Optics and Optical Design. Chapter 5: Electromagnetic Optics. Lectures 9 & 10

Transcription

1 Optics and Optical Design Chapter 5: Electromagnetic Optics Lectures 9 & 1 Cord Arnold / Anne L Huillier

2 Electromagnetic waves in dielectric media

3 EM optics compared to simpler theories

4 Electromagnetic spectrum Electromagnetic optics describes all kinds of EM waves in all possible spectral ranges in possible kinds of media (vacuum, dielectric, conductive, etc.).

5 Example: THz imaging The THz Network,

6 Particle wave X-ray imaging (shadow-graphy) X-ray diffraction X-ray image from the hand of Albert von Koelliker, taken in Wikipedia

7 Maxwell Equations in vacuum Contributions from: Charles Augustin de Coulomb Hans Christian Örsted Carl Friedrich Gauss Jean Baptiste Biot André Marie Ampére Michael Faraday Unified by James Clerk Maxwell in 1861 as set of twenty equations. The current form, termed Maxwell Equations, was compressed by using vector notation by Oliver Heavyside in 1884.

8 Maxwell Equations in a source free medium

9 Boundary conditions

10 Different types of media Linear: Nondispersive: Homogeneous: If P(r,t) is linearly related to Ԑ(r,t). The response is instantaneous. The polarization P(r,t) does not depend on earlier times. The relation between P and Ԑ is no function of space. Isotropic: The relation between P and Ԑ is independent of the direction of Ԑ. Spatially nondispersive: The relation between P and Ԑ is local.

11 Linear, nondispersive, homogeneous, isotropic, sourcefree media

12 Anisotropic, linear, nondispersive media The susceptibility tensor χ can have up to nine independent elements χ ji.

13 Dispersive media

14 Monochromatic electromagnetic waves Introduce monochromatic fields All fields and flux densities can be written in their monochromatic versions accordingly.

15 Transverse electromagnetic (TEM) plane wave E is orthogonal to H. Both are orthogonal to the direction of propagation k.

16 Vectorial spherical wave

17 Example: focusing of vectorial waves

18 Vectorial solutions of the Helmholtz Equation

19 Absorption and dispersion

20 Transmission bands for common materials in optics

21 Implications of dispersion

22 Refractive index for different isotropic materials and crystals

23 The resonant medium

24 The resonant medium

25 Multi resonance media

26 Sellmeier Equation for the refractive index far from resonance

27 Kramers Kronig Relations The Kramers-Kronig relations relate mathematically the real and imaginary parts of the susceptibility to each other. Knowing one determines the other and vice versa.

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29 h t for t and ht is real for t (causal funtion) Causal response function H exp jt h t dt cost h t j sint hdt t Signum function Noncausal odd function h t signum t h t h t o o Causal response function

30 Frequency space imaginary part of a causal response function Frequency space real part of a causal response function H SIGNUM H o H o Real part Imaginary part The real and imaginary parts are related because they originate from the same function and they contain the same information!

31 The Drude Model for conductive media ω<ω p ω>ω p ω=ω p The effective permittivity is negative, β(ω) is imaginary. Light cannot propagate. => Perfect mirror. The effective permittivity is positive. Light can propagate. The refractive index is below 1. β(ω)=. Light cannot propagate. But one can resonantly excite plasma waves. Plasmons!

32 Pulse propagation in dispersive media

33 Dispersive media The field moves in respect to the envelope due to the difference of phase and group velocity The pulse spreads due to group velocity dispersion (GVD)

34 Electric field (a.u.) Temporal and spectral representation of laser pulses and the time bandwidth product Pulsed plane wave t At expj t U FWHM Carrier frequency Pulse envelope (spectrally broad) Time Fourier transform Spectral power (a.u.) Frequency Electric field (a.u.) FWHM FWHM.44 Time-bandwidth product (Gaussian pulse) Spectral power (a.u.) Time Frequency

35 Plane wave propagation ~ A A Laser pulses in dispersive media ~ z, Az, exp j 1 z, t F Az, ~ 1 F F Az, t z Spectral plane wave propagator exp j z Each frequency component evolves with a different wave number Wave number expansion around a carrier ω : c n ' 1! ' '! Group velocity v g 1/ ' Group velocity dispersion s s ' Inverse of a speed '' Inverse of an acceleration m m ' 1 v g, '' Group velocity and group velocity dispersion (GVD) result from dispersion.

36 Group velocity and group index ', ' ' 1 ', n n N N c n n c v n n c n n c n c c c g Depends on the change of the refractive index in respect to the wavelength Group index The speed of a pulse is determined by the rate of change of the refractive index Refractive index for a typical material

37 Group velocity dispersion (GVD) 3 '' '' '' n c D c c Refractive index for fused silica GVD is proportional n ( ), that is the curvature of n( ). 3 ' ', ' ' n c D n c D GVD for fused silica z D z D D D Estimation for dispersive pulse broadening

38 Pulse broadening in dispersive media

39 Dispersive media n<1 n>1 N>1 => v g <v p N>1 => v g <v p Anomalous dispersion Anomalous dispersion Normal dispersion