The spectrogram in acoustics

Transcription

1 Measuring the power spectrum at various delays gives the spectrogram 2 S ω, τ = dd E t g t τ e iii The spectrogram in acoustics E ssssss t, τ = E t g t τ where g t is a variable gating function Frequency Gate Time The gate selects the wave within a preset time interval. The FF 2 of this part of the wave is the instantaneous power spectrum. All together, they form the spectrogram (here, the musical score)

2 The spectrogram in ultrafast optics In practice, we look to measure the spectrum of each temporal slice of the ultrafast pulse The result would look like this:

3 Gate functions in frequency-resolved optical gating (FROG) We measure S ω, τ = E t g t τ e iii dd 2 The narrow-gate limit: if g(t) is much narrower than E t or g t τ δ(t τ), then S ω, τ = E τ e iii 2 = E (τ) 2. The spectrogram is proportional to the envelope in the time domain. The phase lost, just like in intensity AC. Intuitively, a short gate increases the spectrum of E t g t τ making the frequency shifts of say, linear chirps, difficult to detect. The FROG spectrogram becomes insensitive to chirp. A narrow gate is not a good idea. The wide-gate limit: if g t is much wider than E(t), it becomes a constant in S ω, τ = g 2 E t e iii dd g 2 E ω 2 The spectrogram is proportional to the envelope in the frequency domain. The pulse phase is again lost and the measurement becomes insensitive to chirp. Conclusion: the gate should have an extent approximately equal to that of the pulse being measured 2 = Measuring the spectrogram of an ultrafast pulse can be done with several types of optical gates, derived from the pulse itself g t τ = E t τ 2 in the polarization-gating geometry g = E t E (t τ) (self-diffraction) These are examples of χ 3 processes These gates make the spectrogram not necessarily even (a good consequence) but less intense (not good) g = E(t τ) (SHG) This is a χ 2 process, which has more sensitivity, since χ 2 χ 3 E, but (E sss ) FFFF is necessarily even (the pulse reversal ambiguity). Side note: this ambiguity can be resolved by adding positive chirp to the pulse and measuring again.

4 FROG geometries g t τ = E t τ 2 in the polarization-gating geometry The phase e ii t τ of the field does not affect the gate function, as we want a gate that changes with delay τ through the pulse envelope function E 0 t but not with t in the phase g = E t E (t τ) (self-diffraction) The complex conjugation makes the time-dependent phase factor e iii e ii t τ = e iii. The quickly varying ωω phase cancels out. As in the previous case, this assures that the gate (here the grating) seen by the pulse being measured does change as long as the delay τ is held fixed. Two pulses interfering in the crystal change its index of refraction and therefore create a diffraction grating. This grating diffracts the pulse that is measured. The polarization of the pulse passing through the crossed polarizers P is rotated if the delayed pulse arrives at the same time in the crystal. This gives a nonzero intensity at the spectrometer. g = E(t τ) (SHG) has the same geometry as intensity AC

5 ω τ FROG spectrograms of chirped pulses NL chirp Multiple pulses showing beats in τ and ω Because SHG FROG signal is symmetric in time it cannot tell the difference between two reversed pulses SHG FROG and Grenouille

6 From the spectrogram to the pulse shape E sss = E t E t τ 2 We introduce the FT of the function E sss t, τ S ω, τ = E t g t τ e iii dd 2 = E sss t, τ e iii dd 2 = dω E sss t, Ω e iiω e iii dd 2 This reduces the problem to a two-dimensional FT of obtaining E sss t, Ω from S(ω, τ) This problem has an unique solution E sss (t, Ω) In contrast, there is no unique solution in the 1D FT case, because the phase is lost when taking the absolute value. Once E sss t, Ω is known, we can obtain E t from E t E t τ 2 = E sss t, Ω e iτω dω The pulse shape can be uniquely obtained from the spectrogram In practice, an iterative process is used To find the pulse shape from the measured spectrogram, a program calculates E sss t, τ e iii dd from a starting E(t), replaces the magnitude of this complex function with S mmmm ω, τ, and calculates E t from this improved E sss (ω, τ) The process is repeated until convergence is observed (which is assured)

7 Grenouille along the horizontal direction Measuring the spectrum in FROG requires a spectrometer. This is not necessary in a different measurement technique called Grenouille, which applies the SHG-FROG Also, there are no multiple beams to align Beam is widened in cross-section before the 1 st lens. This avoids biases in the analysis, which assumes that the initial beam has an approximately constant intensity along the horizontal and vertical directions. The two beams that combine in the NL crystal are made by bi-prism refraction There is no need to scan one pulse with respect to the other because, as one moves sideways in the crystal, one beam will arrive earlier than the other This means that the delay τ of the spectrogram is mapped onto the horizontal direction The intensity along a camera horizontal direction is proportional to S(ω 0, τ) at fixed frequency ω 0.

8 Grenouille along the vertical direction A thin crystal has a relatively relaxed phase-matching (PM) condition. The orientation of the crystal is important but a slight misalignment will not change its effective length too much and the crystal will still give an intense SHG beam A thick crystal has a more stringent phase matching condition This means that the slightest change in the crystal orientation, which modifies its effective length, would have to be compensated by a change in the wavelength of light In practice, with an incident beam that converges on the crystal, this will result in SHG beams of slightly different frequencies emerging in different directions from the crystal The SHG beam has been spread along the vertical direction according to its wavelength The intensity along the camera vertical direction is proportional to S(ω, τ 0 ) at fixed delay τ 0. This replaces the spectrometer. In summary, the 2D intensity at the camera is proportional to S ω, τ Algorithms are necessary to convert from S ω, τ for E t, just like for SHG FROG The direction of the SHG beam depends on its frequency. The frequency is slightly different because the initial beam has a finite bandwidth

9 The crystal and detecting spatial chirp in a beam Grenouille requires a crystal that (1) spreads the SHG frequencies along the vertical direction. This happens if GGG L = v gg v gg L τ P The group velocity mismatch is defined as GGG = 1 v g λ 2 1 v g λ (2) does not distort the pulse too much with a positive chirp. This happens if GGG L 1 Note: GGG is defined here as GGG = 1 v g λ BB 2 1 v g λ+ BB 2 = Δω 2 k ω2 [with one extra Δω factor compared to class 1] BB If GGG TTT (time-bandwidth product) both conditions can be satisfied with one thickness L GGG Remarkably, this condition can be achieved for typical TTT 1 with a BBO crystal 2-5 mm in thickness The analysis required uniform starting beams along the vertical and horizontal directions One possible distortion is spatial chirp, when the pulse frequency changes slightly across the beam In other words, this condition can be detected with Grenouille A tilted spectrogram indicates spatial chirp (under normal conditions it should be symmetric with respect to a vertical line passing through its center, just as for SHG FROG)

10 SPIDER More complex setups exists A common one (SPIDER) interferes in a crystal two pulses with a variable delay τ with different parts of a highly-stretched version of the initial pulse It finds the spectral phase τ ω In most cases we have TTT 1 and the Grenouille method gives the same results as more complex setups