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Spectral Interferometry

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Definition: a class of interferometric methods, where interference in the frequency domain is exploited

German: spektrale Interferometrie

Categories: methods, optical metrology, light pulses

How to cite the article; suggest additional literature

The intensity profile of the optical spectrum of a signal, such as an ultrashort pulse, can easily be measured using some kind of spectrometer. For the more difficult task of retrieving the spectral phase, methods of Fourier transform spectral interferometry (also called frequency domain interferometry) provide various options.

The basic principle of spectral interferometry is that two pulses are interferometrically combined, and the optical spectrum of that combination is recorded. An essential detail is that one of the pulses is subject to a time delay τ. If the electric fields of the two pulses are described with phasors E1(ω) and E2(ω), the additional time delay translates into a phase factor exp(−i ωτ) for one of the pulses. Apart from irrelevant prefactors, the combined field is thus

optical spectrum in spectral interferometry

from which one can see that the optical spectrum is modulated (see Figure 1). The oscillating term has a phase

phase in spectral interferometry

which shows that there is an oscillation determined by the time delay, the phase of which is also influenced by the difference of the spectral phases of the two pulses.

Pulse Characterization Using a Reference Pulse

It is now possible to characterize an ultrashort pulse, if a reference pulse with a well-characterized spectral phase is available, the spectrum of which covers the full range of the spectrum of the pulse to be characterized. The time delay is chosen such that the period of spectral modulation can be well resolved with the given spectrometer, but is fast enough for an accurate phase determination. The phase of the recorded spectrum can be conveniently and precisely obtained with Fourier transform methods, applied to the spectral intensity curve.

spectral interference

Figure 1: Spectrum of two femtosecond pulses, where one pulse is nonlinearly chirped and delayed by 600 fs. The spectral modulation can be used to reconstruct the spectral phase of the chirped pulse.

A high sensitivity can be achieved, as such a heterodyne method works well even if the signal pulse energy is much lower than that of the reference pulse. Also, no nonlinear crystal is required, thus avoiding possible bandwidth-limiting effects. Finally, an accurate intensity calibration of the spectrometer is not required, since only the fast modulation of the spectrum is of interest, not the slow variation of intensity. However, the need for a suitable reference pulse is clearly a disadvantage.

Spectral Shearing Interferometry

A convenient way of obtaining a reference pulse can be to send a copy of the signal pulse through a sinusoidally modulated phase modulator at a time close to the zero crossing of the phase. (For example, an electro-optic modulator may be inserted in one arm of a Mach–Zehnder interferometer.) The nearly linear temporal phase modulation then corresponds to some spectral shift of magnitude δω. The spectrum of the combined pulses then has a modulation phase

phase in spectral shearing interferometry

where the approximate relation holds if δω is not too large. This shows that the spectral derivative of the phase of the signal pulse can be obtained, which is the frequency-dependent group delay.

This method, called spectral shearing interferometry [1, 10, 11], works well for relatively long pulses. For very short pulses, where the group delay variation within a pulse duration is small, it is difficult to obtain a sufficiently large spectral shift in the modulator. A larger spectral shear can be achieved by using a nonlinear interaction, as discussed in the following section.

Spectral Phase Interferometry for Direct Electric-field Reconstruction

Another method of spectral shearing interferometry, often applied for the complete characterization of ultrashort pulses, is called Spectral Phase Interferometry for Direct Electric-field Reconstruction (SPIDER) [4]. Here, the signal pulse is split into two identical copies with a significant temporal distance, so that there is no temporal overlap. Another pulse, derived from the same input pulse, is strongly temporally broadened by sending it through a highly dispersive optical element, such as a long block of glass or a pair of diffraction gratings.

SPIDER setup

Figure 2: Setup for ultrashort pulse characterization with the SPIDER method.

The long chirped pulse and the two copies of the signal pulse are then combined in a nonlinear crystal, where sum frequency generation occurs (see Figure 2). The two signal pulses overlap with different temporal portions of the chirped pulse, which have different optical frequencies, so that there is also a spectral shear between the two upconverted pulses. Therefore, the optical spectrum of the sum frequency signal reveals the temporally resolved group delay in a way which is analogous to that discussed above. From the group delay, it is easy to retrieve the frequency-dependent spectral phase, so that complete pulse characterization is achieved.

In comparison with frequency-resolved optical gating (FROG), the SPIDER method has various advantages and disadvantages:

A comprehensive comparison of SPIDER and FROG techniques is difficult, because there are many variants of both methods, which have specific advantages under certain circumstances.

Bibliography

[1]V. Wong and I. A. Walmsley, “Analysis of ultrashort pulse-shape measurement using linear interferometers”, Opt. Lett. 19 (4), 287 (1994)
[2]E. Tokunaga et al., “Femtosecond continuum interferometer for transient phase and transmission spectroscopy”, J. Opt. Soc. Am. B 13 (3), 496 (1996)
[3]C. X. Yu et al., “Direct measurement of self-phase shift due to fiber nonlinearity”, Opt. Lett. 23 (9), 679 (1998)
[4]C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses”, Opt. Lett. 23 (10), 792 (1998)
[5]L. Gallmann et al., “Characterization of sub-6-fs optical pulses with spectral phase interferometry for direct electric-field reconstruction”, Opt. Lett. 24 (18), 1314 (1999)
[6]L. Gallmann et al., “Techniques for the characterization of sub-10-fs optical pulses: a comparison”, Appl. Phys. B 70, S67 (2000)
[7]L. Gallmann et al., “Spatially resolved amplitude and phase characterization of femtosecond optical pulses”, Opt. Lett. 26 (2), 96 (2001)
[8]C. Dorrer and I. A. Walmsley, “Accuracy criterion for ultrashort pulse characterization techniques: application to spectral phase interferometry for direct electric field reconstruction”, J. Opt. Soc. Am. B 19 (5), 1019 (2002)
[9]W. Kornelis et al., “Single-shot kilohertz characterization of ultrashort pulses by spectral phase interferometry for direct electric-field reconstruction”, Opt. Lett. 28 (4), 281 (2003)
[10]C. Dorrer and I. Kang, “Highly sensitive direct characterization of femtosecond pulses by electro-optic spectral shearing interferometry”, Opt. Lett. 28 (6), 477 (2003)
[11]I. Kang et al., “Implementation of electro-optic spectral shearing interferometry for ultrashort pulse characterization”, Opt. Lett. 28 (22), 2264 (2003)
[12]A. S. Radunsky et al., “Simplified spectral phase interferometry for direct electric-field reconstruction by using a thick nonlinear crystal”, Opt. Lett. 31 (7), 1008 (2006)
[13]J. R. Birge et al., “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization”, Opt. Lett. 31 (13), 2063 (2006)
[14]S. P. Gorza et al., “Spectral shearing interferometry with spatially chirped replicas for measuring ultrashort pulses”, Opt. Express 15 (23), 15168 (2007)

(Suggest additional literature!)

See also: spectral phase, interferometers, spectrometers, optical heterodyne detection, frequency-resolved optical gating
and other articles in the categories methods, optical metrology, light pulses

In the RP Photonics Buyer's Guide, 5 suppliers for equipment for spectral interferometry are listed.

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