Linear pulse propagation

Transcription

1 Ultrafast Laser Physics Ursula Keller / Lukas Gallmann ETH Zurich, Physics Department, Switzerland Linear pulse propagation Ultrafast Laser Physics ETH Zurich

2 Superposition of many monochromatic waves U. Keller, ULP, Chap. 2, p. 1

3 Plane wave, monochromatic Fourier transformation Inverse Fourier Transformation (for every position in space) Fourier Transformation U. Keller, ULP, Chap. 2, p. 3

4 Superposition of monochromatic waves Light pulse Wave packet U. Keller, ULP, Chap. 2, p. 8-10

5 Monochromatic plane wave Vacuum Dispersive material Frequency Period Phase velocity Wave number Wavelength U. Keller, ULP, Chap. 1

6 Optical dispersion U. Keller, ULP, Chap. 1

7 Helmholtz equation Fourier transformation U. Keller, ULP, Chap. 2, p. 5

8 Analogy to the Schrödinger equation Helmholtz equation: Wave equation in the spectral domain Time independent Schrödinger equation: free particle particle in a potential field U. Keller, ULP, Chap. 2, p. 5

9 Dispersion for quantum mechanical particles Photon in vacuum: Photon in medium: Free electron: Electron in conduction band: Phonon in 1-dimensional lattice: U. Keller, ULP, Chap. 2, Appendix

10 Linear system theory Input System Output impulse response Examples of linear systems: pulse propagation in dispersive media photo detector (impulse response links photo current with light power or intensity) active light modulator in the linear regime image propagation through a lens systems stochastic processes such as amplitude and phase noise (linearized as a perturbation on a much stronger signal) U. Keller, ULP, Chap. 2, p. 6

11 Linear system theory Input System Output impulse response Input System Output transfer function U. Keller, ULP, Chap. 2, p. 6

12 Linear system theory Input System Output impulse response Input System Output transfer function Spectral power density: Input System Output U. Keller, ULP, Chap. 2, p. 6

13 Linear system example: dispersion What is the impulse response or transfer function? What is the pulse form during propagation? How is the spectrum of the pulse changing? Solution: Chapter 2, page 7

14 Dispersive pulse broadening Linear pulse broadening: Transform-limited pulse is broadening in the time domain but its spectrum remains unchanged.

15 Light pulse U. Keller, ULP, Chap. 2, p. 10

16 Laser pulse U. Keller, ULP, Chap. 2, p. 8-11

17 Laser pulse U. Keller, ULP, Chap. 2, p. 8-11

18 Example: Gaussian pulse U. Keller, ULP, Chap. 2, p. 12

19 Chirped Gaussian pulse U. Keller, ULP, Chap. 2, p. 13

20 Time-bandwidth products U. Keller, ULP, Chap. 2, p. 12

21 Spectral phase yielding shortest pulse : center of gravity rms pulse duration : shortest pulse for: U. Keller, ULP, Chap. 2, p. 15

22 Inappropriateness of FWHM definition Temporal FWHM of pulse with nonlinear spectral phase is shorter than transform-limited pulse FWHM is the standard in the community but one has to be aware of its limitations

23 Optical dispersion Positive dispersion: Negative dispersion: U. Keller, ULP, Chap. 1

24 Dispersive pulse broadening

25 Helmholtz equation: Linear pulse propagation slowly varying envelope approximation Very simple equation of motion for the pulse envelope in the spectral domain with a very easy solution (i.e. a phase shift for each frequency component) U. Keller, ULP, Chap. 2, p

26 First and second order dispersion Taylor expansion around the center frequency : First order dispersion: Second order dispersion: U. Keller, ULP, Chap. 2, p. 18

27 Phase and group velocity U. Keller, ULP, Chap. 2, p

28 Dispersive pulse broadening Gaussian pulse: (initially unchirped pulse) Approximation for (strong pulse broadening) U. Keller, ULP, Chap. 2, p

29 U. Keller, ULP, Chap. 2, p. 27

30 Example: fused quartz Fused 800 nm Example: What is the final pulse duration: 10 fs pulse through 10 lenses, each 1 cm thick? 10 fs pulse through 10 cm fused quartz? U. Keller, ULP, Chap. 2, p. 22

31 Optical communication U. Keller, ULP, Chap. 2, p. 31

32 Higher order dispersion First order dispersion (group delay) Second order dispersion (group delay dispersion - GDD) Third order dispersion (TOD) U. Keller, ULP, Chap. 2, p. 37

33 Wigner representation of ultrashort pulses Wigner distribution: window function Time-frequency representation / windowed Fourier transform / instantaneous spectrum vs time Example: Properties: Linear propagation in non-absorbing medium conserves area ( minimum time-bandwidth product)

34 Effect of dispersion orders on Wigner trace Everything calculated for an initially 10-fs long Gaussian pulse After 100 fs 2 of GDD: Pulse remains Gaussian in time! Chirp is linear

35 Effect of dispersion orders on Wigner trace Everything calculated for an initially 10-fs long Gaussian pulse After 1000 fs 3 of TOD: Beating of simultaneous frequencies causes post-(pre-)pulses

36 Effect of dispersion orders on Wigner trace Everything calculated for an initially 10-fs long Gaussian pulse After fs 4 of FOD: Even dispersion orders yield symmetric pulse distortions in time (for a symmetric spectrum)

37 Pulse after 3 mm of fused silica Everything calculated for an initially 10-fs long Gaussian pulse After 3 mm of fused silica (800 nm center wavelength): Chirp is dominantly linear however, influence of higher orders clearly visible Pulse is not Gaussian in time anymore!

38 Higher order dispersion U. Keller, ULP, Chap. 2, p. 38

39 Higher order dispersion First order dispersion (group delay) Second order dispersion (group delay dispersion - GDD) Third order dispersion (TOD) GDD becomes important, when: TOD becomes important, when: Dispersion lengths: Gaussian pulse Soliton pulse U. Keller, ULP, Chap. 2, p. 37