Deconvolving Auto- and Crosscorrelation Traces

When extracting the pulse width from an autocorrelation curve, two points have to be taken into account:

  1. the relation between the value obtained for the corresponding fit parameter and the full-width-half-maximum (FWHM) of the curve,
  2. and the relation between the width of a curve and the width of the curve obtained by convoluting the curve with itself.

1 Gaussian width and FWHM

When fitting a Gaussian curve, the model function is usually

t−t0 2 f (t) = A0e −( τ ) + Y0
(1)

where τ is the parameter controlling the width. The relationship to the FWHM is given by the condition

( ) − τ1∕τ2 2 ! 1 e = 2 (2) ( τ1∕2)2 τ = ln2 (3) √ ---- τ1∕2 = τ ln 2. (4)
The FWHM ΔT is then given by
√ ---- ΔT = 2τ1∕2 = τ2 ln2 ≈ τ ⋅ 1.67.
(5)

2 Convolving a Gaussian with itself

Assuming a laser pulse with a Gaussian envelope

2 I(t) = I e−(tτ) , 0
(6)

the autocorrelation curve can be described as

+∫∞ 2 −(t′)2 − (t′−t)2 ′ A (t) = I0 e τ e τ dt (7) −∞ +∫∞ ( ) 2 2t′2 − 2t′t + t2∕2 + t2∕2 ′ = I0 exp − ------------2---------- dt (8) −∞ τ + ∞ ( √ -- √ -)2 2 − 1 t2 ∫ 2 ⋅ t′ − t∕ 2 ′ = I0e 2( τ) exp − -------------- dt . (9) − ∞ τ
Since this integral is infinite, a constant offset in the integration variable does not change the value of the integral. Thus, the integral does not depend on t. In consequence, the obtained function is again a Gaussian curve (time dependent term before the integral) and a constant factor, given by the value of the infinite integral
1 t 2 −(√-t-)2 A (t) ∝ I02e−2(τ) = I20e 2⋅τ .
(10)

The width of this Gaussian curve is thus √ -- 2 times larger than the original curve.

Therefore, the laser field expressed by the FWHM ΔT reads

(2√ln2⋅t)2 I(t) = I0e − ΔT .
(11)

When fitting a Gaussian to an autocorrelation curve, the relation between fit parameter τ as given in (1) and FWHM ΔT of the pulse is

√ -- √ ----√ -- √ ----- ΔT = ΔA ∕ 2 = τ2 ln2∕ 2 = τ 2ln2 ≈ τ ⋅ 1.18.
(12)

3 Third Order Autocorrelation

In a third order autocorrelation one of the fields enters twice into the expression and the measured signal becomes

+∫ ∞ ( ′)2 ( ′ )2 ( ′)2 A (t) = I3 e−2 tτ e− t−τt dt′ ∝ e− 23 tτ 3 0 − ∞
(13)

where the integral has been evaluated as before. As FWHM ΔT3 for the deconvoluted pulse width we get therefore

∘ ---- √ ---∘ ---- ∘ -------- ΔT3 = ΔA ∕ 3∕2 = τ2 ln2 2∕3 = τ2 2ln2∕3 ≈ τ ⋅ 1.36.
(14)

4 Cross Correlation

A cross correlation of two pulses with Gaussian profile of duration τ1 and τ2, respectively, is described by a convolution of the two Gaussians

+∫∞ (t′)2 (t′−t)2 C (t) = I0,1I0,2 e− τ1 e− -τ2- dt′ (15) τ1τ2π −∞ +∫∞ [ 2 2 ′2 2′ 22 ] = I0,1I0,2 exp − (τ2-+-τ1)t--−-2τ1-tt +-τ1-t dt′ (16) τ1τ2π τ12τ22 −∞ [ ] I0,1I0,2 τ12t2-−-τ41t2∕(τ21 +-τ22) = τ τ π exp − τ 2τ 2 1 2 ⌊ ( ∘ -1-2---- ∘ -------)2⌋ +∫ ∞ t′ τ 2+ τ2 − τ2t∕ τ2+ τ 2 × exp |⌈− ------1----2----1------1---2---|⌉ dt′ (17) τ12τ22 − ∞ [ ] +∫ ∞ ( ∘ --------)2 I0,1I0,2 21-−-τ21∕(τ21-+-τ22) ′--τ12+-τ22- ′ = τ1τ2π exp − t τ2 exp − t τ1τ2 dt (18) 2 − ∞ I0,1I0,2 ( t2 ) = ∘------------exp − -2----2- . (19) π(τ21 + τ22) τ1 + τ2
The result is a Gaussian with a width given by the geometrical mean of the widths of the two convoluted pulses. Its amplitude is strictly monotonously increasing with decreasing pulse duration of either pulse when keeping the other one fixed. In case of τ1 ≫ τ2 the width is equal to τ1 and the peak intensity scales with 1∕τ1, and vice versa.