Presentation on theme: "Outline Index of Refraction Introduction Classical Model"— Presentation transcript:
1 Outline Index of Refraction Introduction Classical Model Jing LiOutlineIntroductionClassical ModelTypical measurement methodsApplicationReference
2 Definition of Index of Refraction In uniform isotropic linear media, the wave equation is:They are satisfied by plane wavey=A e i(k r- wt)y can be any Cartesian components of E and HThe phase velocity of plane wave travels in the direction of k is
3 Definition of Index of Refraction We can define the index of refraction asMost media are nonmagnetic and have a magnetic permeability m=m0, in this caseIn most media, n is a function of frequency.
4 Classical Electron Model ( Lorentz Model) +-w0XELet the electric field of optical wave in an atom beE=E0e-iwtthe electron obeys the following equation of motionX is the position of the electron relative to the atomm is the mass of the electronw0 is the resonant frequency of the electron motiong is the damping coefficient
5 Classical Electron Model ( Lorentz Model) The solution isThe induced dipole moment isa is atomic polarizabilityThe dielectric constant of a medium depends on the manner in which the atoms are assembled. Let N be the number of atoms per unit volume. Then the polarization can be written approximately asP = N p = N a E = e0 c E
6 Classical Electron Model ( Lorentz Model) The dielectric constant of the medium is given bye = e0 (1+c) = e0 (1+Na/ e0)If the medium is nonmagnetic, the index of refraction isn= (e /e0)1/2 = (1+Na/ e0 )1/2If the second term is small enough then
7 Classical Electron Model ( Lorentz Model) The complex refractive index isat w ~w0 ,Normalized plot of n-1 and k versus w-w0
8 For more than one resonance frequencies for each atom, Classical Electron Model ( Drude model)If we set w0=0, the Lorentz model become Drude model. This model can be used in free electron metals
9 Relation Between Dielectric Constant and Refractive Index By definition,We can easily get:
10 An Example to Calculate Optical Constants Aleksandra B. Djuris¡ic and E. H. Li, JOURNAL OF APPLIED PHYSICS, 85(1999)2848The symbols in these two graphs are not clear to me. I assume they are the real part and the imaginary part of the dielectric constant on the left, but I do not know what is n and k– the real part only?Real and imaginary part of the index of refraction of GaN vs. energy;
11 Kramers-Kronig Relation The real part and imaginary part of the complex dielectric function e (w) are not independent. they can connected by Kramers-Kronig relations:P indicates that the integral is a principal value integral.K-K relation can also be written in other form, likeJ. Appl. Phys. 75 (1994)4176
12 A Method Based on Reflection Typical experimental setup( 1) halogen lamp;(2) mono-chromator; (3) chopper; (4) filter;(5) polarizer (get p-polarized light); (6) hole diaphragm;(7) sample on rotating support (q); (8) PbS detector(2q)Alibert et al. J. Appl. Phys., Vol. 69(1991)3208
13 Calculation In this case, n1=1, and n2=nr+i n i z E1' qE1E1'n1=1zxn2=nr+i n iSnell Law become:Reflection coefficient:Explain the fitting procedure: how many unknowns and typically how many angles are usedReflectance:R(q1, l, nr, n i)=|r p|2From this measurement, they got R, q for each wavelength l, Fitting the experimental curve, they can get nr and n i .Reflection of p-polarized light
14 Results Based on Reflection Measurement Single effective oscillator model(Eq. 1)(Eq. 2)Need to explain the figures– Eq.(1), (2) quoted in the figure, can you explain them? Refractive index is the real part only?FIG. 2. Measured refractive indices at 300 K vs. photon energy of AlSb and AlxGa1-xAsySb1-y layers lattice matched to GaSb (y~0.085 x).Dashed lines: calculated curves from Eq. ( 1);Dotted lines: calculated curves from Eq. (2)E0: oscillator energyEd: dispersion energyEG: lowest direct band gap energy
15 Use AFM to Determine the Refractive Index Profiles of Optical Fibers The basic configuration of optical fiber consists of a hair like, cylindrical, dielectric region (core) surrounded by a concentric layer of somewhat lower refractive index( cladding).n1n22aS. T. Huntington, J. Appl. Phys., 82(1997) say more about the fiber and the measurement method with AFM and the principle—how the AFM has anything to do with the dielectric constantModern communication cables are optical waveguides in the form of glass fibers. The basic configuration consists of a hair like, cylindrical, dielectric region surrounded by a concentric layer of somewhat lower refractive index( cladding). An additional layer of optically lossy material (jacket) may also be present.Fiber samples wereCleaved and mounted in holderEtched with 5% HF solutionMeasured with AFMThere is no way for AFM to measure refractive index directly.People found fiber material with different refractive index have different etch rate in special solution.
16 AFMThe optical lever operates by reflecting a laser beam off the cantilever. Angular deflection of the cantilever causes a twofold larger angular deflection of the laser beam.The reflected laser beam strikes a position-sensitive photodetector consisting of two side-by-side photodiodes.The difference between the two photodiode signals indicates the position of the laser spot on the detector and thus the angular deflection of the cantilever.Because the cantilever-to-detector distance generally measures thousands of times the length of the cantilever, the optical lever greatly magnifies motions of the tip.
18 A Method Based on Transmission For q=0, input wave function a e if ,tm=aTT’R’2m-1 e i(f-(2m-1)d ) (m=1,2…)d=2pdn/lThe transmission wavefunction is superposed by all tma T = a T T’ e if S m(R’2m-1 e-i(2m-1)d )=(1-R2)a e i(f-d) /(1-R2e-i2d)(TT’=1-R2 ; R’=-R)If R<<1, thena T =a e i(f-d)maximum condition is 2d=2pm= 4pdn/ln(lm)=m lm/2ddqq1t1t2t3r1r2r3n
20 ApplicationIn our lab., we have a simple system to measure the thickness of epitaxial GaN layer.
21 Thickness Measurement n(lm)=m lm/2dSteps to calculate thicknessGet peak position lmd=(lm lm-1)/2/[lm-1 n(lm) - lm n(lm-1)]Average dget m min= n(l max)*2d/ l maxCalculate d : d=m lm/2/n(lm) (from m min for each peak)Average d againLimitMinimum thickness:~500/nError<l/2n
22 ReferencePochi Yeh, "Optical Wave in Layered Media", 1988, John Wiley & Sons IncE. E. Kriezis, D. P. Chrissoulidis & A. G. Papafiannakis, Electromagnetics and Optics, 1992, World Scientific Publishing Co.,Aleksandra B. Djurisic and E. H. Li, J. OF Appl. Phys., 85 (1999) 2848 (mode for GaN)C. Alibert, M. Skouri, A. Joullie, M. Benounab andS. Sadiq , J. Appl. Phys., 69(1991)3208 (Reflection)Kun Liu, J. H. Chu, and D. Y. Tang, J. Appl. Phys. 75 (1994)4176 (KK relation)G. Yu, G. Wang, H. Ishikawa, M. Umeno, T. Soga, T. Egawa, J. Watanabe, and T. Jimbo, Appl. Phys. Lett. 70 (1997) 3209Jagat, (AFM)Authors needed to be listed in the references– before you conclude, can you say anything about the pro and con of the methods mentioned?J. Appl. Phys., Vol. 73 (1993)1732S. T. Huntington, J. Appl. Phys., 82(1997)2730