Dispersion and Ultrashort Pulses II

Transcription

1 Dispersion and Ultrashort Pulses II Generating negative groupdelay dispersion angular dispersion Pulse compression Prisms Gratings Chirped mirrors

2 Chirped vs. transform-limited A transform-limited pulse: satisfies the equal sign in the relation ν τ C is as short as it could possibly be, given the spectral bandwidth has an envelope function which is REAL (phase Φ(ω) = 0) has an electric field that can be computed directly from S(ω) exhibits zero chirp: the same period A chirped pulse: satisfies the greater than sign in the relation ν τ C is longer than it needs to be, given the spectral bandwidth has an envelope function which is COMPLEX (phase Φ(ω) 0) requires knowledge of more than just S(ω) in order to determine E(t) exhibits non-zero chirp: not the same period

3 Pulse propagation and broadening After propagating a distance z, an (initially) unchirped Gaussian of initial duration t G becomes: E( t, z) exp 2 τ exp ( ) ( 1 i z ) = ξ ξ = t + ξ z pulse duration increases with z ( t z / V ( )) 2 g ωp t 2 G + 2 ik " z G 1 tg chirp parameter β 2 k " For normal dispersion (i.e., most of the time), β is positive. 3 2 pulse duration t G (z)/t G propagation distance ξz

4 So how can we generate negative GDD? This is a big issue because pulses spread further and further as they propagate through materials. We need a way of generating negative GDD to compensate. Negative GDD Device

5 Prisms disperse light. Because the refractive index depends on wavelength, the refraction angle also depends on wavelength. Because n generally decreases with wavelength (dn/dλ < 0), smaller wavelengths experience greater refraction angles. Differentiating implicitly with respect to λ: dθt dn cos( θt ) = sin( θi ) dλ dλ We obtain the prism dispersion: dθt dn sin( θi ) = dλ dλ cos( θ ) t

6 Angular dispersion yields negative GDD. Suppose that an optical element introduces angular dispersion. Optical element α Input beam Optic axis Here, there is negative GDD because the blue precedes the red. We ll need to compute the projection onto the optic axis (the propagation direction of the center frequency of the pulse).

7 Negative GDD Taking the projection of onto the optic axis, a given frequency ω sees a phase delay of ϕ(ω): z ϕ( ω) = k ( ω) r optic axis k ( ω) = k( ω) z cos[ θ ( ω)] = ( ω / c) z cos[ θ ( ω)] θ(ω) θ(ω) << 1 Optic axis We re considering only the GDD due to the angular dispersion θ(ω) and not that of the prism material. Also n = 1 (that of the air after the prism). dϕ / dω = ( z / c) cos( θ ) ( ω / c) z sin( θ ) dθ / dω d ϕ z dθ z dθ z dθ z d θ = sin( θ) sin( θ) ω cos( θ) sin( ) 2 ω θ 2 dω c dω c dω c dω c dω But θ << 1, so the sine terms can be neglected, and cos(θ) ~ 1.

8 Angular dispersion yields negative GDD. Copying the result from the previous slide: 2 2 d ϕ ω0 z dθ 2 dω c dω ω ω0 0 The GDD due to angular dispersion is always negative! Also, note that it doesn t matter what kind of device gave rise to the angular dispersion (i.e., whether it was a prism or a diffraction grating or whatever).

9 A prism pair has negative GDD. Assume Brewster angle incidence and exit angles. How can we use dispersion to introduce negative chirp conveniently? Let L prism be the path through each prism and L sep be the prism separation. d 2 3 ϕ λ 4L 0 dn dω 2 sep ω 0 2πc 2 dλ λ0 Always negative! This term assumes that the beam grazes the tip of each prism 2 + L prism λ 0 3 2πc 2 d 2 n dλ 2 λ 0 This term allows the beam to pass through an additional length, L prism, of prism material. Always positive (in visible and near-ir) We can independently vary L sep or L prism to tune the GDD!

10 Pulse Compressor This device, which also puts the pulse back together, has negative group-delay dispersion and hence can compensate for propagation through materials (i.e., for positive chirp). Angular dispersion yields negative GDD. It s routine to stretch and then compress ultrashort pulses by factors of >1000.

11 What does the pulse look like inside a pulse compressor? If we send an unchirped pulse into a pulse compressor, it emerges with negative chirp. Note all the spatio-temporal distortions.

12 What does the pulse look like inside a pulse compressor? If we send a positively pulse into a pulse compressor, it emerges unchirped. Note all the spatio-temporal distortions.

13 Adjusting the GDD maintains alignment. Any prism in the compressor can be translated perpendicular to the beam path to add glass and reduce the magnitude of negative GDD. Remarkably, this does not misalign the beam. The output path is independent of prism position. Input beam Output beam

14 Incorporating a four-prism pulse compressor into the laser cavity was a revolutionary advance. Appl. Phys. Lett. 38, 671 (1981)

15 The required separation between prisms in a pulse compressor can be large. The GDD the prism separation and the square of the dispersion. Different prism materials Compression of a 1-ps, 600-nm pulse with 10 nm of bandwidth (to about 50 fs). Kafka and Baer, Opt. Lett., 12, 401 (1987) It s best to use highly dispersive glass, like SF10, or gratings. But compressors can still be > 1 m long.

16 Four-prism pulse compressor Also, alignment is critical, and many knobs must be tuned. Prism Wavelength tuning Wavelength tuning Prism Wavelength tuning Prism Prism Fine GDD tuning Coarse GDD tuning (change distance between prisms) Wavelength tuning All prisms and their incidence angles must be identical.

17 Pulse-compressors have alignment issues. Pulse compressors are notorious for their large size, alignment complexity, and spatio-temporal distortions. Pulsefront tilt Spatial chirp Unless the compressor is aligned perfectly, the output pulse has significant: 1. 1D beam magnification 2. Angular dispersion 3. Spatial chirp 4. Pulse-front tilt

18 Why is it difficult to align a pulse compressor? The prisms are usually aligned using the minimum deviation condition. Deviation angle Prism angle Minimum deviation Angular dispersion The variation of the deviation angle is 2 nd order in the prism angle. But what matters is the prism angular dispersion, which is 1 st order! Using a 2 nd -order effect to align a 1 st -order effect is a bad idea.

19 Two-prism pulse compressor Coarse GDD tuning Periscope Wavelength tuning Roof mirror Prism Wavelength tuning Prism Fine GDD tuning This design cuts the size and alignment issues in half.

20 Single-prism pulse compressor Periscope Prism Corner cube Roof mirror Wavelength tuning GDD tuning

21 Diffraction-grating pulse compressor The grating pulse compressor also has negative GDD. 3 λ L 0 sep c d ω 2 d ϕ dω 2π cos ( β ) 0 Grating #2 ω ω ο where d = grating spacing (same for both gratings) L sep Note that, as in the prism pulse compressor, the larger L sep, the larger the negative GDD. Grating #1

22 2nd- and 3rd-order phase terms for prism and grating pulse compressors Grating compressors yield more compression than prism compressors. ϕ'' ϕ''' Piece of glass Note that the relative signs of the 2nd and 3rd-order terms are opposite for prism compressors and grating compressors.

23 Compensating 2nd and 3rd-order spectral phase Use both a prism and a grating compressor. Since they have 3rd-order terms with opposite signs, they can be used to achieve almost arbitrary amounts of both second- and third-order phase. Prism compressor Grating compressor Given the 2nd- and 3rd-order phases of the input pulse, ϕ input2 and ϕ input3, solve simultaneous equations: ϕ input 2 + ϕ prism 2 + ϕ grating 2 = 0 ϕ input 3 + ϕ prism 3 + ϕ grating 3 = 0

24 Pulse Compression: Simulation Using prism and grating pulse compressors vs. only a grating compressor Resulting intensity vs. time with only a grating compressor: Note the cubic spectral phase! Resulting intensity vs. time with a grating compressor and a prism compressor: Brito Cruz, et al., Opt. Lett., 13, 123 (1988).

25 Pulse Compression: Results The prisms + gratings pulse compressor design was used by Fork and Shank in 1987 to compress pulses to six femtoseconds a record that stood for over a decade. 60 fsec input pulse single-mode optical fiber chirped but spectrally broader pulse 6 fsec output pulse

26 Short optical pulses progress is amazing Charles Shank Erich Ippen Shortest laser pulse (sec) one nanosecond one picosecond visible light one femtosecond Year consumer electronics world's fastest transistor } This ten-year gap happened (in part) because nobody thought to try to figure out a way to compensate the fourth-order phase. x-rays Mikio Yamashita Ferenc Krausz

27 The grism pulse compressor has tunable third-order dispersion. A grism is a prism with a diffraction grating etched onto it. The (transmission) grism equation is: a [sin(θ m ) n sin(θ i )] = mλ Note the factor of n, which does not occur for a diffraction grating. A grism compressor can compensate for both 2 nd and 3 rd -order dispersion due even to many meters of fiber.

28 Chirped mirrors A mirror whose reflection coefficient is engineered so that it has the form: so that r ( ω) = 1 r ( ω) ( ) = e iφ ω and φ(ω) is chosen to cancel out the phase of the incident pulse. TiO 2 layers SiO 2 layers

29 Chirped mirror coatings Longest wavelengths penetrate furthest. Doesn t work for < 600 nm

30 Chirped mirrors for extra-cavity dispersion control Each bounce off of a mirror adds the chosen spectral phase φ(ω) to the pulse. One can accumulate large changes in φ(ω) through multiple bounces.

31 Chirped mirrors for extra-cavity dispersion control

32 Chirped mirrors intra-cavity