Ultrashort Laser Pulse Phenomena, 2nd Edition

Pulse Amplitude and Phase Reconstruction 477

9.4.3. Retrieval from Correlation and Spectrum

While being always symmetric, the shape of an intensity and interferometric

autocorrelation is (somewhat) sensitive to the pulse shape. It is conceivable that

the pulse spectrum can complement the information provided by the symmetric

second-order autocorrelations, to determine the signal shape. As an illustration

of this, Table 9.1 shows analytical expressions [2] for the pulse spectrum, the

intensity correlation, and the envelope of the interferometric correlation for vari-

ous pulse shapes. For some typical shapes of the temporal intensity profile given

in the first column, the spectral intensity (column 2) is used to compute the

duration–bandwidth product τ

p

v listed in column 3. The unit for the time t

is such that the functional dependence takes the simplest form in column 1.

The inverse of that time unit is used as unit of frequency . The most often

quoted parameter is the ratio of the FWHM τ

ac

of the intensity autocorrela-

tion (column 4) to the pulse duration τ

p

, and is given in column 5. Finally, the

upper and lower envelopes of the interferometric autocorrelation G

2

(τ) can be

reconstructed from the expressions given in column 6.

Table 9.1

Typical pulse shapes, spectra, intensity, and interferometric autocorrelations.

To condense the notation, x has been substituted for

2

3

τ , y for

4

7

τ , ch for cosh,

sh for sinh. τ

ac

is the FWHM of the intensity autocorrelation. τ

p

is the FWHM of

the pulse intensity given in column 1. In the last column,

Q =±4[τ ch2 τ −

3

2

ch

2

xshx(2 − ch2x)]/[sh

3

2x].

E

2

(t) |E()|

2

τ

p

vA

c

(τ) τ

ac

/τ

p

G

2

(τ) −[1 +3A

c

(τ)]

e

−t

2

e

−

2

0.441 e

−τ

2

/2

1.414 ±e

−(3/8)τ

2

sech

2

(t) sech

2



π

2



0.315

3τ(chτ − shτ)

sh

3

τ

1.543 ±

3(sh2τ − 2τ)

sh

3

τ

[e

t/(t−A)

+

e

−t/(t+A)

]

−1

A =

1

4

1 +1/

√

2

ch

15π

16

 + 1/

√

2

0.306

1

ch

3

8

15

τ

1.544 ±4

⎛

⎜

⎝

ch

4

15

τ

ch

8

15

τ

⎞

⎟

⎠

3

A =

1

2

sech

2π

4

 0.278

3sh4x −8τ

4sh

3

4

3

τ

1.549 ±Q

A =

3

4

1 −1/

√

2

ch

7π

16

 − 1/

√

2

0.221

2ch4y + 3

5ch

3

2y

1.570 ±4

ch

3

y(6ch2y −1)

5ch

3

2y

478 Diagnostic Techniques

As we have seen in the previous section, the interferometric autocorrelation

also carries information about the pulse chirp. At the same time these correla-

tion functions are one of the data sets that require relatively little experimental

effort. It is therefore tempting to explore the feasibility of obtaining ampli-

tude and phase of the optical pulse from such measurements. Indeed some

of the earliest successful retrievals of the complex field of fs pulses were

based on the simultaneous fitting of spectrum and interferometric autocorrela-

tion [2,4]. The reconstruction was facilitated after replacing the autocorrelation

by a cross-correlation of pulses of different duration. The result of that cross-

correlation approximated the longest of the two pulses. Figure 9.10 shows an

example of a Michelson interferometer unbalanced with a phase-only filter

(block of glass), and the recorded intensity and interferometric correlation

functions.

In most cases the problem reduces to the task of measuring the pulse spectrum

and suitable correlations and finding an amplitude and time-dependent phase that

fits the data best. Care has to be taken to guarantee a unique retrieval, which

is to avoid ambiguities hidden in the data sets (see, for example, Problem 3

at the end of this chapter). For this reason unbalanced correlators where a

Glass

D

a

KDP

400 0

Delay (fs)

400

0

Delay (fs)

300300

b

t

t

t

Downchirped

Pulse

Figure 9.10 Sketch of an asymmetric correlator to record second-order correlation functions

G

2

through SHG. An example of intensity (a) and interferometric (b) cross-correlation measured

with this setup is also shown. The input was an asymmetric downchirped pulse, which is compressed

in the arm containing a 5-cm BK7 block. Adapted from Diels et al. [2].

Pulse Amplitude and Phase Reconstruction 479

Measure spectrum:

Measure correlation(s):

Guess a spectral phase:

S()

C

k,m

()

()

Complex spectrum

S()S()e

i()

~~~

~

Electric field

E(t )FT

1

{S ()}

Search algorithm to find

new spectral phase

()

Calculate correlations

C

k,r

()

Retrieved pulse

E(t)



c

?

Compare calculated and

measured correlations

M

k1

 

k

Figure 9.11 Schematic diagram of the retrieval of phase and intensity from correlation and spectrum

only (PICASO) [36, 51].

linear optical element of known transfer function is inserted into one arm of the

correlator have been implemented [34, 36].

A possible retrieval algorithm is sketched in Figure 9.11. The pulse spec-

trum, S() =|E()|

2

, and a pulse correlation of type k or an ensemble of

M correlations, C

k,m

are measured. The retrieval starts by guessing a spectral

phase, ϕ(), which combined with the measured spectrum results in an initial

pulse



S()e

iϕ()

. This pulse is used to calculate the correlation(s) C

k,r

(τ

i

) that

are recorded in the measurement. A root mean square deviation of measured and

calculated correlation can be defined by



k

=

M



k=1

;

<

<

=

1

N

N



i=1

6

C

k,r

(τ

i

) − C

k,m

(τ

i

)

7

2

, (9.38)