Fields, wave and electromagne3c pulses. fields, waves <- > par0cles pulse <- > bunch (finite in 0me),

Transcription

1 Fields, wave and electromagne3c pulses fields, waves <- > par0cles pulse <- > bunch (finite in 0me), 1

2 Op3cs ray or geometric op0cs: ABCD matrix, wave op0cs (used e.m. field to describe the op0cal field): applica0on to diffrac0on, nonlinear op0cs, polariza0ons, quantum op0cs: light- mater interac0on, harmonic genera0ons, 2

3 Ray op3cs postulates Postulates: Light travels in form of rays Medium characterized by an index of refrac0on n defined as the ra0o of velocity of light in vacuum over velocity of light in medium Fermat s principle: op0cal rays traveling between two points A and B follow a path such that the 0me of travel between two points is an extremum rela0ve to the neighboring paths. This extremum is usually a minimum: so light goes from A to B along the path of least 0me. This is the op0cs equivalent to the least ac0on principle 3

4 Ray op3cs (see Lectures 1+2) same as charged par0cle: used ABCD formalism 4

5 Complex representa3on of an electromagne3c wave The E field is used to describe an electromagne0c wave r =(x, y, z) E(r, t) =E 0 (r)e i[k.r!t] direc0on of E is the polariza0on. field is real k =(k x,k y,k z ) E(r, t) =E 0 (r) cos[k.r!t] 5

6 Example of a plane wave For a plane wave in free space: E(r, t) =E 0 e i[kz!t]ˆxˆxˆx this wave travels along the +ẑẑẑ direc0on. the field is constant at in the (x,y) plane The B- field can be obtained from Maxwell s equa0ons 6

7 Helmoltz s equa3on start with the wave equa0on considering a medium with index of refrac0on n the wave equa0on becomes considering we can rewrite the wave equa0on as with k 2 n2! 2. E(r, t) =E(r)e i!t r 2 E + k 2 E =0 c 2 r 2 E µ d2 E dt 2 =0 r 2 E + n2 d 2 E c 2 dt 2 =0 Helmoltz s equa3on 7

8 Plane wave is solu3on of Helmoltz s equa3on consider a more general plane wave (only spa0al func0on): E(r) =Ae ik.r points located at contant phase describe the wavefront: k.r =2 n integer 8

9 spherical and paraboloidal waves are also solu3on spherical wave: constant phase E(r) = A r eikr kr =2 n integer amplitude of k and r no dot product here Paraboloidal wave where we used E(r) = A z eikz e ik x 2 +y 2 2z r ' z + x2 + y 2 2z 9

10 Paraxial Helmholtz equa3on Start with Helmholtz equa0on Consider the wave Complex amplitude Complex envelope which is a plane wave (propaga0ng along z) transversely modulated by the complex amplitude A. Assume the modula0on is a slowly varying func0on of z (slowly here mean slow compared to the wavelength) A varia0on of A can be writen as So that 10

11 Paraxial Helmholtz equa3on So Expand the Laplacian Transverse Laplacian The longitudinal deriva0ve is Plug back in Helmholtz equa0on Which finally gives the paraxial Helmholtz equa0on (PHE): 11

12 Gaussian beams The paraboloid wave is solu0on of the PHE Doing the change give a shiaed paraboloid wave (which is s0ll a solu0on of PHE) If ξ complex, the wave is of Gaussian type and we write where z 0 is the Rayleigh range We also introduce Wavefront curvature Beam width How is W related to the rms size? 12

13 Gaussian beams R and W can be related to z and z 0 Explici0ng in U one gets 13

14 Intensity distribu3on Transverse intensity distribu0on at different z loca0ons - 4z 0-2z 0 z/z 0 - z 0 0-4z 0-2z 0 - z 0 0 Corresponding profiles

15 Intensity distribu3on (cnt d) On- axis intensity as a func0on of z is given by z/z 0 z/z 0

16 Wavefront radius The curvature of the wavefront is given by

17 Beam width and divergence Beam width

18 Depth of focus A depth of focus can be defined from the Rayleigh range 2 2z 0

19 Phase The argument as three terms Phase associated to plane wave Guoy phase shia On axis (ρ=0) the phase s0ll has the Guoy shia Spherical distor0on of the wavefront At z 0 the Guoy shia is π/4 Varies from - π/2 to +π/2

20 how to characterize an op3cal beam in general? E field is representa0ve of the op0cal- intensity distribu0on at a given loca0on introduce a concept similar to phase space representa0on called Wigner func0on (adapted from Quantum mechanics): W f (q, p) = dimensionality N Z K 2 f q complex conjugate q 0 f q + q0 q e ikq0 q 0p d N q 0 observable constant 20

21 consider the transverse phase space usually (q, p)! (x, p x )=(x, ~k x ) $ (x, k x ) and use the electric field as the observable so that W E (x, k x )= 1 2 Z E x note that the Wigner func0on can be nega0ve x 0 2 E x + x0 e ik xx 0 dx

22 Wigner func3on proper3es integrals are Z Z W E (x, k x )dxdk x = Z E(x) 2 dx propor0onal to e.m. energy Z Z W E (x, k x )dx = E(k x ) W E (x, k x )dk x = E(x) spectrum field spa0al distribu0on 22

23 Moments of the Wigner func3on as for phase space distribu0on one can define 1 st (n+m=1) and 2 nd order moments (n+m=2): RR dxdkx x n k m x W E (x, k x ) hx n,k m x i = RR dxdkx W E (x, k x ) these are representa0ve of the beam size and spread in spa0al frequency (~ divergence) 23