*2.6.2.1 Interferometric autocorrelation*

Setup for an interferometric autocorrelator is similar to the field autocorrelator above, with the following optics added: L: converging lens, SHG: secondharmonic generation crystal, F: spectral filter to block the fundamental wavelength. A nonlinear crystal can be used to generate the second harmonic at the output of a Michelson interferometer in a collinear geometry. In this case, the signal recorded by a slow detector (**Figure 12**).

$$I(\tau) = \int\_{-\infty}^{+\infty} |E(t) + E(t - \tau)|^2 dt \tag{28}$$

*I*ð Þ*τ* is called the interferometric autocorrelation. It contains some information about the phase of the pulse: the fringes in the autocorrelation trace wash out as the spectral phase becomes more complex (**Figure 13**).

#### **Figure 12.**

*Setup for an interferometric autocorrelator, similar to the field autocorrelator above, with the following optics added: L: Converging lens, SHG: Second-harmonicgeneration crystal, F: Spectral filter to block the fundamental wavelength.*

**Figure 13.**

*FROG reconstruction scheme. When both E1 and E2 have been unknown, then we dealwith blind-FROG problem. When E*<sup>1</sup> ¼ *E*<sup>2</sup> *then we have to do with SHG-FROG problem.*

#### *2.6.2.2 Spider*

The Spectral Phase Interferometry for Direct Electric-field Reconstruction technique (SPIDER) is based on spectral interferometry and needs no components which has to be shifted over the measurement process. From the signal *E t*ð Þ that should be characterized, the copy-signal is being generated by beam splitter

$$E(t-\tau)\exp\left[iw\_0\tau\right] \tag{29}$$

The time between the signal and copy itself has been established through fixed position at the optical delay-line. Then the copy of signal goes through phase filter (dispersive medium, for instance SF10 glass), so arises the signal

$$E\_M(t) = F^{-1}\left\{\tilde{E}(w) \exp\left[i\mathcal{Q}\_M(w)\right]\right\} \tag{30}$$

EM electrical field at point M

∅*M*: phase of electrical field

Through the phase filter electric field *E w* <sup>~</sup>ð Þ get additional spectral phase <sup>∅</sup>*M*ð Þ *<sup>w</sup>* , which corresponds to temporal extension *E t*ð Þ. From the signals

$$E\_M(t) \text{ and } E(t-\tau) \exp\left[iw\_0\tau\right] + E(t) \tag{31}$$

*2.6.2.2.1 Examples*

*Generation of two sheared replicas of the input pulse by non-linear interaction with a chirped pulse.*

*Femtosecond Laser Pulses: Generation, Measurement and Propagation*

*DOI: http://dx.doi.org/10.5772/intechopen.95978*

**Figure 15.**

**17**

**Figure 14.**

*Main steps of SPIDER technique.*

SFG-Signal can be created

$$E\_{\rm SFG}(t) \propto E\_M(t) \left\{ E(t-\tau) \exp\left[i w\_0 \tau\right] + E(t) \right\} \tag{32}$$

$$\dot{\rho} = E\_M(t)E(t-\tau)\exp\left[i\nu\_0\tau\right] + E\_M(t)E(t) \tag{33}$$

As square law detectors are not sensitive to the phase, the measurement of the intensity (whether it is spatial or spectral) is an easy task but the measurement of the phase needs indirect solutions (**Figure 14**).

Spectral interferometry allows us obtain difference between two spectral phases. To the spectral interferometry spectrum, one should apply fast Fourier transform and as the product one will achieve in form of one center peak and two sidebands lower peaks in the time domain (**Figure 15**).

Centered peak contains only spectrum information. One filter out two peaks and from existing one can receive spectral phase difference by applied inverse fast Fourier transform. To the main advantages of the SPIDER method can be counted following properties: pulse retrieval is direct (non-iterative), minimal data are required: only one spectrum yields spectral phase [20].

*Femtosecond Laser Pulses: Generation, Measurement and Propagation DOI: http://dx.doi.org/10.5772/intechopen.95978*

**Figure 14.** *Main steps of SPIDER technique.*

*2.6.2.2 Spider*

**Figure 13.**

The Spectral Phase Interferometry for Direct Electric-field Reconstruction tech-

The time between the signal and copy itself has been established through fixed position at the optical delay-line. Then the copy of signal goes through phase filter

Through the phase filter electric field *E w* <sup>~</sup>ð Þ get additional spectral phase <sup>∅</sup>*M*ð Þ *<sup>w</sup>* ,

As square law detectors are not sensitive to the phase, the measurement of the intensity (whether it is spatial or spectral) is an easy task but the measurement of

Spectral interferometry allows us obtain difference between two spectral phases. To the spectral interferometry spectrum, one should apply fast Fourier transform and as the product one will achieve in form of one center peak and two sidebands

Centered peak contains only spectrum information. One filter out two peaks and

from existing one can receive spectral phase difference by applied inverse fast Fourier transform. To the main advantages of the SPIDER method can be counted following properties: pulse retrieval is direct (non-iterative), minimal data are

*E t*ð Þ � *τ exp iw*½ � <sup>0</sup>*τ* (29)

*EM*ðÞ¼ *<sup>t</sup> <sup>F</sup>*�<sup>1</sup> *E w* <sup>~</sup>ð Þ*exp i*½ � <sup>∅</sup>*M*ð Þ *<sup>w</sup>* (30)

*EM*ð Þ*t and E t*ð Þ � *τ exp iw*½ �þ <sup>0</sup>*τ E t*ð Þ (31)

*ESFG*ð Þ*t* ∝*EM*ð Þ*t* f g *E t*ð Þ � *τ exp iw*½ �þ <sup>0</sup>*τ E t*ð Þ (32) ¼ *EM*ð Þ*t E t*ð Þ � *τ exp iw*½ �þ <sup>0</sup>*τ EM*ð Þ*t E t*ð Þ (33)

nique (SPIDER) is based on spectral interferometry and needs no components which has to be shifted over the measurement process. From the signal *E t*ð Þ that should be characterized, the copy-signal is being generated by beam splitter

*FROG reconstruction scheme. When both E1 and E2 have been unknown, then we dealwith blind-FROG*

(dispersive medium, for instance SF10 glass), so arises the signal

*problem. When E*<sup>1</sup> ¼ *E*<sup>2</sup> *then we have to do with SHG-FROG problem.*

*Recent Advances in Numerical Simulations*

which corresponds to temporal extension *E t*ð Þ. From the signals

EM electrical field at point M ∅*M*: phase of electrical field

SFG-Signal can be created

the phase needs indirect solutions (**Figure 14**).

lower peaks in the time domain (**Figure 15**).

**16**

required: only one spectrum yields spectral phase [20].

*Generation of two sheared replicas of the input pulse by non-linear interaction with a chirped pulse.*
