Spectral Phase

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

The electric field of an optical pulse may be described in the time domain or in the frequency domain. In the frequency domain, it can be of interest to know not only the power spectral density (i.e., the intensity spectrum) but also the spectral phase. This is defined as the phase of the electric field in the frequency domain, i.e., the complex phase of the function

Complete pulse characterization includes measuring not only the optical spectrum, i.e. the squared modulus of <$E(\nu )$>, but also the spectral phase, which contains additional information. This is possible e.g. with the methods of frequency-resolved optical gating (FROG) and spectral phase interferometry for direct electric-field reconstruction (SPIDER, → spectral phase interferometry).

Note that there are different sign conventions in wave optics; the equation above is for physicists' convention.

Spectral Phase and Group Delay

The group delay for light in an optical component or setup can be defined as the derivative of the spectral phase delay with respect to angular optical frequency:

That can be understood by considering a light pulse, where the peak intensity is found at a time where all spectral components are in phase. After passage through an optical component, leading to frequency-dependent phase changes, that condition is no longer fulfilled at the original time of the pulse peak, but at a later time, where the spectral components again acquire the same phase. That temporal shift of the pulse is determined by the group delay provided that the underlying linear approximation is valid – i.e., possibly not for broadband pulses experiencing more complex changes in spectral phase.

Examples

It is instructive to consider the changes in spectral phase associated with certain operations:

• A constant change in temporal phase translates directly into the same change in the spectral phase (for time-dependent phase changes, the relation is much less obvious), and to no group delay.
• A time delay by <$T$> corresponds to a change in spectral phase which is <$2\pi \: \nu \: T$>, i.e. proportional to the optical frequency.
• Chromatic dispersion directly affects the spectral phase and also causes a group delay. For example, the effect of third-order dispersion corresponds to adding a term to the spectral phase which varies with the third power of the frequency offset.

When the spectral phase of a pulse is constant or depends linearly on the frequency, the pulse is unchirped, which implies that it is at the transform limit. A chirp in the time domain is associated with a nonlinear frequency dependence of the spectral phase. A dispersive pulse compressor basically has the task of applying spectral phase shifts so that the resulting spectral phase is constant (or changes only linearly with frequency). The deviations from a flat spectral phase are more informative measure of the quality of pulse compression than e.g. just the pulse duration achieved.

The spectral phase can be useful for understanding the phenomenon of spectral interference. For example, consider two identical pulses with a relative time delay <$T$>. The difference in spectral phase, which is linear in frequency (see above), causes a spectral modulation. See the article on spectral phase interferometry for more details.

Modifying the Spectral Phase

There are pulse shapers which can be used to modify the spectral phase of pulses. Such a setup consists of, e.g., a first diffraction grating to separate different frequency components spatially, a liquid crystal modulator for applying position-dependent phase shifts, and a second diffraction grating to recombine the frequency components.

By properly adjusting all the phase values, it is possible e.g. to obtain transform-limited pulses, being as short as the given spectral width allows, or to form longer pulses with complicated temporal shapes. Conditions for such capabilities are that the full optical bandwidth can be processed, and that the spectral resolution (related to the maximum occurring group delay) is sufficiently high. On the other hand, a fast optical modulator is not required.

More to Learn

Encyclopedia articles:

• optical phase
• chirp
• transform limit
• pulse characterization
• spectral phase interferometry

Bibliography

(Suggest additional literature!)

Suppliers

The RP Photonics Buyer's Guide contains 31 suppliers for pulse characterization instruments. Among them:

Edmund Optics

Our compact ultrafast autocorrelator is used to characterize ultrafast laser pulses originating from Ti:sapphire and Yb:doped lasers. Featuring a built-in two-photon absorption (TPA) detector, this autocorrelator is ideal for measuring ultrafast femtosecond and picosecond laser pulses at wavelengths from 700 to 1100 nm. The highly sensitive TPA detector allows for measurements of ultrafast laser pulses with high sensitivity by eliminating the need for angle tuning of the SHG nonlinear crystal.

ALPHALAS

Ultrafast photodetectors from ALPHALAS in combination with high-speed oscilloscopes are the best alternative for measurement of optical waveforms with spectral coverage from 170 to 2600 nm (VUV to IR). For example, photodetectors with rise time 10 ps and bandwidth 30 GHz in combination with 50 GHz sampling oscilloscope can be successfully used to measure optical pulse widths down to 10 ps using deconvolution. Configurations of the photodetectors include free-space, fiber receptacle or SM-fiber-pigtailed options and have compact metal housings for noise immunity. The UV-extended versions of the Si photodiodes are the only commercial products that cover the spectral range from 170 to 1100 nm with a rise time < 50 ps. For maximum flexibility, most models are not internally terminated. A 50 Ohm external termination supports the specified highest speed operation.

RPMC Lasers

Serving North America, RPMC Lasers offers a compact, state-of-the-art, real-time, ultrashort laser pulse characterization device, designed to provide high-resolution measurements for ultrafast oscillators and amplifiers. With input specifications including 80 fs – 4 ps pulses, 1 kHz – 200 MHz rep. rate, and 1010 nm – 1060 nm wavelength range, it offers unparalleled precision in ultra-short pulse measurements, providing flexibility for a wide range of laser characterization. A high-resolution spectrometer provides detailed spectral information about their pulses, while real-time data acquisition & display help users monitor and adjust processes on the fly.

APE

APE offers a range of products for pulse characterization in the picosecond and femtosecond domain:

• autocorrelators
• FROG
• SPIDER
• spectrometers

Femto Easy

Femto Easy offers different kinds of devices for the characterization of ultrashort light pulses:

• Single-shot and scanning autocorrelators are easy tools for measuring pulse durations.
• FROG devices allow for full pulse characterization. They are also available in single-shot and scanning versions.

All devices are optimized for easy installation and handling.

Fluence

The Blueback advanced ultrashort laser pulse characterization device is a real-time ultrashort laser pulse characterization device specifically engineered to provide a high-resolution measurement for ultrafast oscillators and amplifiers. It is an essential piece of equipment for everyone who depends on accurate information about properties of their ultrashort pulses. With Fluence Blueback you get more than just a single result. You can watch your pulse evolving in real time.

Thorlabs

The FSAC benchtop interferometric autocorrelator manufactured by Thorlabs is designed to characterize ultrafast pulse durations from 15 – 1,000 fs in the 650 – 1100 nm range. This autocorrelator for use with femtosecond lasers compliments our ultrafast family of lasers, amplifiers, and specialized optics, including nonlinear crystals, chirped mirrors, low GDD mirrors/beamsplitters, and dispersion compensating fiber.

Quantifi Photonics

Quantifi Photonics' IQFROG Frequency-Resolved Optical Gating Pulse Analyzer is a spectrally resolved Second Harmonic Generation (SHG) autocorrelator suitable for intensity and phase measurement for pulses 300 fs to 50 ps long.

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