The Generation of Ultrashort Laser Pulses II

Transcription

1 The Generation of Ultrashort Laser Pulses II The phase condition Trains of pulses the Shah function Laser modes and mode locking 1

2 There are 3 conditions for steady-state laser operation. Amplitude condition threshold slope efficiency Phase condition axial modes homogeneous vs. inhomogeneous gain media Transverse modes Hermite Gaussians the donut mode E(x,y) 2 x y 2

3 Phase condition collection of 4-level systems length L m Round-trip length L rt Steady-state condition #2: Phase is invariant after each round trip angle( E angle( E after before ) ) ( iωl / ) = 1 = exp c rt ω q = 2πc L rt q (q = an integer) An integer number of wavelengths must fit in the cavity. 3

4 Longitudinal modes ω q = 2πc L rt q (q = an integer) axial or longitudinal cavity modes laser gain profile Mode spacing: ν = c/l rt ω 0 fits does not fit But how does this translate to the case of short pulses? 4

5 Femtosecond lasers emit trains of identical pulses. Every time the laser pulse hits the output mirror, some of it emerges. Back mirror Output mirror Laser medium R = 100% R < 100% The output of a typical ultrafast laser is a train of identical very short pulses: I(t) I(t-2T) -3T -2T -T 0 T 2T 3T Intensity vs. time t where I(t) represents a single pulse intensity vs. time and T is the time between pulses. 5

6 The Shah Function The Shah function, III(t), is an infinitely long train of equally spaced delta-functions t III( t) δ ( t m) m= The symbol III is pronounced shah after the Cyrillic character III, which is said to have been modeled on the Hebrew letter (shin) which, in turn, may derive from the Egyptian, a hieroglyph depicting papyrus plants along the Nile. 6

7 The Fourier Transform of the Shah Function Y {III( t )} = III(t) = ( m= = exp( iωm) m= δ t m)exp( iωt) dt = δ ( t m)exp( iωt) dt m= If ω = 2nπ, where n is an integer, the sum diverges; otherwise, cancellation occurs. F{III(t)} t So: Y {III( t)} III( ω / 2 π ) ω 2π

8 The Shah Function and a Pulse Train An infinite train of identical pulses can be written: f(t) f(t-2t) -3T -2T -T 0 T 2T 3T E( t) = f ( t mt ) m= E(t) t where f(t) is the shape of each pulse and T is the time between pulses. But E(t) can also can be written: E( t) = III( t / T ) f ( t) convolution Proof: III( t / T ) f ( t) = δ ( t / T m) f ( t t ) dt To do the integral, set: t /T = m or t = mt m= = m= f ( t mt ) 8

9 The Fourier Transform of an Infinite Train of Pulses An infinite train of identical pulses can be written: E(t) = III(t/T) * f(t) f(t) f(t-2t) -3T -2T -T 0 T 2T 3T E(t) t The Convolution Theorem says that the Fourier Transform of a convolution is the product of the Fourier Transforms. So: F(ω) E% ( ω) E% ( ω) II I( ωt / 2π ) F( ω) ωτ / 2π The spacing between frequencies (modes) is then δω = 2π/T or δν = 1/T. 9

10 The Fourier Transform of a Finite Pulse Train A finite train of identical pulses can be written: E( t) = [III( t / T ) g( t) ] f ( t) where g(t) is a finite-width envelope over the pulse train. g(t) f(t) E(t) Use the fact that the Fourier transform of a product is a convolution F(ω) -3T -2T -T 0 T 2T 3T E% ( ω) [III( ωt / 2 π ) G( ω) ] F( ω) G(ω) E% ( ω) t -6π/T -4π/T -2π/T 0 2π/T 4π/T 6π/T ω 10

11 Actual Laser Modes A laser s frequencies are often called longitudinal modes. They re separated by 1/T = c/2l, where L is the length of the laser. Which modes lase depends on the gain and loss profiles. Intensity Frequency Here, additional narrowband filtering has yielded a single mode. 11

12 Mode-locked vs. non-mode-locked light Mode-locked pulse train: E% ( ω) = F( ω) III( ωt / 2 π ) F( ω) δ ( ω 2 π m / T ) = m= A train of short pulses Non-mode-locked pulse train: E% ( ω) = F( ω) exp( iϕ m) δ ( ω 2 π m / T ) m= Random phase for each mode = F( ω) exp( iϕ m) δ ( ω 2 π m / T ) m= A mess 12

13 Generating short pulses = Mode-locking Locking vs. not locking the phases of the laser modes (frequencies) Intensity vs. time Random phases Light bulb Time Intensity vs. time Locked phases Ultrashort pulse! Time 13

14 Ti:sapphire: how many modes lock? pump Typical cavity layout: length of path from M1 to M4 = 1.5 m 200 nm bandwidth! Wavelength (nm) Ti: sapphire crystal Therefore ν = c/l rt 100 MHz Q: How many different modes can oscillate simultaneously in a 1.5 meter Ti:sapphire laser? A: Gain bandwidth λ = 200 nm ν = (c/λ 2 ) λ ~ Hz ν bandwidth / ν mode = 10 6 modes That seems like a lot. Can this really happen? 14

15 Homogeneous gain media We have seen that the gain is given by: g( ω) = 1 2 N 1+ I / 0 I sat σ0 1+ ζ 2 where ( ω ω ) 2 0 ζ = γ independent of ω Suppose the gain is increased to a point where it equals the loss at a particular frequency, ω q which is one of the cavity mode frequencies. losses ω q 1 ω q ω q+1 laser gain profile Q: Suppose the gain is increased further. Can it be increased so that the mode at ω q+1 oscillates in steady state? A: In an ideal laser, NO! 15

16 Inhomogeneous gain media Suppose that the collection of 4-level systems do not all share the same ω 0 Consider a collection of sites, with fractional number between ω 0 and ω 0 + dω 0 : dn( ω dω 0 ) = Ng( ω0) The modified susceptibility is: 0 ( ω) = dω χ ( ω ω ) ( ) χ g 0 h ; 0 ω0 homogeneous packet inhomogeneous line packets are mutually independent - they can saturate independently! 16

17 Inhomogeneous broadening Lorentian homogeneous line shape ( ω) = dω χ ( ω ω ) χ e 0 h ; 0 ω0 ω 4ln(2) ω 0 2 Gaussian inhomogeneous distribution of width ω No closed-form solution For strong inhomogeneous broadening ( ω >> γ): χ"(ω) = Gaussian, with width = ω (NOT γ) χ'(ω): no simple form, but it resembles χ h '(ω) Examples: Nd:YAG - weak inhomogeneity Nd:glass - strong inhomogeneity Ti:sapphire - absurdly strong inhomogeneity R6G - intermediate case (?) 17

18 Hole burning Q: Suppose the gain is increased above threshold in an inhomogeneously broadened laser. Can it be increased so that the mode at ω q+1 oscillates cw? A: Yes! Each homogeneous packet saturates independently spectral hole burning multiple oscillating cavity modes losses a priori, these modes need not have any particular phase relationship to one another laser gain profile ω q 2 ω q 1 ω q ω q+1 ω q+2 ω q+3 18

19 There are 3 conditions for steady-state laser operation. Amplitude condition threshold slope efficiency Phase condition axial modes homogeneous vs. inhomogeneous gain media Transverse modes Hermite Gaussians the donut mode E(x,y) 2 x y 19

20 Condition on the transverse profile Steady-state condition #3: Transverse profile reproduces on each round trip "transverse modes": those which reproduce themselves on each round trip, except for overall amplitude and phase factors How would we determine these modes? An eigenvalue problem: ( ) ( ) ( ) β E x, y = K x, y;x, y E x, y dx dy nm nm 0 0 nm propagation kernel 20

21 Transverse modes Solutions are the product of two functions, one for each transverse dimension: (, ) = ( ) ( ) E x y u x u y nm n m where the u n s are Hermite Gaussians: Notation: 2 2x x un ( x) = H n exp 2 w w w = beam waist parameter "TEM" = transverse electric and magnetic "nm" = number of nodes along two principal axes 21

22 Transverse modes - examples E(x,y) 2 E(x,y) 2 x y x y 12 mode A superposition of the 10 and 01 modes: the donut mode TEM00 TEM01 TEM02 TEM