SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES
|
|
- Solomon West
- 5 years ago
- Views:
Transcription
1 SECOND ORDER NONLINEAR PROCESSES AT SURFACES AND INTERFACES C.Stancu, R.Ehlch /A1 Boundary condtons of a polarzed sheet Radaton from a polarzed sheet Surface nonlnear response Bulk nonlnear response The equvalence of surface and bulk contrbutons The thn flm geometry - example of an sotropc flm Rotatonal ansotropy; examples
2 Bblography: 1. T.F.Henz, Second-order nonlnear optcal effects at surfaces and nterfaces, n Nonlnear Surface Electromagnetc Phenomena, Eds. H.- E.Ponath and G.I.Stegeman, B.Koopmans, Interface and bulk contrbutons n optcal secondharmonc generaton, Ph.D.thess, N.Bloembergen and P.S.Pershan, Lght waves at the boundary of nonlnear meda, Phys.Rev. 18, 606 (196) 4. P.Guyot-Sonnest, W.Chen and Y.R.Shen, General consderatons on optcal second-harmonc generaton from surfaces and nterfaces, Phys.Rev.B 33, 854 (1986) 5. P.Guyot-Sonnest and Y.R.Shen, Bulk contrbuton n surface secondharmonc generaton, Phys.Rev.B 38, 7985 (1988) 6. J.E.Spe, D.J.Moss and H.M. van Drel, Phenomenologcal theory of optcal second- and thrd- harmonc generaton from cubc centrosymmetrc crystals, Phys.Rev.B 35, 119 (1986) 7. T.F.Henz, M.M.T.Loy and W.A.Thompson, Phys.Rev.Lett. 54, 63 (1985) 8. D.Wlk, D.Johannsmann, C.Stanners and Y.R.Shen, Phys.Rev.B 51, (1994)
3 Applcatons of nonlnear optcal phenomena: - development of optcal-devce technology and laser systems - the use of these phenomena as a tool for materal characterzaton - partcular case: surface specfcty of the second order nonlnear processes for centrosymmetrc materals Wthn the electrc dpole approxmaton the polarzaton can be wrtten as an expanson of the electrc feld: P = τ χ () 1 E + τ χ () :EE + τ χ () 3 :EEE +... The nverson symmetry s reflected n the susceptblty tensor τ χ ( n) whch s nvarant under the transformaton r -r; ths gves τ( n) τ( ) χ ( ) n n = χ any even-order response s forbdden wthn the electrc dpole approxmaton for a centro-symmetrcal materal At the nterface between two centro-symmetrc meda, the nverson symmetry s broken by defnton, gvng rse to an ED allowed second order susceptblty second order processes are useful nondestructve technques for the study of surfaces and bured nterfaces, wth a resoluton better than the nherent penetraton depth of the probe
4 Boundary condtons of a polarzed sheet ε ε 1 ε Ω k 1 (Ω) k (Ω) Ω Medum 1 Medum k 1 (ω 1 ) k 1 (ω ) ω 1 ω τ ( ) χ q τ ( ) χ s τ ( ) χ q The nonlnearty of the nterface s treated as a sheet of generalzed nonlnear source polarzaton. From Maxwell s equatons wth a nonlnear source polarzaton P nls occupyng a fnte volume: D = -4π P nls c E + B t = 0 B = 0 c H - D = 4 π nls P t t the boundary condtons for the electrc and magnetc felds on ether nls sde of ths polarzed sheet P s (x,y) (wth a volume polarzaton nls P s (x,y)δ (z)) are: nls Dz = -4π Ps Et = - 4π nls t P s, z ε ' B z = 0 Ht = 4π nls Ps z ) c t
5 Radaton from a polarzed sheet nls P s (x,y,t) = P s e px- Ω t + c.c. The reflected beam s gven by the wave vector k 1 = p x ) ) - q 1 z, wth q 1 = [ε1k - p ] 1/ ; smlarly, the transmtted beam s gven by k = p x ) ) + q 1 z. From the boundary condtons, the radated felds are (wth ε1 = ε = ε = 1): E (x,y,z,t) = E (Ω )e k r- Ω t + c.c. where: E = πk [P q s - k ) ( k ) Ps )] The Fresnel correctons are made for the waves propagatng towards the nterface by replacng e ) ) wth e = F e j, where the Fresnel transformaton has the dagonal elements (for sotropc meda): q xx F = ε j j q + ; ε εq F yy q = j ; q + q F zz = εε q 1 j ε' ε q + εq j j j j j The radated feld can be wrtten lke ) e πk E = sec θ ( e P) 1 / s ε / ε ( ) where sec θ ( Ω) Ω 1 = Ω defnes, for a lossless medum, the angle cq ( Ω) between the wave at Ω and the surface normal.
6 Surface nonlnear response When consderng the nonlnear polarzed sheet as arsng from the surface nonlnear susceptblty tensor τ ( ) χ s : P s (Ω) = τ ( ) χ s (Ω=ω 1 +ω ):E(ω 1 )E(ω ) or (Ω=ω 1 +ω ):e 1 (ω 1 )e (ω )E 1 (ω 1 )E (ω ) P s (Ω) = τ ( ) χ s The radated felds can be now wrtten as ) πωsec θ ( Ω) ( ) e( Ω) E ( Ω) = [( e Ω) χ :( e ω )( e ω )] E ( ω ) E ( ω ) 1 / s cε ( Ω) When absorpton s present n the bulk meda, the electrc feld ampltudes correspond to the ncomng or outgong felds near the nterface. For an arbtrary geometry of the pump beams, the drecton of the radated beam s gven by the n-plane wave vector components p(ω) = p(ω1) + p(ω) - the nonlnear Snell s law In terms of rradances (I = cε 1/ E /π for a plane wave n a medum ε), the felds are gven by I ( Ω) 8 3 π Ω sec θ ( Ω) ( e ) ( ) = Ω χ :( e ω )( e ω ) I ( ω ) I ( ω ) 3 s c [ ε ( Ω) ε ( ω ) ε ( ω ) 1 1
7 Bulk nonlnear response The nonlnear source polarzaton n the bulk of a materal, as a multpole expresson n succesve degrees of nonlocalty, s: P nls (Ω ) = χ () (Ω = ω 1 +ω ):E(ω 1 )E(ω ) + 1 τ ( χ q )(Ω = ω 1 +ω ):E(ω 1 ) E(ω ) + 1 τ ( χ q )(Ω = ω 1 +ω ):E(ω ) E(ω 1 ) +... The relatve contrbutons of the succesve terms vary lke (ka), wth a the typcal atomc dmenson. For centrosymmetrc meda, the leadngorder response conssts of the electrc-quadrupole and magnetc-dpole terms comprsed of products of E and the spatal dervatves of E. Up to the frst order spatal dervatves of the electrc feld, we can express the generalzed polarzaton lke: P () (Ω ) = () Pd (Ω ) - Q τ c ( ) (Ω ) + Ω M () (Ω ) where, n the case of SHG, we can wrte: P () d = τ τ χ D :E(ω )E( ω ) + χ P :E(ω ) E( ω τ ) Q ( ) = τ χ Q :E(ω )E( ω ) M () = τ χ M :E(ω )E( ω ) τ wth χ D descrbng a local response, whle τ χ P, nonlocal. τ χ Q and τ χ M are For nonmagnetc materals (nverson symmetry n both space and tme), M () (Ω ) = 0.
8 The effectve SH polarzaton can be wrtten as: P (Ω ) = γ (E E) + ( δ - β - γ )(E )E + β ( E)E + ζ E E β, γ and δ descrbe the sotropc response of the medum. * for a homogeneous medum, E = 0 * (δ-β-γ)(e )E = 0 when only a sngle plane wave s present n the medum (reflectons from deeper lyng nterfaces are gnored) * n an sotropc materal, ζ = 0 * the only bulk term contrbutng to the sgnal n a smple measurement of an sotropc medum remans γ (E E) whch s a longtudnal feld for any pump feld E, gvng rse to radaton felds ndstngushable from those of an equvalent surface nonlnear response
9 The equvalence of surface and bulk contrbutons For a bulk nonlnear source polarzaton gven by a sngle wave vector k b : P(x,y,z,t) = P b (Ω,kb )e k b x-ω t ϑ (z) + c.c. one can defne an equvalent nonlnear polarzaton sheet at the nterface between meda 1 and : P s eq (x,y,z,t) = P s eq (Ω,kb )e k b x-ω t δ (z) + c.c. Takng the bulk polarzaton as a successon of polarzed sheets and consderng the propagaton of the nonlnear wave n medum, the equvalent polarzaton sheet s gven by P s eq (Ω,kb ) = q q P k x ) P k y ) ε'( Ω) P k z ) [ ( Ω, ) + ( Ω, ) + ( Ω, ) ] bx, b by, b bz, b + ε ( Ω) b where q b s the z-component of k b and q s the z-component of the Ω- wave n medum. The bulk nonlnear polarzaton P b (Ω ) = γ (E E) excted by a sngle plane wave can be wrtten as P b (Ω ) = γ ks [E(ω ) E( ω )]. Defnng an effectve surface polarzaton P s eq (Ω ) correspondng to ths term and lookng at the p-polarzed radaton whch t produces, one fnds: e p P eq s (Ω) = ) C[q x+ ε ε ' pz) ] {[- γ (E E)(qs +q ) -1 ][p x+ ) ε ' ε = -Cγ p(e E) = e p [- γ ε ' (E E) z ) ] ε γ = e p P s ( Ω) where we have defned a surface polarzaton γ P s ( Ω) = - γ ε ' (E(ω) E(ω)) z ) ε qz ) ]}
10 By specfyng the radated feld n terms of a surface polarzaton γ ) P s z one can reproduce the angular and polarzaton dependence of the bulk longtudnal polarzaton, wthout ntroducng the vector k s. It s therefore mpossble to separate the bulk longtudnal polarzaton from the z-component of the surface polarzaton. Ths equvalence has also been demonstrated for an arbtrary pump feld, n sotropc as well as n cubc materals. The SH response from an nterface between two bulk centrosymmetrc meda can be descrbed by an effectve surface nonlnear tensor ( ) ( ) ( ) χ seff, = χ s + χ s, γ wth the radated feld expressed lke ) e πωsec θ ( Ω) ( ) ( Ω) E ( Ω) = [( e Ω) χ :( e ω )( e ω )] E ( ω ) E ( ω ) 1 / seff, cε ( Ω)
11
12 Thn flm geometry * offers addtonal possbltes for the problem of the SHG vs. bulk separaton. * mportant for the study of bured nterfaces n case of overlayers havng thcknesses of the order of the wavelength of lght The total SH feld generated by a nonlnear flm s gven by: E(Ω ) = E B (Ω ) + E S (Ω ) + E I (Ω ) wth a total SH output: 3πω 3 ( ) S(Ω ) = θ χ I ω AT 3 1/ eff ηc ε ( Ω) ε ( ω) sec ( ) Ω q where τ ( ) τ τ τ ( ) χ = e ( Ω) L( Ω, z = 0): χ : L( ω, z = 0) ( e ω) L( ω, z = 0) ( e ω) eff S τ τ τ τ ( ) + e ( Ω) L( Ω, z = d): χ : L( ω, z = d) ( e ω) L( ω, z = d) ( e ω) I d τ τ τ τ ( ) + e ( Ω) L( Ω, z'): χ : L( ω, z') ( e ω) L( ω, z') ( e ω) dz' B 0 q θ Ω s the ext angle, A the beam cross secton, τ T the pulse wdth, e ( Ω) the unt vector of the feld at Ω and L( ω, z) s a dagonal tensor descrbng the feld nsde the flm, wth the elements: p k fzz p k fz d z f L (, z) t ( e r e ) cos θ ( ) ω = xx 1 I cos s k fzz s k fz ( d z ) L ( ω, z) = t ( e + r e ) yy 1 p k fzz p k fz ( d z ) ε1 L ( ω, z) = t ( e + r e ) zz 1 I ε t h ts = h 1 rre h 1 S I h k d fz I 1 / 1 / f θ ω
13 τ L, the bulk term n χ eff Because of the phase factors n wll exhbt a composte nterference pattern as the thckness d vares; the nterface, surface and bulk term can have relatve phase dfferences and ther nterference also changes wth d. By fttng S(ω ) versus d one can get to the dfferent surface and bulk susceptbltes. The effectve bulk SH polarzaton s wrtten as: P (Ω ) = γ (E E) + ( δ - β - γ )(E )E + β ( E)E + ζ E E For an sotropc medum, one can express the parameters β, γ, δ and ζ as a functon of the non-vanshng bulk susceptbltes: β = ( τ P χ jj γ = τ P χ jj - τ Q χ jj - τ Q χ jj ) δ = ( τ P χ - τ Q χ ) ζ = 0 δ - β - γ = ( τ P χ jj where τ Q χ jj = τ Q χ jj - τ Q χ jj and ) τ PQ, χ = τ PQ, χ jj + τ PQ, χ jj + τ PQ, χ jj Snce γ s ndstngushable from surface contrbuton, t wll be accounted for by takng effectve surface susceptbltes for the z drecton. There wll be 7 ndependent susceptblty elements, probed for varous polarzatons combnatons: s/p m/s p/p m/p χ S,zyy - γ χ I,zyy + γ χ S,zzz - γ χ I,zzz + γ χ S,yzy χ S,yzy δ - β - γ ( )
14 D.Wlk, D.Johannsmann, C.Stanners and Y.R.Shen, Phys.Rev.B 51, (1994)
15 Rotatonal ansotropy P (Ω ) = γ (E E) + ( δ - β - γ )(E )E + ζ E E ζ 0 The cartesan coordnate axes concde, n ths expresson, wth the prncpal crystallographc axes. When estmatng the values of the generated felds n dfferent polarzaton combnatons, one needs to transform χq() nto the laboratory axes. As a general rule, rotatonal measurements of the L-th order multpole contrbuton to the N-th order NLO process of a materal wth q-fold rotatonal symmetry can only show ansotropc response f N + L q. For cubc centrosymmetrc materals (m3m- or 43-symmetry), the varaton of the SH felds wth an arbtrary azmuthal angle can be wrtten as E p (Ω) = a + b (m) cos(mϕ) E s (Ω) = c (m) sn(mϕ) where m = 3 for (111) crystal faces and m = 4 for (001) crystal faces. For a (110) face, E p (Ω) = a + b () cos(ϕ) + b (4) cos(4ϕ) E s (Ω) = c () sn(ϕ) + c (4) sn(4ϕ)
16 Photopolymerzaton of a crystallne C 60 flm unpolymerzed polymerzed p-p sgnal I SHG (arb.u.) 0,35 0,30 0,5 0,0 0,15 0,10 0,05 0,00 0,35 0,30 0,5 0,0 0,15 0,10 0,05 0, Azmuthal angle (deg) ps sgnal unpolymerzed polymerzed
17 Reconstructon of S(111) surfaces The polarzaton dependence for a SH sgnal assocated only wth the surface nonlnear susceptblty, for normal ncdence: * S(111)-7x7: ( ) I x (Ω) = A χ s, xxx cos ϕ I y (Ω) = A χ ( ) s, xxx sn ϕ * S(111)-x1 I x (Ω) = A χ ( ) s, xxx I y (Ω) = A χ ( ) s, xxx sn ϕ cos ϕ + χ ( ) s, xyysn ϕ T.F.Henz, M.M.T.Loy and W.A.Thompson, Phys.Rev.Lett. 54, 63 (1985)
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationBoundaries, Near-field Optics
Boundares, Near-feld Optcs Fve boundary condtons at an nterface Fresnel Equatons : Transmsson and Reflecton Coeffcents Transmttance and Reflectance Brewster s condton a consequence of Impedance matchng
More informationLecture 6. P ω ω ε χ ω ω ω ω E ω E ω (2) χ ω ω ω χ ω ω ω χ ω ω ω (2) (2) (2) (,, ) (,, ) (,, ) (2) (2) (2)
Lecture 6 Symmetry Propertes of the Nonlnear Susceptblty Consder mutual nteracton of three waves: ω, ω, ω = ω + ω 3 ω = ω ω ; ω = ω ω 3 3 P ω ω ε ω ω ω ω E ω E ω n + m = 0 jk m + n, n, m j n k m jk nm
More information( ) + + REFLECTION FROM A METALLIC SURFACE
REFLECTION FROM A METALLIC SURFACE For a metallc medum the delectrc functon and the ndex of refracton are complex valued functons. Ths s also the case for semconductors and nsulators n certan frequency
More informationCHAPTER II THEORETICAL BACKGROUND
3 CHAPTER II THEORETICAL BACKGROUND.1. Lght Propagaton nsde the Photonc Crystal The frst person that studes the one dmenson photonc crystal s Lord Raylegh n 1887. He showed that the lght propagaton depend
More informationSUPPLEMENTARY INFORMATION
do: 0.08/nature09 I. Resonant absorpton of XUV pulses n Kr + usng the reduced densty matrx approach The quantum beats nvestgated n ths paper are the result of nterference between two exctaton paths of
More informationLecture 3. Interaction of radiation with surfaces. Upcoming classes
Radaton transfer n envronmental scences Lecture 3. Interacton of radaton wth surfaces Upcomng classes When a ray of lght nteracts wth a surface several nteractons are possble: 1. It s absorbed. 2. It s
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationFrequency dependence of the permittivity
Frequency dependence of the permttvty February 7, 016 In materals, the delectrc constant and permeablty are actually frequency dependent. Ths does not affect our results for sngle frequency modes, but
More informationWaveguides and resonant cavities
Wavegudes and resonant cavtes February 8, 014 Essentally, a wavegude s a conductng tube of unform cross-secton and a cavty s a wavegude wth end caps. The dmensons of the gude or cavty are chosen to transmt,
More informationProblem 1: To prove that under the assumptions at hand, the group velocity of an EM wave is less than c, I am going to show that
PHY 387 K. Solutons for problem set #7. Problem 1: To prove that under the assumptons at hand, the group velocty of an EM wave s less than c, I am gong to show that (a) v group < v phase, and (b) v group
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationECE 107: Electromagnetism
ECE 107: Electromagnetsm Set 8: Plane waves Instructor: Prof. Vtaly Lomakn Department of Electrcal and Computer Engneerng Unversty of Calforna, San Dego, CA 92093 1 Wave equaton Source-free lossless Maxwell
More informationHomework 4. 1 Electromagnetic surface waves (55 pts.) Nano Optics, Fall Semester 2015 Photonics Laboratory, ETH Zürich
Homework 4 Contact: frmmerm@ethz.ch Due date: December 04, 015 Nano Optcs, Fall Semester 015 Photoncs Laboratory, ETH Zürch www.photoncs.ethz.ch The goal of ths problem set s to understand how surface
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More information16 Reflection and transmission, TE mode
16 Reflecton transmsson TE mode Last lecture we learned how to represent plane-tem waves propagatng n a drecton ˆ n terms of feld phasors such that η = Ẽ = E o e j r H = ˆ Ẽ η µ ɛ = ˆ = ω µɛ E o =0. Such
More informationSupporting Information
Supportng Informaton Water structure at the ar-aqueous nterface of dvalent caton and ntrate solutons Man Xu, Rck Spnney, Heather C. Allen* allen@chemstry.oho-state.edu Fresnel factors and spectra normalzaton
More information) is the unite step-function, which signifies that the second term of the right-hand side of the
Casmr nteracton of excted meda n electromagnetc felds Yury Sherkunov Introducton The long-range electrc dpole nteracton between an excted atom and a ground-state atom s consdered n ref. [1,] wth the help
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationRate of Absorption and Stimulated Emission
MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld
More informationQuantum Mechanics I Problem set No.1
Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t
More informationNonlinear Optics. Office: Tien s Photonic Research Hall 412. Tel : E
Nonlnear Optcs Offce: Ten s hotonc Research Hall 1 Tel : 5731975 mal:jyhuang@facultynctuedutw Ths course s amed to help students masterng the prncples and techncal materals of a graduate-level Nonlnear
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationWaveguides and resonant cavities
Wavegudes and resonant cavtes February 26, 2016 Essentally, a wavegude s a conductng tube of unform cross-secton and a cavty s a wavegude wth end caps. The dmensons of the gude or cavty are chosen to transmt,
More informationIntroduction to Antennas & Arrays
Introducton to Antennas & Arrays Antenna transton regon (structure) between guded eaves (.e. coaxal cable) and free space waves. On transmsson, antenna accepts energy from TL and radates t nto space. J.D.
More informationSupplementary Information for Observation of Parity-Time Symmetry in. Optically Induced Atomic Lattices
Supplementary Informaton for Observaton of Party-Tme Symmetry n Optcally Induced Atomc attces Zhaoyang Zhang 1,, Yq Zhang, Jteng Sheng 3, u Yang 1, 4, Mohammad-Al Mr 5, Demetros N. Chrstodouldes 5, Bng
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationGeneralized form of reflection coefficients in terms of impedance matrices of qp-qp and qp-qs waves in TI media
Generalzed form of reflecton coeffcents n terms of mpedance matrces of q-q and q-q waves n TI meda Feng Zhang and Xangyang L CNC Geophyscal KeyLab, Chna Unversty of etroleum, Bejng, Chna ummary Reflecton
More information2010 Black Engineering Building, Department of Mechanical Engineering. Iowa State University, Ames, IA, 50011
Interface Energy Couplng between -tungsten Nanoflm and Few-layered Graphene Meng Han a, Pengyu Yuan a, Jng Lu a, Shuyao S b, Xaolong Zhao b, Yanan Yue c, Xnwe Wang a,*, Xangheng Xao b,* a 2010 Black Engneerng
More information4.2 Chemical Driving Force
4.2. CHEMICL DRIVING FORCE 103 4.2 Chemcal Drvng Force second effect of a chemcal concentraton gradent on dffuson s to change the nature of the drvng force. Ths s because dffuson changes the bondng n a
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationTHEOREMS OF QUANTUM MECHANICS
THEOREMS OF QUANTUM MECHANICS In order to develop methods to treat many-electron systems (atoms & molecules), many of the theorems of quantum mechancs are useful. Useful Notaton The matrx element A mn
More information1 Rabi oscillations. Physical Chemistry V Solution II 8 March 2013
Physcal Chemstry V Soluton II 8 March 013 1 Rab oscllatons a The key to ths part of the exercse s correctly substtutng c = b e ωt. You wll need the followng equatons: b = c e ωt 1 db dc = dt dt ωc e ωt.
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More informationIntroductory Optomechanical Engineering. 2) First order optics
Introductory Optomechancal Engneerng 2) Frst order optcs Moton of optcal elements affects the optcal performance? 1. by movng the mage 2. hgher order thngs (aberratons) The frst order effects are most
More informationEffect of Losses in a Layered Structure Containing DPS and DNG Media
PIERS ONLINE, VOL. 4, NO. 5, 8 546 Effect of Losses n a Layered Structure Contanng DPS and DNG Meda J. R. Canto, S. A. Matos, C. R. Pava, and A. M. Barbosa Insttuto de Telecomuncações and Department of
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mt.edu 6.13/ESD.13J Electromagnetcs and Applcatons, Fall 5 Please use the followng ctaton format: Markus Zahn, Erch Ippen, and Davd Staeln, 6.13/ESD.13J Electromagnetcs and
More informationNote on the Electron EDM
Note on the Electron EDM W R Johnson October 25, 2002 Abstract Ths s a note on the setup of an electron EDM calculaton and Schff s Theorem 1 Basc Relatons The well-known relatvstc nteracton of the electron
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationChapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
More informationExchange bias and coercivity of ferromagnetic/antiferromagnetic multilayers. Klaus D. Usadel
Exchange bas and coercvty of ferromagnetc/antferromagnetc multlayers Klaus D. Usadel Unverstät Dusburg-Essen, Germany Contents: exchange bas: ntroducton doman state model: results from MC smulatons mean
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule Dee Dagram Expermentally determned energy
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationTitle: Radiative transitions and spectral broadening
Lecture 6 Ttle: Radatve transtons and spectral broadenng Objectves The spectral lnes emtted by atomc vapors at moderate temperature and pressure show the wavelength spread around the central frequency.
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationFresnel's Equations for Reflection and Refraction
Fresnel's Equatons for Reflecton and Refracton Incdent, transmtted, and reflected beams at nterfaces Reflecton and transmsson coeffcents The Fresnel Equatons Brewster's Angle Total nternal reflecton Power
More informationPHY2049 Exam 2 solutions Fall 2016 Solution:
PHY2049 Exam 2 solutons Fall 2016 General strategy: Fnd two resstors, one par at a tme, that are connected ether n SERIES or n PARALLEL; replace these two resstors wth one of an equvalent resstance. Now
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationA PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.
Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR
More informationNon-interacting Spin-1/2 Particles in Non-commuting External Magnetic Fields
EJTP 6, No. 0 009) 43 56 Electronc Journal of Theoretcal Physcs Non-nteractng Spn-1/ Partcles n Non-commutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, Ile-Ife, Ngera
More informationAdvanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)
Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed
More informationσ τ τ τ σ τ τ τ σ Review Chapter Four States of Stress Part Three Review Review
Chapter Four States of Stress Part Three When makng your choce n lfe, do not neglect to lve. Samuel Johnson Revew When we use matrx notaton to show the stresses on an element The rows represent the axs
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationLevel Crossing Spectroscopy
Level Crossng Spectroscopy October 8, 2008 Contents 1 Theory 1 2 Test set-up 4 3 Laboratory Exercses 4 3.1 Hanle-effect for fne structure.................... 4 3.2 Hanle-effect for hyperfne structure.................
More informationDepartment of Chemistry Purdue University Garth J. Simpson
Objectves: 1. Develop a smple conceptual 1D model for NLO effects. Extend to 3D and relate to computatonal chemcal calculatons of adabatc NLO polarzabltes. 2. Introduce Sum-Over-States (SOS) approaches
More informationApplied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus
.101 Appled Nuclear Physcs (Fall 004) Lecture 3 (1/3/04) Nuclear Reactons: Energetcs and Compound Nucleus References: W. E. Meyerhof, Elements of Nuclear Physcs (McGraw-Hll, New York, 1967), Chap 5. Among
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationComputational Electromagnetics in Antenna Analysis and Design
Computatonal Electromagnetcs n Antenna Analyss and Desgn Introducton It s rare for real-lfe EM problems to fall neatly nto a class that can be solved by the analytcal methods presented n the precedng lectures.
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More information2924 J. Acoust. Soc. Am. 110 (6), December /2001/110(6)/2924/22/$ Acoustical Society of America
Extncton theorem for object scatterng n a stratfed medum Purnma Ratlal and Ncholas C. Makrs Massachusetts Insttute of Technology, Cambrdge, Massachusetts 0239 Receved 20 December 2000; revsed 3 June 200;
More informationLecture 4. Macrostates and Microstates (Ch. 2 )
Lecture 4. Macrostates and Mcrostates (Ch. ) The past three lectures: we have learned about thermal energy, how t s stored at the mcroscopc level, and how t can be transferred from one system to another.
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationSELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:
SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be
More informationNONLINEAR OPTICS OF FERROELECTRICS MATERIALS. School of Physics, Universiti Sains Malaysia, USM, Penang, Malaysia. Prefecture , Japan.
Sold State Scence and Technology, Vol., No & (5) 5-6 ISSN 8-789 NONLINEAR OPTICS OF FERROELECTRICS MATERIALS Junadah Osman, T.Y. Tan, D.R. Tlley, Y. Ishbash, and R. Murgan School of Physcs, Unverst Sans
More information7. Products and matrix elements
7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More informationCHAPTER 5: Lie Differentiation and Angular Momentum
CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly,
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More informationAmplification and Relaxation of Electron Spin Polarization in Semiconductor Devices
Amplfcaton and Relaxaton of Electron Spn Polarzaton n Semconductor Devces Yury V. Pershn and Vladmr Prvman Center for Quantum Devce Technology, Clarkson Unversty, Potsdam, New York 13699-570, USA Spn Relaxaton
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationSolutions to Problem Set 6
Solutons to Problem Set 6 Problem 6. (Resdue theory) a) Problem 4.7.7 Boas. n ths problem we wll solve ths ntegral: x sn x x + 4x + 5 dx: To solve ths usng the resdue theorem, we study ths complex ntegral:
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum
More informationImplementation of the Matrix Method
Computatonal Photoncs, Prof. Thomas Pertsch, Abbe School of Photoncs, FSU Jena Computatonal Photoncs Semnar 0 Implementaton of the Matr Method calculaton of the transfer matr calculaton of reflecton and
More informationThermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)
hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationPhysics 443, Solutions to PS 7
Physcs 443, Solutons to PS 7. Grffths 4.50 The snglet confguraton state s χ ) χ + χ χ χ + ) where that second equaton defnes the abbrevated notaton χ + and χ. S a ) S ) b χ â S )ˆb S ) χ In sphercal coordnates
More information