l is typical for many materials Next if we convert our single atom response to polarization per volume we can link rco to Xo giving us a microscopic model for Xco aco and nco Fundamentals of Optical Science Spring 2006 Class 5 slide 11 From microscopic polarizability to Poa First relate singe atom ro to single atom dipole moment 1a ei7a This gives a 262E or 5 a a if a with or a tensor describing the general atomic or molecular polarizability ln tensor form it can describe anisotropic materials We will use the scalar form 1 2 6 giving us the Lorentz polarizability aa 2 2 m 00 a 1Fa To convert this single molecule response to the polarization per unit volume sum over all atoms in a volume V then divide by volume For a N atomscm3 this gives NV atoms average dipole moment per atom Hm gmm wm or PltcogtNgltcogtEltcogt average atomic polarizability Fundamentals of Optical Selence Spring 2006 Class 5 slide 12 Lorentz model complex susceptibility Remember that we previously defined Wu Xlw Elu Pa NaaEa which we can now relate to the microscopic model 2 Y e 1 which with 050 2 N 0560 m mg wZ 139Fa mo 0 mo N62 1 es glv 60m 002 602 1170 2 N62 where the prefactor IS related to the plasma frequency cop through mp 60 m Assumptions restoring force linear in r damping linear in drdt averagedmacroscopic fields and local fields equal dipoles all point along the applied field isotropic Fundamentals of Optical Science Spring 2006 Class 5 slide 13 Lorentz model real and imaginary susceptibility We can now rewrite Xco as lt gt 0 a Z wOZ wZ l39Fw make the denominator real by imilliplying with quotl General trick un lderstand amp n iemorizel 2 mp wOZ a2zl a 2 2 2 2 wO a lFa wO a 11 4 maximum near 030 resonance which gives us the real and imaginary components of complex susceptibility Xco 2 2 y 2 0 0 0 Fa and Zquot0 02 002 6022F202 p 2 002 a2 1 2a2 Fundamentals of Optical Science Spring 2006 Class 5 slide 14 Lorentz model resonance approximation The first term in the denominator can be written as 002 02 00 caa0 0 Near the resonance frequency we have a0 0 ltlt 00 so we can write 002 02 z 200 00 0 which gives 2 2 39 z cop coo a d H g 0p F2 Z 0 2600 a wgt2 ltr2gt2 3 l w 2 ltw0 my r m2 I 1 1 On resonance 600 a2 r22 r22 F FWHM of 5quot Near resonance 1 1 at pomp coo m2r222F22 X has a Lorentzian line shape with a Full Width at Half Maximum FWHM off Fundamentals of Optical Science Spring 2006 Class 5 slide 15 Lorentz model line shape L 1 quotJ L 5i I I I I I I 1 I I I I I I I I 35 Fundamentals of Optical Science Spring 2006 Class 5 slide 16 dielectric functions 2 2 2 00 0 Zya wp 2 002 02 F2602 HX39zu amp o H 2 Fa Z Op 002 22F202 an X w 6 Lorentz model example broad resonance 6 I III P I I I 8a 8018 1 I2 II2 I 110 3 Jar 8r 8r Ka 8r128rll28rl 060 2 2K Ra I l l2 K2 n12lt2 re ectance from air at normal incidence 1 I I I I I Roa 05 Fundamentals of Optical Science Spring 2006 Class 5 slide 17 Lorentz model examples narrow resonance 0304 03p8 lquot03 Fundamentals of Optical Science Spring 2006 Class 5 slide 18 G sltwso1ze mm a 8 m2 we aa21lt Ra I l l2 K2 n12lt2 R0 05 re ectance from air at normal incidence Lorentz model R013 05 0304 03p8 lquot03 Fundamentals of Optical Science Spring 2006 Class 5 slide 19 typical frequency dependence Four distinct spectral regions Transmissive 0 lt 030 F2 Absorptive m0 F2 lt 0 lt 030 Reflective 030 F2 lt 0 lt mp Transmissive 0 gt 0 P frequency ranges are approximate The regions are more distinct for smaller F and larger 03p Real atoms quantum mechanical picture 6 Real atoms up to tens of electrons many 394 i3 electronic levels gt many resonant tranSItions j2 99 X j1 AEl All Ne2 f Using perturbation theory 1a Z 2 2 60m 1 mf a le0 Ground State with ooj the resonance frequency of a transition with AEJ hm ijis the oscillator strength and 1 is the decay rate of statej The ThomasReichKuhn sum rule states that Z Z J39 Fundamentals of Optical Science Spring 2006 Class 5 slide 20 Real atoms quantum mechanical picture 6 Idealized case purely radiative relaxation J4 J23 decay rate F units S4 is related to the i2 FWHM of an emission line f F1 fast decay high damping gt f f3 4 broad resonance broad emission spectrum 2 f1 F 4 Ground State In molecules solutions solids broadened absorption bands Energy levels broaden due to vibrational and rotational states interactions between electrons on neighboring atoms Doppler shifts gt broad emission absorption line is not always due to fast decay Fundamentals of Optical Science Spring 2006 Class 5 slide 21 Real world example a KCI crystal Reflectance for single transition R0 05 one broad reflection band Reflectance for polar solid KCI Below 7 eV transmissive and normal dispersion Higher energy several transitions Reflectance Vol High resonance frequencies KCI looks transparent at visible frequencies hveV Fig 36 The spectral dependence of the re ectance of KCl The region of transparency extends to about 7 eV Above 7 eV there are a number of sharp peaks related to narrow energy bands and excitons From H R Philipp and H4 Ehrenreich Phys Rev 131 2016 1963 FLIHUEHIBHLHIS UI UPLICHI OCIEHUE Dpllilg AUUO mass I slide 22 Refractive index vs frequency Silicon 7H1 Refractive index n 6 Absorption index K 39 50 t I g5 4 0 C 39 x g4 M 3 05 3 a o a 20 c 2 lt1 3910 1 I I I I 00 Photon energy hv eV At visible frequencies below resonance transmissive BUT high refractive index still yields high reflectivity 3050 see Wooten gt all visible frequencies reflected strongly gt Si looks gray metallic I ig jgg enta39s 0f Opt39ca39 Sc39ence Spr39 9 2006 39 0393 5 From httpwwwiofferssiruSVANSMnk Refractive index vs frequency Si3N4 and SiO2 2 8 I I I I l I I I I L I I I I I 27 t 25 39 39 2 160 E 2 5 s E E E 2 4 39 39 g 1 55 o 5 g 23 g g SD 22 5 150 21 39 NWv 8 I3 N 4 S 0 20 145 2 I I I I I I I I I I I 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 Photon energy by EU Photon energy by eV Common dielectrics typically one resonance relevant in experiments used at low frequencies gt transmissive spectral region Fundamentals of Optical Science Spring 2006 Class 5 mtgWWW ioffe rssi rUSVANSMnk slide 24 39 39 39 39 Isolated absorbers in a transparent host material 3 vsblei Nd Ions In glass Distinct absorption lines within host glass transmissive region or o I Describe P as superposition Ptot Phost P b o I Percanl ohmrplion I dopant 2 mp Bot 80 Zhost 2 2 00 0 le 4000 5000 6000 7000 3000 9000 10000 Wavelength A Fig 29 Absorption versus wavelength of Nd glass Material EDZ thickness 63 mm with Xhost assumed constant Around an absorption line the dielectric function can be described by 2 2 P P a w1zh r O a a2 iFa or g a a A a a2 iFa where quotL is known as the high frequency dielectric constant 2 At zero frequency this gives a static dielectric constant ast 800 2 Fundamentals of Optical Science Spring 2006 Class 5 0 slide 25 Next lectu re Re ne our model of P03 to account for differences between the microscopic world of atoms amp molecules and the macroscopic world of optical materials amp optical measurements Local eld corrections Averaging over random orientations of molecules Averaging over statistical variations in atomsmolecules Inhomogeneous broadening Fundamentals of Optical Science Spring 2006 Class 5 slide 26 What you will learn this semester 1 Electromagnetic waves can move charges around 2 Accelerating charges emit electromagnetic waves and along the way we may also lean about free charge vacuum displacement polarization bound electron polarization and magnetization the vector potential the meaning of susceptibility Kramers Kronig relations dipole radiation Rayleigh scattering the Lorentz oscillator electronic transitions in atoms molecular rotationalvibrational transitions the Drude model for metals energy bands excitons impurities color centers optical properties of atoms molecules metals dielectrics and semiconductors lasers black body radiation Einstein coefficients population inversion rate equations gain gain saturation the anharmonic classical oscillator model harmonic generation nonlinear absorption amp refraction and some other things Today overview of course contents Fundamentals of Optical Science Spring 2006 Class 01 slide 2 January 23 in solids of a plane wave interacting with a polarizable oscillating dipole March 8 April 12 Sorry this is a no frills site I will try to improve the Aesthetics in the next week or so no promises Slides used for classes will be available for download before each class Also since I write on my slides I will save a version that l have written on after the class Notes that contain some other examinable material are posted on the web site In fact most of the notes for the entire course are there now But these will be revised as I prepare classes so while you are encouraged to read ahead please be sure to check for changes as we go through the course Homeworks will be posted on the site and handed out at classes Fundamentals of Optical Science Spring 2006 Class 01 Light is an electromagnetic EM wave coupled E and B In vacuum EM waves travel at light speed c 3E8 ms accurate to 01 gt free space wavelength given by 7t cf Photon energy E 2 ha 2 hx or approximately EeV s 124 Mum Mum as 124 EeV HeNe laser 633 nm Fundamentals of Optical Science Spring 2006 Class 01 slide 4 visible light violet 400 nm energy 12404 z 3 eV red 600nm energy 12406 z 2 eV f 3E8 6E7 5E14 Hz f 3E8 4E7 75E14 Hz Typical thermal energies E kT Room temperature E 24 meV Wavelength of thermal radiation order I 124 0024 50 um Optical data transmission rates THz your PC GHz your current heart rate 39 mum mg g f u l K J a 102 102 1015 10 gt ID 109 I06 103 Inc I I J I I T T i l I T 39 l l I l 1777 IF 39i I 39v i i g Gamma rays 3 lnlrared Ultramle 5 AE Xrays Microwaves TY F M EM Long radio WW I I I II J I II I I I I I i I I I I I I I I I l I I I 1013 1017 10 I I0 6 m 3 10 103 106 10 1 fm pm 1 nm I urn 1 mm I m 1 km L Wavelength ml 75 E Fundamentals of Optical Science Spring 2006 Class 01 slide 5 A lot of interesting physics happens at energies of a few eV Not accidentally this is the range that our eyes can detect enough energy to affect chemistry in our eyes and partly accidentally this is the type of radiation the reaches the earth s surface few eV energy can induce electronic transitions in atomsmolecules gt we can store discrete energies good for plants pfh wns per Un area r good for solar cells unit umz e good for photography bad for high energy LEDs crystal damage bad for Rembrandt paintings bleaching Fundamentals of Optical Science Spring 2006 Class 01 slide 6 solar llux energy EV solar spectrum on the ground EM waves move charges and accelerating charges emit EM waves Optical properties of materials determined by 1 how easy it is to move charges using an EM wave 2 how many charges are available atoms ions optical transition between electron orbits absorption lines ionization energies molecules same plus vibrations meV range optical excitation of molecular vibrations Raman metals large concentration of free electrons high reflectivity strong absorption insulators electrons strongly bound to atoms Egap gt 4 eV low absorption refraction semiconductors electrons somewhat bound EgalD lt 4 eV transparent in IR absorbing in visUV light can generate excitons bound electronhole pairs Fundamentals of Optical Science Spring 2006 Class 01 slide 7 j nilquotl tl lili39li 9 flll tll tir tenl 11 l in 15 Electromagnetic behavior of continuous linear media a e 95 V E 7quot 1 diverging E fields relate to charges 0 a 2 diverging magnetic flux doesn t exist o 3 rotating E fields are related to changing currents A 33 T x E 3 4 rotating B flux relates to changing E and several currents gt gt E A39Ztu II v x3 p n P 3 Vtyxwrgil a free current densit magnetization current density polarization current density Fundamentals of Optical Science Spring 2006 Class 01 slide 8 039 illilfwl u ilif infill Utility Ml7 ll39llll T infill a U UCT E F 1 15 VIE 539 fol9371 P9 SG l Z 3P5 11 Says high curvature of E in space correspondes to fast change in E over time Low frequency High frequency ELQVQH WW Solutions oftheform 41 E0 OJ u k l for wave propagation along the zdirection Fundamentals of Optical Science Spring 2006 Class 01 slide 9 ii illrmn 9397 quotTilile ll ll illl xiii We will derive relations that link what appear to be independent qualities real susceptibility X to a spectrum of the imaginary susceptibility X 09 z u quot039 X W 7577 0 real refractive index n to an absorption spectrum a co 5 u39J 39 Mu l n v rbz do 0 a phase shift upon reflection p to a reflectivity spectrum R u a cwum alw39 11 J u z I o w Fundamentals of Optical Science Spring 2006 Class 01 slide 10 s Illuminate atom E 9 i I Charge displacement E E E Equation of motion ma Em Average electron position 70 All forces acting on the electrons E Felecmcal F friction mere Gives charge motion dipole moment for a e2 1 a a glven drIVIng force or polarizablllty on m 6002 602 Ira Fundamentals of Optical Science Spring 2006 Class 01 slide 11 R03 05 Fundamentals of Optical Science Spring 2006 Class 01 slide 12 Four distinct spectral regions Transmissive 0 lt 030 F2 Absorptive m0 F2 lt 0 lt 030 Reflective 030 F2 lt 0 lt mp Transmissive 0 gt 0 P frequency ranges are approximate The regions are more distinct for smaller F and larger mp Inside a dense material atoms cannot be described as individual oscillators Many neighbors near atom affect local field We will estimate total effect of all neighbors Fundamentals of Optical Science Spring 2006 Class 01 slide 13 Look at how things change when electrons are not bound Low frequency response very different from Lorentz model Drude model predicts monotonous decrease of llLr for decreasing frequency 2 8r39a1 a Z with llLr crossing zero at up Fundamentals of Optical Science Spring 2006 Class 01 slide 14 Magnetization along the polarization occurs in chiral materials Example helical polarizable molecule Oscillating E field will induce a time varying rotating current with corresponding magnetization along the direction of the E field Definition of chirality by Kelvin quotI call any geometrical gure or group of points chiral and say it has chirality if its image in a plane mirror ideally realizea cannot be brought to coincide with itself We will find that in this case polarized light will rotate inside the material lefthanded righthanded Fundamentals of Optical Science Spring 2006 Class 01 slide 15 Linear polarization tries to induce linear charge motion but Lorentz force prevents this from being purely linear O W B 39EHQ39 Fr a L 39 I e 1 t o 3 j v rt 8 Ely quot l no applied B gt ve and E parallel vc 39 Charge oscillation along x introduces small Ey component 2 field rotation Vb B K 4 note this plot represents undriven motion of the Lorentz oscillator Fundamentals of Optical Science Spring 2006 Class 01 slide 16 9 j l l l ll ll ifllil Interaction of light with electronic states in semiconductors causes many effects PHOTON ENERGY eV Fundamentals of Optical Science Spring 2006 Class 01 slide 17 5 Core E Possible I lnterband quot39quotquotquot l electron drband absorption 3 3 region absorption region 39 gt eststrahlen regiongt 3 WAVELENGTH 39 7 001 OJA 1 1039 100 1000 b l o x m l l 10 C E 88 3 9 108 5 E I 5 E 2 quot a 0 1 105 g 2 S g t 104 L I 393 Lu g E 5 8 3 E s A 10 3 3 e Z I O g 2 a V 9 2 1 3 5 5 E l 10 E E g 0 D E 8 o 8 U L J I g 101 E 8 L 39 U E g D u lt 1 l g l l 103 1o2 101 10 10 10 2 10 3 so l quotaf rm Ln nil in list 39 xiiljl l39ri ijil jwi ji if Transverse mode low k acoustic branch Not excited by light 99929996 000 W Transverse mode k0 or very close to it optical branch Excited by light 00009 o a V u i g 3 0 o o o b 39 Transverse mode low k optical branch Fundamentals of Optical Science Spring 2006 Class 01 slide 18 We will calculate emission spectrum of black body at thermal equilibrium hn g pm hi III 1I391 I391T39I mm WWIka in mm Fundamentals of Optical Science Spring 2006 Class 01 slide 19 e Two level system two states with absorption spontaneous emission and stimulated emission Calculate populations N2t and N1t vs PIoumlo I l l l 3 a u I a u a Photon flux p my variable Ru a P absorption R1 1 P stimulated emission P A 1w Fundamentals of Optical Science Spring 2006 Class 01 slide 20 If pushed to the limit optical materials show new behavior Large amplitude binding Can be described as sum of potential looks anharmonic harmonic oscillations or colors Result anharmonic oscillation 5 Fundamentals of Optical Science Spring 2006 Class 01 slide 21

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