The structure of laser pulses

Transcription

1 1 The structure of laser pulses

2 2 The structure of laser pulses Pulse characteristics Temporal and spectral representation Fourier transforms Temporal and spectral widths Instantaneous frequency Chirped pulses Gaussian and chirped Gaussian pulses Temporal properties Spatial properties Linear systems and filters The chirp filter

3 3 Why pulsed lasers? Power = Energy / time Concentrating the optical energy in a (short) pulse increases the power of the laser. 1 µs microsecond 10-6 s 0.3 km 1 ns nanosecond 10-9 s 0.3 m 1 ps picosecond s 0.3 mm 1 fs femtosecond s 0.3 µm 1 as attosecond s 0.3 nm Ultrashort laser pulses are the shortest events produced by mankind

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6 6 Progress in short-pulse generation Alexandra A. Amorim, FCUP

7 7 How to represent a basic optical pulse? 1. Choose the central frequency ν 0 2. Define the complex envelope A(t) exp(i 2πν 0 t) = exp(iω 0 t) A(t ) = A(t ) exp[ iϕ(t )] amplitude phase 3. Multiply them U(t) = A(t)e iω 0 t = A(t)e i( ω 0 t +ϕ(t ))

8 8 Temporal representation of an optical pulse Re{ U(t )} = A(t ) cos[ ω 0 t + ϕ(t )] I(t) = U(t) 2 = A(t) 2

9 9 Intensity and types of pulses The usual definition applies: I(t) = U(t) 2 = A(t) 2 Some frequent types of laser pulses (τ is called the width): Gaussian Lorentzian Hyperbolic secant I(t ) exp( 2t 2 / τ 2 ) I(t) 1 1+ t 2 / τ 2 I(t ) sech 2 = ( t / τ) 2 e t /τ t /τ + e 2

10 10 A pulse can be described in the time or frequency domains They are fully equivalent. To switch between one description and the other we use Fourier transforms (FT): U(t ) = A(t )e i ( ω 0t +ϕ(t )) amplitude time phase V (ω) = U(t )exp( iωt ) spectral amplitude frequency [ ] = V (ω) exp iψ(ω) Fourier transform spectral phase FT s are always applied to amplitude (or electric field), not intensity. This concept is essential for understanding laser pulses.

11 11 Temporal and spectral descriptions time U(t ) = A(t ) exp[ iϕ(t )] frequency V (ω) exp[ iψ(ω) ]

12 12 The spectral amplitude and intensity Just like for the time description we can also define a spectral intensity: I(t ) = U(t ) 2 S(ω) = V (ω) 2 The spectral intensity has a concrete physical meaning: it is the value measured by a spectrometer S(ω)

13 13 How does a spectrometer work? Input beam thin slit (object) 1 st mirror collimate Diffraction grating wavelength-dependent output angle Output beam thin slit for each wavelength (image) 2 nd mirror focus

14 14 What is the spectral phase? The spectral phase is the phase of each of the frequencies that compose the function. V (ω) exp[ iψ(ω) ] ω 1 In this example we have ψ(ω) = 0. ω 2 ω 3 ω 4 ω 5 ω 6 0 t This leads to the formation of a pulse at t = 0 (constructive interference) and zero elsewhere (destructive interference).

15 15 Fourier transforms The principle is that you can express any function A(t) as a superposition of harmonic functions of different amplitude and phase: f (t ) = f (ω)exp(iωt ) Coefficient Harmonic function The (complex) coefficients can be determined by calculating the integral f (ω) = f (t )exp( iωt ) Inverse Fourier transform

16 16 Examples of important FT s uniform ( ) f ( ν ) f t 1 δ( ν) impulse rectangular exponential Gaussian hyperbolic secant δ( t ) rect( t ) exp( t ) exp( πt 2 ) sech( πt ) 1 sinc( ν) 2 ( ) πν exp( πν 2 ) sech( πν)

17 17 Interesting properties of FT s Linearity Scaling Time translation τ Frequency transl. ω 0 F(f 1 + f 2 ) = F(f 1 ) + F(f 2 ) F[f (t )] = f (ω): F[f (t / τ)] = f (τω) / τ F[f (t τ)] = f (ω)e iωτ F[f (t )e iω 0 t ] = f (ω ω 0 ) Parseval s theorem f (t ) 2 dt = f (ω) 2 d ω

18 18 Example: time translation F[f (t τ)] = f (ω)e iωτ ψ(ω 1 ) = 0 ψ(ω 2 ) = 0.2 π ψ(ω 3 ) = 0.4 π ψ(ω 4 ) = 0.6 π ψ(ω 5 ) = 0.8 π ψ(ω 6 ) = π t

19 19 Temporal and spectral widths = widths of the temporal and spectral intensities. τ FWHM A commonly used measure is the Full Width at Half Maximum (FWHM): Δν full width ½ maximum

20 20 Relation between temporal and spectral widths Because of the Fourier transform relationship the following relationship holds (Δω = 2πΔν): τ FWHM Δν ~ 1 The exact value depends of the pulse shape (e.g for a Gaussian temporal shape Homework!) This is similar to the result for a mode-locked laser It is also similar to a Heisenberg uncertainty principle This is a fundamental result especially for very short pulses

21 21 Exercise: spectral width in frequency and in wavelength Normally it is more practical to express a spectral width in terms of a wavelength range (e.g. nm) than in terms of a frequency range (e.g. THz). a) Show that for Δν ν the following relationship between Δλ and Δν holds: Δλ λ 0 2 c Δν b) For a spectral width Δν = 10 THz, calculate Δλ at 0.55 µm and 1.1 µm. (Answer: 10 nm; 40 nm)

22 22 Relation between widths for a Gaussian pulse In terms of the wavelength width: Δλ λ 2 0 c Δν depends on the wavelength! τ FWHM Δν 0.44

23 23 Instantaneous frequency what is the instant color of a pulse? U(t ) = A(t )e i ( ω 0 t +ϕ(t ) ) The instantaneous frequency is defined as the time derivative of the temporal phase: ω i = d dt ν i = ν π [ ω 0 t + ϕ(t )] = ω 0 + dϕ dϕ dt dt A particular case of interest is the linearly varying phase: ϕ(t ) = αt ω i = ω 0 + α This simply corresponds to a fixed frequency shift.

24 24 Chirped pulses Another very important case is when the instantaneous frequency varies with time. Such pulses are called chirped. ϕ ""(t ) = d 2 ϕ % > 0 up chirped dt 2 & ' < 0 down chirped The simplest case is a linearly varying instantaneous frequency. This corresponds to a quadratic temporal phase: ϕ(t ) = ω 0 t + at 2 / τ 2 ω i = ϕ = ω 0 + (2a / τ 2 )t chirp parameter 1 a = d 2 φ dt ϕ = 2a / 2 τ2 2 ϕ "" τ 2

25 25 Pulses with linear chirp τ = 20 fs, ν 0 = 300 THz up-chirped down-chirped Linearly chirped pulses are the basis for the technique of chirped pulse amplification (CPA) which is used in modern high power lasers.

26 26 Gaussian pulses Describing pulses using the Gaussian function is very practical, even if in reality a pulse may not be exactly Gaussian. A(t ) = A 0 exp( t 2 /τ 2 ) I(t ) = I 0 exp( 2t 2 /τ 2 ) Amplitude and intensity of a transform-limited Gaussian pulse (n.b. the phase is constant) τ = FWHM t = τ : I(t ) = I 0 exp( 2) I 0

27 27 Exercise transform-limited Gaussian pulse properties Demonstrate the following relations for a transform limited Gaussian pulse A(t ) = A 0 exp( t 2 /τ 2 ) I(t ) = I 0 exp( 2t 2 /τ 2 ) τ FWHM = 2ln 2τ 1.18τ ' S(ω) exp τ2 ( 2 ω ω 0) 2 * ), ( + Δν = 0.375/τ = 0.44/τ FWHM

28 28 Why transform-limited? A transform-limited Gaussian pulse obeys the relation τ FWHM Δν = 2ln 2 π 0.44 This is the minimum duration bandwidth product allowed by the Fourier transform. The transform-limited Gaussian pulse is the ideal pulse. We can also say that it is Fourier-transform limited. For a chirped Gaussian pulse we always have τ FWHM Δν > 2ln 2 π

29 29 Chirped Gaussian pulses More generally, a Gaussian pulse may have the form: A(t ) = A 0 exp( αt 2 ) α = (1 ia)/τ 2 This is called a chirped Gaussian pulse A(t ) = A 0 exp( t 2 /τ 2 )exp( iat 2 /τ 2 ) Intensity Phase Instant. frequency ( ) = A(t ) 2 = A 0 exp( 2t 2 /τ 2 ) ( ) = at 2 /τ 2 ( ) = ϕ &( t ) = 2at /τ 2 I t ϕ t ω i t

30 30 Transform-limited and chirped Gaussian pulses: properties transform limited I( t ) I 0 exp 2t 2 / τ 2 chirped ( ) A(t) = A 0 exp( t 2 / τ 2 ) A(t) = A 0 exp (1 ia)t 2 / τ 2 ( ) I 0 exp( 2t 2 / τ 2 ) τ FWHM 2 ln2τ 1.18τ 2 ln2τ 1.18τ ' S(ω) exp τ2 ( 2 ω ω 0 ) 2 * ' ( ) ( +, exp τ2 ω ω ) 2 ) 0 2 ( 1+ a 2 ) ( ) 0.44 Δν 0.44 / τ FWHM 1+ a 2 τ FWHM ω i ω 0 ω 0 + ( 2a / τ 2 )t *, +,

31 31 Transform-limited and chirped Gaussian pulses: properties Example for a λ 0 = 1 µm, τ FWHM = 5 fs pulse. Chirped pulse with a = ±2. a = 0 a = 2 a = 2

32 32 Spatial properties of laser pulses In free space, or in a linear, homogeneous and non-dispersive medium, an optical pulse obeys the wave equation. Using the same considerations as we did for wave optics the solutions are of the type: ( ) = A t z c ( ) = A 2 ( t ) U r,t I t ( )e iω 0 t z c ( ) A( ρ,z,t ) = g ( t z c) iz 0 exp i z + iz 0 π λ 0 ρ 2 z + iz 0

33 33 What does a Gaussian pulse look like? This is a pulse that is Gaussian both in space and in time. cτ p cτ p L w cτ p w 1 ns 0.3 m 1 ps 0.3 mm 1 fs 0.3 µm TL Gaussian pulse (up-)chirped Gaussian pulse

34 34 Linear systems* The transmission of an optical pulse through an arbitrary linear optical system can be described in terms of the theory of linear systems. (*see Fund. Photonics Appendix B) U 1 (t) V 1 (ω) Linear system U 2 (t) V 2 (ω) V 2 (ω) = H (ω)v 1 (ω) Transfer function of the system

35 35 Linear filtering of an optical pulse For an optical pulse centered around ω 0 we only need H(ω) at frequencies inside the spectral width Δω. If Δω ω 0 it is more practical to work with the complex envelope A(t) instead of the wavefunction U(t). U( t ) = A( t )e iω 0 t V ω A 2 (δω) = H e (δω) A 1 (δω) ( ) = ( ) A ( δω) A ω ω 0 envelope transfer function (Note that we can also write in the time domain: the FT of a product is a convolution, and H e (t) is the impulse response function = the inverse FT of the envelope transfer function.) A 2 (t ) = H e (t ) A 1 (t ) = H e (t t $ )A 1 ( t $ )d t $ H e (t ) = FT 1 { H e (ω)}

36 36 Example: using an equalizer to attenuate sound frequencies

37 37 The ideal filter An ideal filter preserves the shape of the input pulse multiplies by a constant (G > 1: amplifier; G < 1: attenuator) may delay the pulse by a fixed time τ d (remember the time-delay property of FT s) H e (δω) = H 0 exp( i δωτ d ) G = H 0 2 Example: a slab of ideal material with thickness d, ref. index n and distributed attenuation coefficient α (m -1 ): H e (ω) = exp( αd /2)exp( i ωnd /c 0 ) A neutral density filter H 0

38 38 The chirp filter It s the most important type of filter in optics. Its phase is a quadratic function of frequency: b = chirp coefficient (s 2 ) b > 0: up-chirping b < 0: down-chirping H % e (δω) = exp i b δω2 ( ' & 4 * ) H e ( t ) = 1 ib /2 exp % i t 2 ( ' & b * ) Chirp filters can be cascaded: the sum of two chirp filters with coeff. b 1 and b 2 is equal to a single filter b = b 1 + b 2. Pulse dispersion by a prism may be described in terms of chirped filters.

39 39 Ideal vs. chirp filters H 0 exp( i δωτ d ) % exp i b δω2 ( ' & 4 * )

40 40 Arbitrary filter chirped filter + ideal filter If the magnitude and phase of an arbitrary filter H e (ω) vary slowly within the spectral width of the pulse, we may assume the following: ( ) H ( ω 0 ) H 0 ( ) ψ 0 + ' H ω 0 + δω ψ ω 0 + δω H e δω ψ δω ( ) H 0 exp i ψ 0 + ' constant phase ψ '' δω 2 (Taylor series around ω 0 ) [ ( ψ δω + 1 ψ 2 '' δω 2 )] linear filter chirped filter τ d = ψ # b = 2 ψ ""

41 41 Arbitrary filter chirped filter + ideal filter ( ) H 0 exp i ψ 0 + ' H e δω [ ( ψ δω + 1 ψ 2 '' δω 2 )] A treatment based on ideal + chirped filters is enough to understand several important phenomena related to the propagation and modification of short laser pulses. (more complex systems may require higher order terms: ψ (3), ψ (4), etc)

42 42 Exercise: chirp filtering of a Gaussian pulse Calculate and interpret the effect of a chirp filter with chirp parameter b on a transform limited Gaussian pulse of amplitude A 10 and width τ 1. Determine the new duration, chirp parameter and amplitude. H e ( δω) = exp( i bδω 2 /4) A 1 ( t ) = A 10 exp( t 2 2 /τ ) 1 Steps: 1) Calculate the Fourier-transform à 1 (δω) 2) Multiply by the chirped filter to obtain à 2 (δω) 3) Compare à 2 (δω) with a general chirped Gaussian pulse

43 43 Chirp filtering of a Gaussian pulse: results Width Chirp parameter τ 2 = τ 1 1+ b 2 /τ 1 4 a 2 = b /τ 1 2 Amplitude A 20 = A ib /τ 1 2 The temporal width increases by a factor depending on b. b = τ 1 2 τ 2 = 2τ 1 b >> τ 1 2 τ 2 b /τ 1 The initial TL pulse becomes chirped with chirp parameter proportional to b. The spectral width remains the same.

44 44 Chirp filtering of a Gaussian pulse: results pulse width original transf. limited pulse chirp

45 45 Chirp filtering of a Gaussian pulse In this case the chirped filter is applied to an already chirped Gaussian pulse with chirp a 1 H e ( δω) = exp i bδω 2 /4 A 1 t ( ) ( )t 2 /τ 1 2 ( ) = A 10 exp 1 ia 1 ( ) Width Chirp parameter τ 2 = τ a 1 b / τ ( 2 1+ a 1 )b 2 4 / τ 1 a 2 = a 1 + ( a 1 )b / τ 1 The great difference now is that the output pulse can be shorter than the original one. The minimum width τ 0 and required chirp coeff. b are: τ 0 = τ 1 1+ a 1 2 b min = a 1 τ 2 0 = a 1 1+ a τ

46 46 Example: Compression and expansion of laser pulses (1) Input pulse: A 1 t ( )t 2 2 ( /τ 1 ) ( ) = A 10 exp 1 ia 1 pulse width a 1 = -1 b < 0 b > 0 a 1 = 1 b min = a 1 1+ a τ = τ 2 1 τ 0 = τ 1 1+ a 1 2 = τ 1 2 chirp b min = τ 0 2

47 47 Example: Compression and expansion of laser pulses (2) Input pulse: A 1 t ( )t 2 2 ( /τ 1 ) ( ) = A 10 exp 1 ia 1 pulse width b < 0 b > 0 a 1 = 1 b min = a 1 1+ a τ = τ 2 1 τ 0 = τ 1 1+ a 1 2 = τ 1 2 chirp b min = τ 0 2

48 48 In the mid-1980 s high power (~TW) lasers were very large Intensity = Energy Area time

49 49 Application: Chirped pulse amplification CPA was introduced in 1986 and represented a revolution in the amplification of very short laser pulses. TL pulse Stretching Amplification Compression

50 50 Typical CPA setup

51 Evolution of peak laser intensity 51

52 52 Examples of implementation of chirp filters Material dispersion Angular dispersion Interferometric dispersion Others: polarization dispersion, nonlinear dispersion

53 53 Pulse shaping Masks grating grating To generate pulses that control chemical reactions or other phenomena To generate trains of pulses for telecommunications To precompensate for distortions that occur in dispersive media