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Chapter 1 Electromagnetic Fields Lecture 1 Maxwell’s equations 1.1 Maxwell’s equations and boundary conditions Maxwell’s equations describe how the electric field and the magnetic field are generated, and how they change in space and time. Lights are electromagnetic waves which obey Maxwell’s equations. Maxwell’s equations have three major equivalent forms. 1) General form of Maxwell’s equations: 1.It is also called Maxwell’s equations in vacuum. 2.It actually applies to all cases, either in vacuum or in a medium. It is thus called the general form of Maxwell’s equations. E t 0 B E t B 0 E B 0 J t 0 t Here total free bound , J total J free J bound . 1 Definitions of the auxiliary fields: D 0E P H B / 0 M , B 0 (H M) Our text was wrong in the definition of H. Bound charge and bound current density: b P P Jb M t Refer to an electrodynamics textbook. E.g., Griffiths. E – Electric field H – Magnetic field D – Electric displacement B – Magnetic induction We then come to 2) Maxwell’s equations in matter: 1.It contains no more physics than the general form, except that the charge density and the current density are decomposed into free and bound. 2.It will be extensively used in our course. D f B E t B 0 D H Jf t – Electric charge density J – Electric current density – Permittivity – Permeability 0 – Permittivity of vacuum 0 – Permeability of vacuum P – Electric polarization We then drop the “f” subscripts. M – Magnetization 2 Material equations (Constitutive relations): D E H 1B, (B H) and are of essential importance in our course. They characterize the materials and are something you can explore all through your life. 1. They are position dependent in an inhomogeneous medium. 2. They are tensors in an anisotropic medium. 3. They depend on the field strength (E and H) in a nonlinear medium. We then come to 3) Maxwell’s equations in a homogeneous medium: 1.It is valid only in a homogeneous medium. 2.It takes the same general form, with the changes 0 , 0 , t f , J t J f . f E B t B 0 E B J f t E 3 Lecture 2 Energy of the electromagnetic fields Boundary conditions: Boundary conditions are the relations between the electromagnetic fields on two sides of the interface that separates two media. Gauss’ theorem: FdV F dS Stokes’ theorem: F dS F dl V S V S B 0 n (B 2 B1 ) 0 B2 n B1n D n (D D ) D D 2 1 2n 1n B E n (E2 E1 ) 0 E2 t E1t t D H J n (H 2 H1 ) K H 2 t H1t K t When 0, K 0, we have BB22nn BB11nn D D11nn D22nn D E22tt E E11tt E H H22tt H H11tt n – Unit normal to the surface – Surface charge density K – Surface current density 4 1.2 Poynting’s theorem and conservation laws Electromagnetic waves carry energy and momentum. Conservation of the energy of the electromagnetic fields requires: Energy flowing in through the boundary surface of a volume = Increase of the energy of the electromagnetic fields inside the volume + Work done by the electromagnetic fields on the electric charges inside the volume Considering energy transfer in a unit volume and in a unit time: S U W t t S U S – Energy flow through a unit area in a unit time U – Energy density of the electromagnetic fields W W – Work done by the electromagnetic fields on the electric charges in a unit volume Question: What are the expressions for S, U and W ? 5 Work done by the electromagnetic fields on the charges in a unit volume in a unit time: W D E v B v v E J E E H t t J E E H E D H B t t B E H H E E H H E H t This suggests S E H, U e E D, and U m H B. D U e ( E, D) E D is a function of the final E and D fields. 0 D Choosing a straight path of D in the final D direction, then U e ( E, D) E D. 0 Assuming linear medium, i.e., and are independen t of E and H, then 1 E D E (E) E E D E E D ( E D). 2 U S J E t 1 U U e U m E D B H 2 S E H U – Energy density of the electromagnetic fields. Linear response assumed. Otherwise D B 0 0 U E D H B. S – Poynting vector. Energy flow through a unit area in a unit time. Instantaneous intensity of light. Continuity equation. Poynting’s theorem. 6 Lecture 3 Wave equations 1.3 Complex function formulism Lights are most often described by sinusoidal time-varying fields: a(t ) A cos(t ) Complex representation: a(t ) Re[ A ei (t ) ] Re[ Aeit ], where A A ei . ~(t ) Aeit A ei (t ) represent the field, and we always mean its real part. Let a Complex representations have no problems with linear mathematical operations: ~ ~ ~ ~ ~ ~ E3 E1 E2 Re[ E3 ] Re[ E1 E2 ] E3 E1 E2 . It does not work for the product of fields: U e The correct way is: U e 1 ~ 1 ~~ ED U e ED 2 2 1 1 ~ ~* 1 ~ ~ * EE DD 2 2 2 This method is often needed in nonlinear optics. 7 Time averaging of two sinusoidal products with the same frequency: a (t ) A cos(t ), b(t ) B cos(t ) ~ a~(t ) Aeit A ei (t ) , b (t ) Be it B ei (t ) a ( t )b( t ) 1 ~ ~* 1 ~ ~* a a 2 b b 2 1 ~ ~ ~ ~* ~* ~ ~* ~* ab ab a b a b 4 1 ~ ~* ~* ~ 1 ~ 1 ~ a b a b Re[ a~(t )b * (t )] Re[ a~* (t )b (t )] 4 2 2 Examples: Time-averaged Poynting vector and the energy density: 1 ~ ~ S Re[ E H * ] 2 1 ~ ~ ~ ~ U Re[ E D* B H * ] 4 Conventions: We then drop the tidal sign, but we need to be careful in 1)Multiplying two fields, and 2)Multiplying a field with a scalar or tensor which is possibly a complex number. 8 1.4 Wave equations and monochromatic plane waves Each of the field vectors of light oscillates in space and time, which satisfies a wave equation. Assume the medium is isotropic ( and are scalars), linear ( and are independent of fields), and there is no free charge or free current ( =0, J =0). B 1 t E H 0 t 2 B H 1 E E 2 0 t D H 2E t H 2 0 t t D E 1 1 1 E E E, E E 2E E 2E E 2 log E E 0 t D 0 (E) E E 2 2E E 2 log E E log 0. t 2H 2 H 2 log H H log 0. t 2 The Maxwell’s equations and the materials equations are mathematically symmetric in the exchanges of E −H, B D, and −. 9 Further suppose the medium is homogeneous ( and does not change in space under a translational shift), the wave equations are 2E E 2 0 t 2H 2 H 2 0 t 2 A partial, linear, second order, homogeneous differential equation Solution to the wave equations: Plane waves Aei (t k r ) Wavelength (e i (k r t ) is also good.) Amplitude : A Phase : t k r Angular frequecy : 2 2 / 2 Wave vector : k v 1 c Phase velocity : v n Period Index of refraction : n / 0 0 10 Vector nature of the fields of electromagnetic waves (in an isotropic medium): E u1E0ei (t kr ) H u 2 H 0ei (t kr ) E 0 k u1 0 Transverse waves H 0 k u2 0 kˆ u1 u 2 B E t kE0 H 0 H 0 E0 / u1 , u 2 , kˆ are mutually orthogonal , and u1 u 2 kˆ . E and H are in phase if and are real. In an anisotropic medium ( and are tensors), only D and B are perpendicular to k. Time-averaged Poynting vector (intensity of light) and energy density: S 1 1 1 1 2 2 2 Re[ E H* ] E0 kˆ E0 v kˆ , I E0 v 2 2 2 2 1 1 1 2 2 2 Re[ E D* B H* ] E0 H 0 E0 4 4 2 1 2 U e U m E0 4 U 11 Lecture 4 Propagation of a laser pulse 1.5 Propagation of a laser pulse; group velocity Intense ultra-short laser pulses (Dt ~ 10-15 second, femto-second) are particularly important in exploring the dynamic structure of atoms and molecules, and their interaction with light. Bandwidth of a laser pulse: A laser pulse can be treated as a sum of many plane waves with different wavelengths. The pulse has a bandwidth in frequency domain, which satisfies the uncertainty rule: DDt 1. In the language of Fourier transform, at a fixed point in space E (t ) 1 2 E ( ) exp it d 2 E (t ) dt is the energy in dt. E ( ) 1 2 E (t ) exp it dt D E ( ) d is the energy in d. 2 0 12 Example: A Gaussian laser pulse 2 ln 2 t 2 E (t ) E0 exp i t 0 , 2 T T is the fwhm pulse dura tion for the light intensity. 1 E ( ) 2 1 2 2 ln 2 t 2 E exp i t 0 dt 0 T 2 2 ln 2 iT 2 2 T 2 2 0 0 E exp t dt 0 2 T 4 ln 2 8 ln 2 2 ln 2 0 2 E0 exp 4 ln 2 / T 2 2 ln 2 / T 2 2 2 ln 2 0 2 2 ln 2 E0 exp 2 T The Fourier transform of a Gaussian pulse is again a Gaussian function. All E() are in phase. This is a transform-limited pulse. 4 ln 2 is the fwhm bandwidth. T T 4 ln 2 2.77 in intensity, DDt 5.55 2 in amplitude distributi on. 13 Lecture 5 Group velocity Group velocity: A laser pulse normally has a carrier and an envelope. Group velocity is the velocity at which the envelope of the laser pulse travels. E ( x, t ) 1 2 1 2 E ( ) exp{i[t k ( ) x ]}d D E ( ) exp{i[( )t (k k ) x ]}d exp[ i( t k x )] D 0 dk 1 d 2k k k0 ( 0 ) d 0 2 d 2 1 E ( x, t ) 2 0 0 0 ( 0 ) 2 . For small D , 0 dk D E ( ) exp i( 0 ) t d x d exp[ i(0t k0 x )] 0 1 dk f x t exp[ i (0t k0 x )] d 0 Envelope Carrier t =0 vg vp t =t 1 dk Group velocity : vg d 0 0 c Phase velocity : v p k0 n(0 ) 14 More about group velocity: dn v v for normal dispersion 0 . g p d 1 dk dn vg v g v p for anomalous dispersion 0 . d d c c . dn dn dn 1. v g c when n n n d d d d nk Here vg is not the pulse velocity any more because of the higher order of d n . Group velocity dispersion: E ( x, t ) 1 2 E () exp{i[( )t (k k ) x]}d 0 D dk 1 d 2k k k0 ( 0 ) d 0 2 d 2 0 exp[ i (0t k0 x )] 1 d 3k ( 0 ) ( 0 )3 . 3 6 d 2 0 0 d 2k 3 d 2 n • Group velocity dispersion broadens or compresses a laser pulse. d 2 2c 2 d2 d 3k • distorts a laser pulse. d 3 15 Lecture 6 Dispersion 1.6 Dispersion and the Sellmeier equations n 0 0 non magnetic ( ) n( ) . How do we get n ()? 0 0 Electric polarization: The electric dipole moment per unit volume induced by an external electric field. For an isotropic and linear material, P D 0E ( 0 )E n( ) n( ) ( ) D P x(t ) Atom = electron cloud + nucleus. How is an atom polarized ? Restoring force: F kE x Natural (resonant) frequency: 0 k E me E + External force: FE qe E (t ) qe E0 exp( it ) dx Damping force: me (does negative work) dt dx d 2x 2 me 2 Newton’s second law of motion: qe E0 exp( it ) me0 x me dt dt 16 dx d 2x qe E0 exp( it ) me x me me 2 dt dt 2 0 Solution: x(t ) x0 exp( it ) qe E0 / me qe / me exp( i t ) E (t ) 02 2 i 02 2 i x0 is frequency dependent Electric polarization (and thus and n ) is frequency dependent n(). Electric polarization (= dipole moment density): Nqe2 / me P(t ) Nqe x(t ) 2 E (t ) 0 2 i Nqe2 / me P(t ) 0 0 2 E (t ) 0 2 i Dispersion equation: Nqe2 1 2 n ( ) 1 2 0 0 me 0 i 2 n 2 1 N 17 Quantum theory: 0 is the transition frequency. Nqe2 For a material with several transition frequencies: n ( ) 1 0 me Oscillator strength: f j 1 2 j fj 2 0j 2 i j j Normal dispersion: n increases with frequency. Anomalous dispersion: n decreases with frequency. n n'in" 1.5 2 2 exp( ikx) exp i n' x n' ' x 1.0 Re (n) 0.5 n' Phase velocity n" Absorption (or amplification) 0.0 0.5 1.5 0 2.0 Im (n) 0.5 18 Sellmeier equation: An empirical relationship between refractive index n and wavelength for a transparent medium: Nqe2 Refer to n ( ) 1 0 me 2 j fj 2 0j 2 i j . • Sellmeier equations work fine when the wavelength range of interests is far from the absorption of the material. dn d 2 n , , • The beauty of Sellmeier equations: n( ), d d2 are obtained analytically. • Sellmeier equations are extremely helpful in designing various optical devices. Examples: 1) Control the polarization of lasers. 2) Control the phase and pulse duration of ultra-short laser pulses. 3) Phase-match in nonlinear optical processes. Example: BK7 glass Coefficient Value B1 1.03961212 B2 2.31792344×10 B3 1.01046945 C1 6.00069867×10 μm C2 2.00179144×10 μm C3 1.03560653×10 μm −1 −3 2 −2 2 2 2 19