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Particle nature of light waves in dielectric media
ARTICLE IN PRESS Physica B 404 (2009) 3880–3885
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Particle nature of light waves in dielectric media C.Z. Tan Bowei Geophysics Ltd., Xiangtan National High-Tech. Center, 411104 Xiangtan, Province Hunan, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 31 January 2009 Accepted 5 July 2009
Wave-particle duality is a foundation for modern science. The speed of light waves in dielectric media is less than c. The corresponding particles thus have mass. Combining wave-particle duality with the theory of relativity, an exactly solvable problem was proposed, concerning the transition from photons in vacuum to particles in dielectric media. The rest mass, the momentum, and the total energy of material particles are shown to be the functions of the refractive index of the medium and the wavelength of the incident light. The proposed relationships were applied to study the wavelengthdependent index of refraction of dielectrics and the correlation of the refractive indices of anisotropic crystals, which were confirmed by the experimental results. Variation of the refractive index with wavelength is found to obey the proposed relation. The refractive indices of anisotropic crystals are shown to be the correlated quantities. & 2009 Elsevier B.V. All rights reserved.
PACS: 04.30.Nk 42.25.Lc 42.25.Bs 78.20.Ci 03.30.+p 91.60.Mk Keywords: Wave-particle duality Theory of relativity Dispersion Birefringence
1. Introduction Study of interaction of light waves with matter has a long history in science and technology. Light traveling in a transparent isotropic medium is normally treated as the propagation of the electromagnetic waves at a speed c/n, where c is the speed of light in vacuum, and n is the refractive index of the medium [1]. Light waves in vacuum correspond to photons of zero mass and the energy hc/l, as well as the de Broglie momentum h/l, where h is Planck’s constant, and l is the wavelength of light in vacuum [2]. However, the expressions for the energy and the momentum of light waves in the medium remain an unsolved problem [3]. Electromagnetic waves carry momenta (linear and angular parts) and thus exert forces and torque on dielectrics [4,5]. When the light wave is incident from vacuum into the medium, its electromagnetic momentum is changed. In dispersive media, the speed of light depends on the wavelength of the incident light and is less than c. The corresponding particles in the medium thus have mass. Combining wave-particle duality with Einstein’s theory of relativity, the relationships are proposed in the present work to describe the dependence of the rest mass, the momentum, and the energy of the particles of light waves in the medium on the index of refraction and the wavelength of light. In
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anisotropic crystals, the incident light is divided into the ordinary (o-ray) and the extraordinary (e-ray) waves [1]. Classically, indices of refraction are considered to be the independent quantities. The proposed relationships are applied to study the nature of dispersion of dielectrics and the correlation of the refractive indices of anisotropic crystals, which are confirmed by the experimental results.
2. Particle nature of light waves in dielectric media When the light wave of the wavelength l is incident from vacuum into a dielectric medium, its frequency remains unchanged. However, the wavelength is changed to l/n [1]. This observation is a starting point of the present work. According to wave-particle duality, a corresponding particle in the medium thus has the momentum of p¼
nh
l
:
ð1Þ
Light in the medium is considered to consist of a stream of particles. Suppose a particle strikes the interface between two dielectric media. The component of momentum of the particle perpendicular to the interface is transferred to a medium. Light waves thus exert forces on dielectrics [4,5]. The material in the vicinity of the interface affects the component of momentum perpendicular to the interface. However, the parallel component
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Fig. 2. The calculated rest masses at different wavelengths of the incident light for the representative crystals of silicon (Si) and germanium (Ge). The indices of refraction are taken from Refs. [8,9].
Fig. 1. Schematic depiction of the transition from a photon in vacuum to a particle in the medium.
combining Eqs. (1), (4), and (5), which read m¼
remains unchanged, as schematically shown in Fig. 1. According to the conservation of momentum, the component of momentum parallel to the interface for a single particle obeys the form of: pi sin(yi) ¼ pt sin(yt), where pi and pt are the incident and transmitted momenta, yi and yt are the incidence and the refraction angles, respectively. Combining this form with pi ¼ h/l, and pt ¼ nh/l we obtain Snell’s law [1]: sin(yi) ¼ n sin(yt), which, in turn, confirms p ¼ nh/l. Eq. (1) thus satisfies the conservation of momentum. On the other hand, in the theory of relativity, the total energy, W, and the momentum, p, of the corresponding particle traveling at the velocity, u, are given by [6] mc2 W ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi; u2 1 2 c
l
¼ W mc2 :
W¼
ð7Þ
Due to the contribution of the rest energy mc2, the total energy of the particle is larger than the photon energy hc/l. The propagation velocity u of the particle through the medium is u¼
2nc : n2 þ 1
ð8Þ
ð3Þ
p¼
ð4Þ
The incident photons excite material particles in the medium. A particle absorbs a photon of energy of hc/l, and propagates at the velocity u. After propagating through the medium, the particle transfers the energy hc/l to the transmitted photon, and keeps the rest energy mc2 in the medium. Consider the conservation of energy, namely the incident photon energy hc/l is equal to the relativistic kinetic energy, Wmc2, of the corresponding particle hc
ðn2 þ 1Þhc : 2l
and
The energy flux accompanying with traveling of the particle can be written as uW. Combining Eq. (7) with (8), the energy flux is given by nhc2/l. Finally, the relationship between p and W is given by the following equation:
where m is the rest mass of the particle. From Eqs. (2) and (3) we see that the energy and the momentum are related by W 2 ¼ p2 c2 þ m2 c4 :
ð6Þ
ð2Þ
and mu p ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi; u2 1 2 c
ðn2 1Þh ; 2cl
ð5Þ
The expressions of the rest mass and the total energy of the particle as the functions of n and l can be obtained after
2nW : ðn2 þ 1Þc
ð9Þ
Note this relation satisfies p/W ¼ u/c2, as obtained by combining Eqs. (2) and (3). Eq. (9) is different from both Abraham and the Minkowski expressions [7]. In normal dispersion region, the index of refraction increases with decreasing wavelength of the incident light. Therefore, the momentum, the rest mass, and the total energy increase with decreasing wavelength l, as described by Eqs. (1), (6), and (7), respectively. For quantitative demonstrations of the rest masses of the particles in dielectric media, Eq. (6) is applied to calculate the rest masses of the particles in the representative crystals of silicon (Si) and germanium (Ge). The evaluated results are shown in Fig. 2. The indices of refraction are taken from Refs. [8,9]. The rest mass of the particles corresponding to light waves in dielectric media is roughly comparable to neutrino mass (1036 kg) [10]. In the present work, we apply Eq. (7) to study the wavelengthdependent index of refraction of dielectric media and the correlation of the refractive indices of anisotropic crystals.
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Eq. (10) can be written as
3. Variation of the index of refraction with wavelength
w In both classical electrodynamics and quantum mechanics, the energy density w of the electromagnetic waves in an isotropic dielectric medium is [1,10,11] w ¼ e0 eE2
ð10Þ
where e0 is the electric permittivity of free space, e is the dielectric constant, and E is the electric field strength in the medium. Consider the light wave of the electric field strength E0 in vacuum is normally incident into the isotropic dielectric medium, as schematically shown in Fig. 3. In this case the transmittance, T, is expressed by T ¼ 4n/(n+1)2, and the value of E2 in the medium is given by E2 ¼ TE20, namely E2 ¼ 4nE20 =ðn þ 1Þ2 [1]. Consequently,
e
¼
4n ðn þ 1Þ2
e0 E20 :
ð11Þ
The term of e0 E20 on the right side of Eq. (11) represents the energy density w0 of the electromagnetic waves in vacuum. In quantum mechanics, the energy density, w0, can be written as N0hc/l, where N0 is the number of photons per unit volume. Accordingly, the energy density, w, of the electromagnetic waves in the medium can be written as NW, where N is the number density, and W is the total energy of the particle. Alternatively, Eq. (11) then becomes NW
e
¼
4n ðn þ 1Þ2
N0
hc
l
:
ð12Þ
Eq. (12) shows the particle character of light waves. Combining Eq. (7) with Eq. (12), we have ðn2 þ 1Þðn þ 1Þ2 N0 : ¼ 8ne N
ð13Þ
The ratio of N0/N is a material quantity, which is independent of the intensity and the wavelength of the incident light. Let k ¼ 8N0/ N. Eq. (13) is then changed to ðn2 þ 1Þðn þ 1Þ2 ¼ k: ne
ð14Þ
The relationship between the dielectric constant and the refractive index of the medium is generally described by Maxwell’s formula [1]: e ¼ n2. From Eq. (14) we have Fig. 3. Normal incidence of light waves into the medium. The electric field E in the medium is given by E2 ¼ 4nE20 =ðn þ 1Þ2 .
ðn2 þ 1Þðn þ 1Þ2 ¼ k: n3
ð15Þ
Fig. 4. Plots of (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light for the representative four cubic substances, silica glass (SiO2), and the crystalline LiF, NaCl, and KI. The symbols represent the experimental results, and the lines are the results of linear regression. The errors in the fitting results are given in the brackets. Experimental data are taken from Refs. [14–17].
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Table 1 The intercept (a) and the slope (k) evaluated by plotting (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light for cubic optical crystals: (n2+1) (n+1)2 ¼ a+kn3. Crystal
Intercept (a)
Slope (k)
Wavelength range (mm)
AgCl AlAs BaF2 C (diamond) CaF2 CdTe CsBr CsCl CsI CuCl GaAs GaP Ge InAs InP KBr KCl KF KI LiF MgAl2O4 MgO NaBr NaCl NaF NaI PbF2 PbS PbSe PbTe Si SiO2 (glass) SrF2 Y3Al5O12 Y2O3 b-ZnS ZnSe ZnTe
1.2276(41) 15.832(40) 2.86780(30) 4.401(21) 2.89794(21) 7.2917(50) 2.5953(14) 2.6811(91) 2.3267(60) 1.2312(45) 28.538(25) 17.355(32) 71.451(34) 26.802(78) 21.373(51) 2.7472(77) 2.8717(24) 2.86160(93) 2.260(42) 2.8899(21) 2.5342(17) 2.4475(17) 2.7647(60) 2.730(16) 2.85743(88) 2.6357(57) 2.4064(44) 77.593(57) 154.533(78) 341.26(42) 33.442(16) 2.8444(24) 2.90076(20) 2.0474(18) 2.075(19) 1.2771(99) 3.495(23) 8.018(67)
5.47150(54) 6.5176(16) 5.16800(10) 5.9723(15) 5.157470(80) 6.13526(27) 5.24191(31) 5.2242(24) 5.2989(12) 5.47185(63) 6.91136(68) 6.5690(12) 7.75674(52) 6.8701(21) 6.7057(17) 5.2074(24) 5.16795(88) 5.17371(45) 5.3250(88) 5.16051(80) 5.25604(37) 5.27436(35) 5.2022(17) 5.2122(45) 5.17701(45) 5.2372(13) 5.28345(91) 7.84785(82) 8.71045(71) 9.9636(23) 7.03677(40) 5.17577(78) 5.156500(76) 5.34907(31) 5.3537(32) 5.73452(88) 5.9065(17) 6.1771(36)
0.54–21.0 0.56–2.2 0.27–10.3 0.225–2.70 0.23–9.7 6.0–22.0 0.36–39 0.18–40 0.29–50 0.43–2.5 1.4–11 0.8–10 2–12 3.7–31.3 0.95–10 0.2–40 0.18–35 0.15–22.0 0.25–50 0.19–9.79 0.35–5.5 0.36–5.4 0.21–34 0.19–27.3 0.15–17 0.25–40 0.3–11.9 3.5–10 5.0–10 4.0–12.5 1.36–11 0.21–3.71 0.21–11.5 0.4–4.0 0.2–12 0.55–10.5 0.55–18 0.55–30
This simple relationship is resulted from wave-particle duality and the theory of relativity. The validity of Eq. (15), in turn, is a direct confirmation of Eqs. (1), (6)–(9). In dispersive media, the index of refraction varies with wavelength. According to the prediction of Eq. (15), a straight line should be demonstrated in the plot of (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light, with the slope of k. The indices of refraction have been accurately determined in wide wavelength regions for numerous optical materials of technical importance [12–17], which can be used to verify the proposed Eq. (15). Fig. 4 shows the plots of (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light for the representative four isotropic materials, silica glass (SiO2), and the crystalline LiF, NaCl, and KI. Experimental results are taken from Refs. [14–17]. As predicted by Eq. (15), linear relations are demonstrated in the plots at different wavelengths. Although the index of refraction varies with the wavelength of the incident light, it is found to obey the general relation of Eq. (15), except for the intercepts in the plots. For most cubic optical crystals, the evaluated results of the intercept, and the slope, k, are listed in Table 1. The errors in the fitting results are given in the brackets. These results are the confirmation of Eq. (15), and in turn, the verification of Eqs. (1), (6)–(9).
Fig. 5. Transition from the o-ray to the e-ray waves of the plane-polarized incident light in a right-angle prism composed of an uniaxial crystal. A right-angle edge is parallel to the optic axis, and an angle y opposite to the right-angle edge parallel to the optic axis satisfies the law of refraction no sin(y) ¼ ne cos(y) or y ¼ arctan(ne/no).
4. Correlation of the refractive indices of anisotropic crystals In anisotropic crystals, the incident light is divided into the ordinary (o-ray) and the extraordinary (e-ray) waves [1]. Classically, indices of refraction are considered to be the independent quantities. Consider the propagation of light waves in a rightangle prism composed of an uniaxial crystal, with its one rightangle edge parallel to the optic axis, and an angle y opposite to the right-angle edge parallel to the optic axis satisfying the law of refraction no sin(y) ¼ ne cos(y) [18,19], as shown in Fig. 5. Note the law of refraction also implies the conservation of the component of momentum parallel to the interface: pi sin(y) ¼ pt cos(y), provided pi ¼ noh/l, and pt ¼ neh/l. Suppose the plane-polarized incident light propagates in the direction of the optic axis, with its vibration direction in the plane of incidence (p-wave). In this case, the incident light is the o-ray waves in the prism. After a total internal reflection, the light beam is turned to propagate in the direction perpendicular to the optic axis, which becomes the e-ray waves [18,19]. The intensity and the polarization state of the o-ray wave remain the same as those of the e-ray wave. In the transition of light waves shown in Figs. 5, both the o-ray and the e-ray waves are with the same electric field strength E. Thus we have a relation of NoWo/eo ¼ NeWe/ee or No ðn2o þ 1Þ=n2o ¼ Neðn2e þ 1Þ=n2e , where Wo and We are the total energies of the particles given by Eq. (7), and No and Ne denote the number densities of the particles of the o-ray and the e-ray waves, respectively. Consequently, we obtain a relationship between the indices of refraction of anisotropic crystals, which reads n2e ðn2o þ 1Þ Ne : ¼ n2o ðn2e þ 1Þ No
ð16Þ
The ratio of Ne/No is a material quantity, which is independent of the intensity and the wavelength of the incident light. Let s ¼ Ne/ No. Eq. (16) then becomes n2e ðn2o þ 1Þ ¼ s: n2o ðn2e þ 1Þ
ð17Þ
According to the prediction of Eq. (17), a straight line should be demonstrated in the plots of n2e ðn2o þ 1Þ versus n2o ðn2e þ 1Þ at different wavelengths of the incident light, with the slope of s. Fig. 6 shows the plots of n2e ðn2o þ 1Þ versus n2o ðn2e þ 1Þ at different wavelengths of the incident light for the representative
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Fig. 6. Plots of n2e ðn2o þ 1Þ versus n2o ðn2e þ 1Þ at different wavelengths of the incident light for the representative four uniaxial crystals, SiO2 (a-quartz), MgF2, CaCO3 (calcite), and TiO2 (rutile). The symbols represent the experimental data, and the lines are the results of linear regression. The representative crystals are of different birefringence ranging from 0.01 to 0.36. Refractive indices of magnesium fluoride (MgF2) are taken from Ref. [20], and the data of the other three crystals are from Refs. [15–17]. Wavelength ranges are given in the brackets.
four uniaxial crystals, SiO2 (a-quartz), MgF2, CaCO3 (calcite), and TiO2 (rutile). Refractive indices of magnesium fluoride (MgF2) are taken from Ref. [20], and the data of the other three crystals are from Refs. [15–17]. Note the representative crystals are of different birefringence ranging from 0.01 to 0.36. The linear relations are demonstrated in the plots of the crystals, as depicted by Eq. (17). The slope of the lines is equal to the constant s. The straight lines are with the intercepts in the plots. For most technical uniaxial crystals, the evaluated results of the intercept and the slope are listed in Table 2. The errors in the fitting results are given in the brackets. Experimental data of the indices of refraction are from Refs. [15–17,20,21]. These results indicate that the refractive indices of anisotropic crystals are the correlated quantities.
5. Discussion The intercepts shown in Figs. 4 and 6 and listed in Tables 1 and 2 are caused by the derivation from Maxwell’s formula: e ¼ n2. In classical dispersion theory, the wavelength-dependent index of refraction is usually given by the Sellmeier equation [1,12–14]
e 1 ¼ n2 1 ¼ l2 p
X fi l2 l2 i i
l2 l2i
;
ð18Þ
where lp is plasma wavelength, li is the resonance wavelength, and fi denotes the oscillator strength. Therefore, the static dielectric constant es can be obtained by using Eq. (18) and extrapolating to l ¼ N, namely [1,12–14] X 2 es ¼ n21 ¼ 1 þ l2 fi li : ð19Þ p i
Experimentally, there exits remarkable difference between the extrapolated es and the dielectric constant measured at low frequencies. As for example, the static dielectric constant evaluated by Sellmeier’s equation is 3.0 for silica glass [14], whereas the dielectric constant measured at 1000 Hz is 3.8 [22]. One can verify that the derivation from Maxwell’s formula leads to the intercepts. Further, nonlinear optical effects can also lead to the intercepts. In this case, nonlinear terms should be added in Eq. (10) to describe the energy density of the electromagnetic waves [23]. Wave-particle duality is a foundation for modern science. Electromagnetic waves propagate at the speed c in vacuum. Hence the photon mass is zero. However, the speed of light waves in the medium is less than c. The corresponding particles thus have mass, according to wave-particle duality and Einstein’s theory of relativity. The transition from a photon in vacuum to a particle in the medium is summarized and schematically shown in Fig. 1. Photons in vacuum have the momentum h/l, and the energy hc/l. The wavelength in the medium becomes l/n. Therefore the de Broglie momentum of the particle is nh/l. According to the law of the conservation of energy, the incident photon energy is equal to the relativistic kinetic energy of the particle. Combining the de Broglie momentum: p ¼ nh/l, the law of the conservation of energy: hc/l ¼ Wmc2, and the relation of W2 ¼ p2c2+m2c4, we obtain the expressions for the total energy and the rest mass of the particles in the medium as the functions of n and l. Combining the total energy W of the particle with the expression of the energy density of the electromagnetic waves in the medium, the general relationships of Eqs. (15) and (17) are proposed to describe the wavelength-dependent index of refraction of dielectric media and the correlation of the refractive indices of anisotropic crystals,
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Table 2 The intercept (b) and the slope (s) evaluated by plotting n2e ðn2o þ 1Þ versus n2o ðn2e þ
6. Conclusions
1Þ for uniaxial crystals: n2e ðn2o þ 1Þ ¼ b þ sn2o ðn2e þ 1Þ.
In conclusions, an exactly solvable problem is proposed in this work, concerning the transition from photons in vacuum to particles in dielectric media. The rest mass, the momentum, and the energy of material particles are shown to be the functions of the refractive index of the medium and the wavelength of the incident light. Variation of the refractive index with wavelength is found to obey the proposed relation. In anisotropic crystals, the indices of refraction are the correlated quantities.
Crystal
Intercept (b)
Slope (s)
Wavelength range (mm)
Ag2AsS3 AgGaS2 AlN Al2O3 (sapphire) AlPO4 b-BaB2O4 (BBO) BaTiO3 BeO CaCO3 (calcite) CaMoO4 CaWO4 CdGeAs2 CdS CdSe CuGaS2 HgS KH2PO4 (KDP) LiIO3 LiNbO3 LiYF4 MgF2 NH4H2PO4 (ADP) NaNO3 PbMoO4 PbTiO3 Se a-SiC SiO2 (a-quartz) SrMoO4 Te TeO2 TiO2 (rutile) Tl3AsSe3 (TAS) YVO4 ZnGeP2 ZnO
0.0170(45) 0.535(32) 0.2204(35) 0.01109(72)
0.97905(79) 1.00731(79) 1.01590(14) 0.99685(62)
0.63–4.6 0.54–21.0 0.22–5.0 0.2–5.5
0.0388(64) 0.307(14) 0.379(17) 0.020(17) 0.843(10) 0.0954(46) 0.073(15) 3.50(10) 0.1729(73) 0.1109(59) 0.2741(95) 0.018(12) 0.197(25) 0.175(13) 0.1600(26) 0.0533(54) 0.0502(20) 0.3428(99)
1.00856(84) 0.9279(14) 0.98343(42) 1.0059(15) 0.8438(11) 1.00663(25) 1.00784(92) 1.01668(55) 0.99777(19) 1.00477(15) 1.00530(22) 1.02254(16) 0.9562(34) 0.95065(87) 0.982490(87) 1.01792(79) 1.01483(38) 0.9358(13)
0.4–2.6 0.22–1.06 0.46–2.1 0.44–7.0 0.2–3.3 0.45–3.8 0.45–4.0 2.4–11.5 0.51–1.4 1–22 0.55–11.5 0.62–11 0.21–1.15 0.35–5.0 0.4–3.1 0.23–2.6 0.2–7.0 0.4–1.06
1.175(33) 1.080(58) 0.25634(93) 2.47(83) 0.22598(30) 0.03286(57) 0.10605(27) 12.47(17) 0.05949(88) 0.2787(80) 0.020(12) 0.044(12) 0.286(11) 0.0020(10)
0.7277(47) 0.9525(15) 0.994660(17) 1.0262(84) 1.0081300(54) 1.007490(66) 1.007300(17) 1.00369(18) 1.018780(25) 1.02092(12) 0.990460(96) 1.04163(56) 1.00502(10) 1.003060(58)
0.43–0.67 0.44–1.1 0.45–1.15 1.06–10.6 0.49–1.06 0.19–2.0 0.45–2.4 4–14 0.4–1.0 0.4–1.5 2–12 0.5–3.0 0.4–12 0.45–4.0
respectively. The proposed relations are confirmed by the experimental results. The validity of the proposed relationships, in turn, is a direct verification of Eqs. (1), (6)–(9).
Acknowledgment This work was supported by the National Natural Science Foundation of China (under Grant 60578033). References [1] M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford, 1984. [2] W. Pauli, General Principles of Quantum Mechanics, Springer, Berlin, 1980. [3] S.R. de Groot, L.G. Suttorp, Foundation of Electrodynamics, North-Holland, Amsterdam, 1972. [4] R.A. Beth, Phys. Rev. 50 (1936) 115. [5] C.Z. Tan, Appl. Phys. B 80 (2005) 875. [6] W. Pauli, Theory of Relativity, Pergamon, New York, 1958. [7] R.N.C. Pfeifer, T.A. Nieminen, N.R. Heckenberg, H. Rubinsztein-Dunlop, Rev. Mod. Phys. 79 (2007) 1197. [8] C.D. Salzberg, I.I. Villa, J. Opt. Soc. Am. 47 (1957) 244. [9] C.D. Salzberg, I.I. Villa, J. Opt. Soc. Am. 48 (1958) 579. [10] C. Kraus, et al., Euro. Phys. J. C 40 (2005) 1434. [11] C.Z. Tan, Phys. B 269 (1999) 373. [12] C.Z. Tan, J. Arndt, J. Phys. Chem. Solids 62 (2001) 1087. [13] C.Z. Tan, H. Li, L. Chen, Appl. Phys. B 86 (2007) 129. [14] I.H. Malitson, J. Opt. Soc. Am. 55 (1965) 1205. [15] W.G. Driscoll, W. Vaughhan (Eds.), Handbook of Optics, McGraw-Hill Book Company, New York, 1978. [16] M. Bass et al.(Ed.), Handbook of Optics, Vol. II, McGraw-Hill, Inc., New York, 1995. [17] P. Klocek, Handbook of Infrared Optical Materials, Marcel Dekker, Inc., New York, 1991. [18] C.Z. Tan, L. Chen, Opt. Lett. 32 (2007) 2936. [19] C.Z. Tan, L. Cao, T.B. Wang, Nucl. Instrum. Methods Phys. Res. B 239 (2005) 267. [20] M.J. Dodge, Appl. Opt. 23 (1984) 1980. [21] W.L. Bond, J. Appl. Phys. 36 (1965) 1674. [22] J. Fontanella, R.L. Johnston, G.H. Siegel, C. Andeen, J. Non-Cryst. Solids 31 (1979) 401. [23] N. Bloembergen, Nonlinear Optics, Benjamin, New York, 1965.
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Physica B journal homepage: www.elsevier.com/locate/physb
Particle nature of light waves in dielectric media C.Z. Tan Bowei Geophysics Ltd., Xiangtan National High-Tech. Center, 411104 Xiangtan, Province Hunan, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 31 January 2009 Accepted 5 July 2009
Wave-particle duality is a foundation for modern science. The speed of light waves in dielectric media is less than c. The corresponding particles thus have mass. Combining wave-particle duality with the theory of relativity, an exactly solvable problem was proposed, concerning the transition from photons in vacuum to particles in dielectric media. The rest mass, the momentum, and the total energy of material particles are shown to be the functions of the refractive index of the medium and the wavelength of the incident light. The proposed relationships were applied to study the wavelengthdependent index of refraction of dielectrics and the correlation of the refractive indices of anisotropic crystals, which were confirmed by the experimental results. Variation of the refractive index with wavelength is found to obey the proposed relation. The refractive indices of anisotropic crystals are shown to be the correlated quantities. & 2009 Elsevier B.V. All rights reserved.
PACS: 04.30.Nk 42.25.Lc 42.25.Bs 78.20.Ci 03.30.+p 91.60.Mk Keywords: Wave-particle duality Theory of relativity Dispersion Birefringence
1. Introduction Study of interaction of light waves with matter has a long history in science and technology. Light traveling in a transparent isotropic medium is normally treated as the propagation of the electromagnetic waves at a speed c/n, where c is the speed of light in vacuum, and n is the refractive index of the medium [1]. Light waves in vacuum correspond to photons of zero mass and the energy hc/l, as well as the de Broglie momentum h/l, where h is Planck’s constant, and l is the wavelength of light in vacuum [2]. However, the expressions for the energy and the momentum of light waves in the medium remain an unsolved problem [3]. Electromagnetic waves carry momenta (linear and angular parts) and thus exert forces and torque on dielectrics [4,5]. When the light wave is incident from vacuum into the medium, its electromagnetic momentum is changed. In dispersive media, the speed of light depends on the wavelength of the incident light and is less than c. The corresponding particles in the medium thus have mass. Combining wave-particle duality with Einstein’s theory of relativity, the relationships are proposed in the present work to describe the dependence of the rest mass, the momentum, and the energy of the particles of light waves in the medium on the index of refraction and the wavelength of light. In
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anisotropic crystals, the incident light is divided into the ordinary (o-ray) and the extraordinary (e-ray) waves [1]. Classically, indices of refraction are considered to be the independent quantities. The proposed relationships are applied to study the nature of dispersion of dielectrics and the correlation of the refractive indices of anisotropic crystals, which are confirmed by the experimental results.
2. Particle nature of light waves in dielectric media When the light wave of the wavelength l is incident from vacuum into a dielectric medium, its frequency remains unchanged. However, the wavelength is changed to l/n [1]. This observation is a starting point of the present work. According to wave-particle duality, a corresponding particle in the medium thus has the momentum of p¼
nh
l
:
ð1Þ
Light in the medium is considered to consist of a stream of particles. Suppose a particle strikes the interface between two dielectric media. The component of momentum of the particle perpendicular to the interface is transferred to a medium. Light waves thus exert forces on dielectrics [4,5]. The material in the vicinity of the interface affects the component of momentum perpendicular to the interface. However, the parallel component
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Fig. 2. The calculated rest masses at different wavelengths of the incident light for the representative crystals of silicon (Si) and germanium (Ge). The indices of refraction are taken from Refs. [8,9].
Fig. 1. Schematic depiction of the transition from a photon in vacuum to a particle in the medium.
combining Eqs. (1), (4), and (5), which read m¼
remains unchanged, as schematically shown in Fig. 1. According to the conservation of momentum, the component of momentum parallel to the interface for a single particle obeys the form of: pi sin(yi) ¼ pt sin(yt), where pi and pt are the incident and transmitted momenta, yi and yt are the incidence and the refraction angles, respectively. Combining this form with pi ¼ h/l, and pt ¼ nh/l we obtain Snell’s law [1]: sin(yi) ¼ n sin(yt), which, in turn, confirms p ¼ nh/l. Eq. (1) thus satisfies the conservation of momentum. On the other hand, in the theory of relativity, the total energy, W, and the momentum, p, of the corresponding particle traveling at the velocity, u, are given by [6] mc2 W ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi; u2 1 2 c
l
¼ W mc2 :
W¼
ð7Þ
Due to the contribution of the rest energy mc2, the total energy of the particle is larger than the photon energy hc/l. The propagation velocity u of the particle through the medium is u¼
2nc : n2 þ 1
ð8Þ
ð3Þ
p¼
ð4Þ
The incident photons excite material particles in the medium. A particle absorbs a photon of energy of hc/l, and propagates at the velocity u. After propagating through the medium, the particle transfers the energy hc/l to the transmitted photon, and keeps the rest energy mc2 in the medium. Consider the conservation of energy, namely the incident photon energy hc/l is equal to the relativistic kinetic energy, Wmc2, of the corresponding particle hc
ðn2 þ 1Þhc : 2l
and
The energy flux accompanying with traveling of the particle can be written as uW. Combining Eq. (7) with (8), the energy flux is given by nhc2/l. Finally, the relationship between p and W is given by the following equation:
where m is the rest mass of the particle. From Eqs. (2) and (3) we see that the energy and the momentum are related by W 2 ¼ p2 c2 þ m2 c4 :
ð6Þ
ð2Þ
and mu p ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi; u2 1 2 c
ðn2 1Þh ; 2cl
ð5Þ
The expressions of the rest mass and the total energy of the particle as the functions of n and l can be obtained after
2nW : ðn2 þ 1Þc
ð9Þ
Note this relation satisfies p/W ¼ u/c2, as obtained by combining Eqs. (2) and (3). Eq. (9) is different from both Abraham and the Minkowski expressions [7]. In normal dispersion region, the index of refraction increases with decreasing wavelength of the incident light. Therefore, the momentum, the rest mass, and the total energy increase with decreasing wavelength l, as described by Eqs. (1), (6), and (7), respectively. For quantitative demonstrations of the rest masses of the particles in dielectric media, Eq. (6) is applied to calculate the rest masses of the particles in the representative crystals of silicon (Si) and germanium (Ge). The evaluated results are shown in Fig. 2. The indices of refraction are taken from Refs. [8,9]. The rest mass of the particles corresponding to light waves in dielectric media is roughly comparable to neutrino mass (1036 kg) [10]. In the present work, we apply Eq. (7) to study the wavelengthdependent index of refraction of dielectric media and the correlation of the refractive indices of anisotropic crystals.
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Eq. (10) can be written as
3. Variation of the index of refraction with wavelength
w In both classical electrodynamics and quantum mechanics, the energy density w of the electromagnetic waves in an isotropic dielectric medium is [1,10,11] w ¼ e0 eE2
ð10Þ
where e0 is the electric permittivity of free space, e is the dielectric constant, and E is the electric field strength in the medium. Consider the light wave of the electric field strength E0 in vacuum is normally incident into the isotropic dielectric medium, as schematically shown in Fig. 3. In this case the transmittance, T, is expressed by T ¼ 4n/(n+1)2, and the value of E2 in the medium is given by E2 ¼ TE20, namely E2 ¼ 4nE20 =ðn þ 1Þ2 [1]. Consequently,
e
¼
4n ðn þ 1Þ2
e0 E20 :
ð11Þ
The term of e0 E20 on the right side of Eq. (11) represents the energy density w0 of the electromagnetic waves in vacuum. In quantum mechanics, the energy density, w0, can be written as N0hc/l, where N0 is the number of photons per unit volume. Accordingly, the energy density, w, of the electromagnetic waves in the medium can be written as NW, where N is the number density, and W is the total energy of the particle. Alternatively, Eq. (11) then becomes NW
e
¼
4n ðn þ 1Þ2
N0
hc
l
:
ð12Þ
Eq. (12) shows the particle character of light waves. Combining Eq. (7) with Eq. (12), we have ðn2 þ 1Þðn þ 1Þ2 N0 : ¼ 8ne N
ð13Þ
The ratio of N0/N is a material quantity, which is independent of the intensity and the wavelength of the incident light. Let k ¼ 8N0/ N. Eq. (13) is then changed to ðn2 þ 1Þðn þ 1Þ2 ¼ k: ne
ð14Þ
The relationship between the dielectric constant and the refractive index of the medium is generally described by Maxwell’s formula [1]: e ¼ n2. From Eq. (14) we have Fig. 3. Normal incidence of light waves into the medium. The electric field E in the medium is given by E2 ¼ 4nE20 =ðn þ 1Þ2 .
ðn2 þ 1Þðn þ 1Þ2 ¼ k: n3
ð15Þ
Fig. 4. Plots of (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light for the representative four cubic substances, silica glass (SiO2), and the crystalline LiF, NaCl, and KI. The symbols represent the experimental results, and the lines are the results of linear regression. The errors in the fitting results are given in the brackets. Experimental data are taken from Refs. [14–17].
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Table 1 The intercept (a) and the slope (k) evaluated by plotting (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light for cubic optical crystals: (n2+1) (n+1)2 ¼ a+kn3. Crystal
Intercept (a)
Slope (k)
Wavelength range (mm)
AgCl AlAs BaF2 C (diamond) CaF2 CdTe CsBr CsCl CsI CuCl GaAs GaP Ge InAs InP KBr KCl KF KI LiF MgAl2O4 MgO NaBr NaCl NaF NaI PbF2 PbS PbSe PbTe Si SiO2 (glass) SrF2 Y3Al5O12 Y2O3 b-ZnS ZnSe ZnTe
1.2276(41) 15.832(40) 2.86780(30) 4.401(21) 2.89794(21) 7.2917(50) 2.5953(14) 2.6811(91) 2.3267(60) 1.2312(45) 28.538(25) 17.355(32) 71.451(34) 26.802(78) 21.373(51) 2.7472(77) 2.8717(24) 2.86160(93) 2.260(42) 2.8899(21) 2.5342(17) 2.4475(17) 2.7647(60) 2.730(16) 2.85743(88) 2.6357(57) 2.4064(44) 77.593(57) 154.533(78) 341.26(42) 33.442(16) 2.8444(24) 2.90076(20) 2.0474(18) 2.075(19) 1.2771(99) 3.495(23) 8.018(67)
5.47150(54) 6.5176(16) 5.16800(10) 5.9723(15) 5.157470(80) 6.13526(27) 5.24191(31) 5.2242(24) 5.2989(12) 5.47185(63) 6.91136(68) 6.5690(12) 7.75674(52) 6.8701(21) 6.7057(17) 5.2074(24) 5.16795(88) 5.17371(45) 5.3250(88) 5.16051(80) 5.25604(37) 5.27436(35) 5.2022(17) 5.2122(45) 5.17701(45) 5.2372(13) 5.28345(91) 7.84785(82) 8.71045(71) 9.9636(23) 7.03677(40) 5.17577(78) 5.156500(76) 5.34907(31) 5.3537(32) 5.73452(88) 5.9065(17) 6.1771(36)
0.54–21.0 0.56–2.2 0.27–10.3 0.225–2.70 0.23–9.7 6.0–22.0 0.36–39 0.18–40 0.29–50 0.43–2.5 1.4–11 0.8–10 2–12 3.7–31.3 0.95–10 0.2–40 0.18–35 0.15–22.0 0.25–50 0.19–9.79 0.35–5.5 0.36–5.4 0.21–34 0.19–27.3 0.15–17 0.25–40 0.3–11.9 3.5–10 5.0–10 4.0–12.5 1.36–11 0.21–3.71 0.21–11.5 0.4–4.0 0.2–12 0.55–10.5 0.55–18 0.55–30
This simple relationship is resulted from wave-particle duality and the theory of relativity. The validity of Eq. (15), in turn, is a direct confirmation of Eqs. (1), (6)–(9). In dispersive media, the index of refraction varies with wavelength. According to the prediction of Eq. (15), a straight line should be demonstrated in the plot of (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light, with the slope of k. The indices of refraction have been accurately determined in wide wavelength regions for numerous optical materials of technical importance [12–17], which can be used to verify the proposed Eq. (15). Fig. 4 shows the plots of (n2+1)(n+1)2 versus n3 at different wavelengths of the incident light for the representative four isotropic materials, silica glass (SiO2), and the crystalline LiF, NaCl, and KI. Experimental results are taken from Refs. [14–17]. As predicted by Eq. (15), linear relations are demonstrated in the plots at different wavelengths. Although the index of refraction varies with the wavelength of the incident light, it is found to obey the general relation of Eq. (15), except for the intercepts in the plots. For most cubic optical crystals, the evaluated results of the intercept, and the slope, k, are listed in Table 1. The errors in the fitting results are given in the brackets. These results are the confirmation of Eq. (15), and in turn, the verification of Eqs. (1), (6)–(9).
Fig. 5. Transition from the o-ray to the e-ray waves of the plane-polarized incident light in a right-angle prism composed of an uniaxial crystal. A right-angle edge is parallel to the optic axis, and an angle y opposite to the right-angle edge parallel to the optic axis satisfies the law of refraction no sin(y) ¼ ne cos(y) or y ¼ arctan(ne/no).
4. Correlation of the refractive indices of anisotropic crystals In anisotropic crystals, the incident light is divided into the ordinary (o-ray) and the extraordinary (e-ray) waves [1]. Classically, indices of refraction are considered to be the independent quantities. Consider the propagation of light waves in a rightangle prism composed of an uniaxial crystal, with its one rightangle edge parallel to the optic axis, and an angle y opposite to the right-angle edge parallel to the optic axis satisfying the law of refraction no sin(y) ¼ ne cos(y) [18,19], as shown in Fig. 5. Note the law of refraction also implies the conservation of the component of momentum parallel to the interface: pi sin(y) ¼ pt cos(y), provided pi ¼ noh/l, and pt ¼ neh/l. Suppose the plane-polarized incident light propagates in the direction of the optic axis, with its vibration direction in the plane of incidence (p-wave). In this case, the incident light is the o-ray waves in the prism. After a total internal reflection, the light beam is turned to propagate in the direction perpendicular to the optic axis, which becomes the e-ray waves [18,19]. The intensity and the polarization state of the o-ray wave remain the same as those of the e-ray wave. In the transition of light waves shown in Figs. 5, both the o-ray and the e-ray waves are with the same electric field strength E. Thus we have a relation of NoWo/eo ¼ NeWe/ee or No ðn2o þ 1Þ=n2o ¼ Neðn2e þ 1Þ=n2e , where Wo and We are the total energies of the particles given by Eq. (7), and No and Ne denote the number densities of the particles of the o-ray and the e-ray waves, respectively. Consequently, we obtain a relationship between the indices of refraction of anisotropic crystals, which reads n2e ðn2o þ 1Þ Ne : ¼ n2o ðn2e þ 1Þ No
ð16Þ
The ratio of Ne/No is a material quantity, which is independent of the intensity and the wavelength of the incident light. Let s ¼ Ne/ No. Eq. (16) then becomes n2e ðn2o þ 1Þ ¼ s: n2o ðn2e þ 1Þ
ð17Þ
According to the prediction of Eq. (17), a straight line should be demonstrated in the plots of n2e ðn2o þ 1Þ versus n2o ðn2e þ 1Þ at different wavelengths of the incident light, with the slope of s. Fig. 6 shows the plots of n2e ðn2o þ 1Þ versus n2o ðn2e þ 1Þ at different wavelengths of the incident light for the representative
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Fig. 6. Plots of n2e ðn2o þ 1Þ versus n2o ðn2e þ 1Þ at different wavelengths of the incident light for the representative four uniaxial crystals, SiO2 (a-quartz), MgF2, CaCO3 (calcite), and TiO2 (rutile). The symbols represent the experimental data, and the lines are the results of linear regression. The representative crystals are of different birefringence ranging from 0.01 to 0.36. Refractive indices of magnesium fluoride (MgF2) are taken from Ref. [20], and the data of the other three crystals are from Refs. [15–17]. Wavelength ranges are given in the brackets.
four uniaxial crystals, SiO2 (a-quartz), MgF2, CaCO3 (calcite), and TiO2 (rutile). Refractive indices of magnesium fluoride (MgF2) are taken from Ref. [20], and the data of the other three crystals are from Refs. [15–17]. Note the representative crystals are of different birefringence ranging from 0.01 to 0.36. The linear relations are demonstrated in the plots of the crystals, as depicted by Eq. (17). The slope of the lines is equal to the constant s. The straight lines are with the intercepts in the plots. For most technical uniaxial crystals, the evaluated results of the intercept and the slope are listed in Table 2. The errors in the fitting results are given in the brackets. Experimental data of the indices of refraction are from Refs. [15–17,20,21]. These results indicate that the refractive indices of anisotropic crystals are the correlated quantities.
5. Discussion The intercepts shown in Figs. 4 and 6 and listed in Tables 1 and 2 are caused by the derivation from Maxwell’s formula: e ¼ n2. In classical dispersion theory, the wavelength-dependent index of refraction is usually given by the Sellmeier equation [1,12–14]
e 1 ¼ n2 1 ¼ l2 p
X fi l2 l2 i i
l2 l2i
;
ð18Þ
where lp is plasma wavelength, li is the resonance wavelength, and fi denotes the oscillator strength. Therefore, the static dielectric constant es can be obtained by using Eq. (18) and extrapolating to l ¼ N, namely [1,12–14] X 2 es ¼ n21 ¼ 1 þ l2 fi li : ð19Þ p i
Experimentally, there exits remarkable difference between the extrapolated es and the dielectric constant measured at low frequencies. As for example, the static dielectric constant evaluated by Sellmeier’s equation is 3.0 for silica glass [14], whereas the dielectric constant measured at 1000 Hz is 3.8 [22]. One can verify that the derivation from Maxwell’s formula leads to the intercepts. Further, nonlinear optical effects can also lead to the intercepts. In this case, nonlinear terms should be added in Eq. (10) to describe the energy density of the electromagnetic waves [23]. Wave-particle duality is a foundation for modern science. Electromagnetic waves propagate at the speed c in vacuum. Hence the photon mass is zero. However, the speed of light waves in the medium is less than c. The corresponding particles thus have mass, according to wave-particle duality and Einstein’s theory of relativity. The transition from a photon in vacuum to a particle in the medium is summarized and schematically shown in Fig. 1. Photons in vacuum have the momentum h/l, and the energy hc/l. The wavelength in the medium becomes l/n. Therefore the de Broglie momentum of the particle is nh/l. According to the law of the conservation of energy, the incident photon energy is equal to the relativistic kinetic energy of the particle. Combining the de Broglie momentum: p ¼ nh/l, the law of the conservation of energy: hc/l ¼ Wmc2, and the relation of W2 ¼ p2c2+m2c4, we obtain the expressions for the total energy and the rest mass of the particles in the medium as the functions of n and l. Combining the total energy W of the particle with the expression of the energy density of the electromagnetic waves in the medium, the general relationships of Eqs. (15) and (17) are proposed to describe the wavelength-dependent index of refraction of dielectric media and the correlation of the refractive indices of anisotropic crystals,
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Table 2 The intercept (b) and the slope (s) evaluated by plotting n2e ðn2o þ 1Þ versus n2o ðn2e þ
6. Conclusions
1Þ for uniaxial crystals: n2e ðn2o þ 1Þ ¼ b þ sn2o ðn2e þ 1Þ.
In conclusions, an exactly solvable problem is proposed in this work, concerning the transition from photons in vacuum to particles in dielectric media. The rest mass, the momentum, and the energy of material particles are shown to be the functions of the refractive index of the medium and the wavelength of the incident light. Variation of the refractive index with wavelength is found to obey the proposed relation. In anisotropic crystals, the indices of refraction are the correlated quantities.
Crystal
Intercept (b)
Slope (s)
Wavelength range (mm)
Ag2AsS3 AgGaS2 AlN Al2O3 (sapphire) AlPO4 b-BaB2O4 (BBO) BaTiO3 BeO CaCO3 (calcite) CaMoO4 CaWO4 CdGeAs2 CdS CdSe CuGaS2 HgS KH2PO4 (KDP) LiIO3 LiNbO3 LiYF4 MgF2 NH4H2PO4 (ADP) NaNO3 PbMoO4 PbTiO3 Se a-SiC SiO2 (a-quartz) SrMoO4 Te TeO2 TiO2 (rutile) Tl3AsSe3 (TAS) YVO4 ZnGeP2 ZnO
0.0170(45) 0.535(32) 0.2204(35) 0.01109(72)
0.97905(79) 1.00731(79) 1.01590(14) 0.99685(62)
0.63–4.6 0.54–21.0 0.22–5.0 0.2–5.5
0.0388(64) 0.307(14) 0.379(17) 0.020(17) 0.843(10) 0.0954(46) 0.073(15) 3.50(10) 0.1729(73) 0.1109(59) 0.2741(95) 0.018(12) 0.197(25) 0.175(13) 0.1600(26) 0.0533(54) 0.0502(20) 0.3428(99)
1.00856(84) 0.9279(14) 0.98343(42) 1.0059(15) 0.8438(11) 1.00663(25) 1.00784(92) 1.01668(55) 0.99777(19) 1.00477(15) 1.00530(22) 1.02254(16) 0.9562(34) 0.95065(87) 0.982490(87) 1.01792(79) 1.01483(38) 0.9358(13)
0.4–2.6 0.22–1.06 0.46–2.1 0.44–7.0 0.2–3.3 0.45–3.8 0.45–4.0 2.4–11.5 0.51–1.4 1–22 0.55–11.5 0.62–11 0.21–1.15 0.35–5.0 0.4–3.1 0.23–2.6 0.2–7.0 0.4–1.06
1.175(33) 1.080(58) 0.25634(93) 2.47(83) 0.22598(30) 0.03286(57) 0.10605(27) 12.47(17) 0.05949(88) 0.2787(80) 0.020(12) 0.044(12) 0.286(11) 0.0020(10)
0.7277(47) 0.9525(15) 0.994660(17) 1.0262(84) 1.0081300(54) 1.007490(66) 1.007300(17) 1.00369(18) 1.018780(25) 1.02092(12) 0.990460(96) 1.04163(56) 1.00502(10) 1.003060(58)
0.43–0.67 0.44–1.1 0.45–1.15 1.06–10.6 0.49–1.06 0.19–2.0 0.45–2.4 4–14 0.4–1.0 0.4–1.5 2–12 0.5–3.0 0.4–12 0.45–4.0
respectively. The proposed relations are confirmed by the experimental results. The validity of the proposed relationships, in turn, is a direct verification of Eqs. (1), (6)–(9).
Acknowledgment This work was supported by the National Natural Science Foundation of China (under Grant 60578033). References [1] M. Born, E. Wolf, Principles of Optics, Pergamon, Oxford, 1984. [2] W. Pauli, General Principles of Quantum Mechanics, Springer, Berlin, 1980. [3] S.R. de Groot, L.G. Suttorp, Foundation of Electrodynamics, North-Holland, Amsterdam, 1972. [4] R.A. Beth, Phys. Rev. 50 (1936) 115. [5] C.Z. Tan, Appl. Phys. B 80 (2005) 875. [6] W. Pauli, Theory of Relativity, Pergamon, New York, 1958. [7] R.N.C. Pfeifer, T.A. Nieminen, N.R. Heckenberg, H. Rubinsztein-Dunlop, Rev. Mod. Phys. 79 (2007) 1197. [8] C.D. Salzberg, I.I. Villa, J. Opt. Soc. Am. 47 (1957) 244. [9] C.D. Salzberg, I.I. Villa, J. Opt. Soc. Am. 48 (1958) 579. [10] C. Kraus, et al., Euro. Phys. J. C 40 (2005) 1434. [11] C.Z. Tan, Phys. B 269 (1999) 373. [12] C.Z. Tan, J. Arndt, J. Phys. Chem. Solids 62 (2001) 1087. [13] C.Z. Tan, H. Li, L. Chen, Appl. Phys. B 86 (2007) 129. [14] I.H. Malitson, J. Opt. Soc. Am. 55 (1965) 1205. [15] W.G. Driscoll, W. Vaughhan (Eds.), Handbook of Optics, McGraw-Hill Book Company, New York, 1978. [16] M. Bass et al.(Ed.), Handbook of Optics, Vol. II, McGraw-Hill, Inc., New York, 1995. [17] P. Klocek, Handbook of Infrared Optical Materials, Marcel Dekker, Inc., New York, 1991. [18] C.Z. Tan, L. Chen, Opt. Lett. 32 (2007) 2936. [19] C.Z. Tan, L. Cao, T.B. Wang, Nucl. Instrum. Methods Phys. Res. B 239 (2005) 267. [20] M.J. Dodge, Appl. Opt. 23 (1984) 1980. [21] W.L. Bond, J. Appl. Phys. 36 (1965) 1674. [22] J. Fontanella, R.L. Johnston, G.H. Siegel, C. Andeen, J. Non-Cryst. Solids 31 (1979) 401. [23] N. Bloembergen, Nonlinear Optics, Benjamin, New York, 1965.