Self-phase Modulation
Author: the photonics expert Dr. Rüdiger Paschotta (RP)
Acronym: SPM
Definition: nonlinear phase modulation of a beam, caused by its own intensity via the Kerr effect
Categories: 
Related: Nonlinear Pulse Compression in a FiberKerr effectKerr lenssolitonscross-phase modulationself-steepeningself-focusingmodulational instabilitySelf-phase Modulation Causes Spectral Broadening – Does it Really?
Opposite term: cross-phase modulation
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DOI: 10.61835/pqf Cite the article: BibTex BibLaTex plain textHTML Link to this page! LinkedIn
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What is Self-phase Modulation?
Due to the Kerr effect, high optical intensity in a medium (e.g. an optical fiber) causes a nonlinear phase delay of the propagating light. That phase delay has the same temporal shape as the optical intensity. In simple models, the effect is described as a nonlinear change in the refractive index:
($\Delta n = {n_2}\;I$)with the nonlinear index ($n_2$) and the optical intensity ($I$). In the context of self-phase modulation, the emphasis is on the temporal dependence of the phase shift, whereas the transverse dependence for some beam profile leads to the phenomenon of self-focusing.
Of course, the nonlinear phase delay comes in addition to the linear phase delay.
Note that the description of self-phase modulation with a time-dependent refractive index is somewhat simplified, and accurate only for pulses which are not too short. In more extreme situations, extended models are required, which also take into account the effect of self-steepening — a change of the temporal pulse shape even in the absence of chromatic dispersion. That may be interpreted as a nonlinear modification of the group velocity. There can also be higher-order nonlinear effects, leading to more complicated propagation phenomena.
Effects on Optical Pulses
Time-dependent Phase Shift
If an optical pulse is transmitted through a medium, the Kerr effect causes a time-dependent phase shift according to the time-dependent pulse intensity. In this way, an initial unchirped optical pulse acquires a so-called chirp, i.e., a temporally varying instantaneous frequency.
The central part of the pulse exhibits an up-chirp.
For a Gaussian beam with beam radius ($w$) in a homogeneous medium with length ($L$), the on-axis phase change per unit optical power is described by the proportionality constant
(${\gamma _{{\textrm{SPM}}}} = \frac{{2\pi }}{\lambda } n_2 L\;{\left( {\frac{\pi }{2}{w^2}} \right)^{ - 1}} = \frac{4 \: n_2 \: L}{{\lambda \;{w^2}}}$)(In some cases, it may be more convenient to omit the factor ($L$), obtaining the phase change per unit optical power and unit length.)
Note that coefficients which are a factor of two smaller sometimes occur in the literature, which can have different reasons:
- An incorrect (2 times too low) equation for the peak intensity of a Gaussian beam is sometimes used. With that, one underestimates the on-axis phase change.
- One may calculate the overall nonlinear phase shift of a Gaussian-shaped waveguide mode, where one gets a kind of averaging between higher- and lower-intensity parts. That reduces the nonlinear phase shift by a factor of 2.
Spectral Changes
The time-dependent phase change caused by SPM is associated with a modification of the optical spectrum:
- If the pulse is initially unchirped or up-chirped, SPM leads to spectral broadening (an increase in optical bandwidth).
- Spectral compression can result if the initial pulse is downchirped (always assuming a positive nonlinear index).
For strong SPM, the optical spectrum can exhibit strong oscillations (see Figure 2). The reason for the oscillatory character is essentially that the instantaneous frequency undergoes strong excursions, so that in general there are contributions from two different times to the Fourier integral for a given frequency component. Depending on the exact frequency, these contributions may constructively add up or cancel each other (see also Ref. [7]).
Simulations on Pulse Propagation
Ultrashort pulses change in complicated ways when propagating through a fiber, for example. The fiber nonlinearity may not only cause self-phase modulation, but also four-wave mixing and Raman scattering, for example. A suitable simulator is essential for getting complete insight — not only on the resulting output pulses, but also on the pulses at any location in your system. The RP Fiber Power software is an ideal tool for such work.
Spatial Effects
For a propagating laser beam, the optical intensity is not only time-dependent, but also spatially dependent. The nonlinear phase changes are then also spatially dependent, and that leads to nonlinear self-focusing.
Self-phase Modulation in Different Situations
Optical Fibers
In optical fibers, SPM can be the dominant effect on an ultrashort pulse if the peak power is high (leading to strong SPM) while the chromatic dispersion is weak, so that the pulse duration remains approximately constant. Figure 3 shows an example case where that assumption is well fulfilled within the first 30 mm of fiber; here, the overall spectral width rises approximately linearly with the propagation distance. Thereafter, it grows faster because anomalous dispersion leads to pulse compression and thus to an increased peak power and an enhanced nonlinear interaction.
In optical fibers with anomalous chromatic dispersion, the chirp from self-phase modulation may be compensated by dispersion; this can lead to the formation of solitons. In the case of fundamental solitons in a lossless fiber, the spectral width of the pulses stays constant during propagation, despite the SPM effect.
In optical fibers with normal dispersion, a modulational instability can occur. That can also contribute to pulse break-up in supercontinuum generation.
Nonlinear Pulse Compression in a Fiber
We explore how we can spectrally broaden light pulses by self-phase modulation in a fiber and subsequently compress the pulses using a dispersive element. A substantial reduction in pulse duration by more than an order of magnitude is easily achieved, while the pulse quality is often not ideal.
Self-phase Modulation in Semiconductors via Carrier Density Changes
The term self-phase modulation is occasionally used outside the context of the Kerr effect, when other effects cause intensity-dependent phase changes. In particular, this is the case in semiconductor lasers and semiconductor optical amplifiers, where a high signal intensity can reduce the carrier densities, which in turn lead to a modification of the refractive index and thus the phase change per unit length during propagation. Comparing that effect with SPM via the Kerr effect, there is an important difference: such carrier-related phase changes do not simply follow the temporal intensity profile because the carrier densities do not instantly adjust to modified intensities. That aspect is pronounced for pulse durations below the relaxation time of the carriers, which is typically in the range of picoseconds to a few nanoseconds.
Mode-locked Lasers
Self-phase modulation has important effects in mode-locked femtosecond lasers. It results mainly from the Kerr nonlinearity of the gain medium, although for very long laser resonators even the Kerr nonlinearity of air can be relevant [5]. Without chromatic dispersion, the nonlinear phase shifts can be so strong that stable operation is no longer possible: the laser would not reach a steady state with well-defined pulses. In that case, (quasi-)soliton mode locking [4] is a good solution, where a balance of self-phase modulation and dispersion is utilized, similar to the situation of solitons in fibers.
Self-phase Modulation via Cascaded Nonlinearities
Strong self-phase modulation can also arise from cascaded ($\chi^{(2)}$) nonlinearities. Basically this means that a non-phase-matched nonlinear interaction leads to frequency doubling, but with subsequent backconversion. In effect, there is little power conversion to other wavelengths, but the phase changes on the original wave can be substantial. This effect may also be used to compensate self-phase modulation from other origins [6].
Frequently Asked Questions
This FAQ section was generated with AI based on the article content and has been reviewed by the article’s author (RP).
What is self-phase modulation (SPM)?
Self-phase modulation is a nonlinear optical effect where an intense light pulse modifies the refractive index of the medium it propagates through. This induces a phase shift on the pulse itself, which has the same temporal shape as the pulse's optical intensity.
What causes self-phase modulation?
SPM is caused by the optical Kerr effect. Due to this effect, the refractive index of a material changes in proportion to the applied optical intensity, a phenomenon described by the nonlinear index ($n_2$).
How does SPM affect an optical pulse?
SPM imposes a time-dependent phase shift on an optical pulse. This results in a time-dependent instantaneous frequency, a phenomenon called chirp. For an initially unchirped pulse, this typically leads to spectral broadening.
What is spectral broadening from SPM?
The chirp induced by SPM means that new frequency components are generated, which increases the optical bandwidth of the pulse. For strong SPM, the broadened spectrum can develop a characteristic oscillatory structure.
How is SPM related to self-focusing?
Both effects arise from the Kerr effect. SPM is the temporal effect, caused by the time-varying intensity of a pulse. Self-focusing is the corresponding spatial effect, caused by the transverse intensity profile of a beam, which creates a focusing lens.
What is the role of SPM in optical fibers?
In optical fibers, SPM's interplay with chromatic dispersion is crucial. In anomalous dispersion regimes, it can be balanced by dispersion to form stable solitons. It is also a key mechanism for supercontinuum generation.
Where is self-phase modulation important?
SPM is a fundamental effect in mode-locked lasers, particularly for soliton mode locking. It is also utilized for pulse compression and is the initial step for supercontinuum generation.
Suppliers
Sponsored content: The RP Photonics Buyer's Guide contains 26 suppliers for pulse compressors. Among them:
SAVANNA-HP, by UltraFast Innovations (UFI®), is a pulse compressor based on a stretched-flexible hollow-core fiber (SF-HCF). It has been developed in collaboration with the Institute for Nanophotonics in Göttingen. It spectrally broadens high-energy femtosecond input pulses by nonlinear interaction with a noble gas of adjustable gas pressure inside a hollow fiber and subsequently compresses the pulse using chirped mirror technology from UltraFast Innovations (UFI).
The state-of-the-art SF-HCF technique allows nearly ideal waveguiding, reducing the losses to a minimum and allowing the application of self-phase modulation over an interaction length of up to 8 m. It can handle extremely intense input pulses with a few tens of mJ pulses and an average power of up to 20 W — with active cooling, even up to several hundred watts. With this, we provide an unmatched compression unit for today's state-of-the-art lasers.
Our pulse stretcher and compressor fiber Bragg gratings are used in chirped-pulse amplification (CPA) systems. They exhibit low insertion loss, allowing for high system efficiency. O/E Land Inc. offers both standard and custom-made pulse stretching and compression fiber Bragg grating products.
The new few-cycle flexible hollow core fiber system allows you to choose various fiber lengths and inner diameters to achieve a desired nonlinear effect. Experimentally measured transmission for multi-mJ femtosecond pulses ranges between 50% and > 90%, depending on the application.
The most versatile choice for laser pulse post compression: The few-cycle hollow core fiber supports input energies from 50 μJ to 100 mJ, up to 20 times compression and transmission > 90% while keeping the footprint and optical path length at a minimum.
We offer different add-on pulse compression modules (MIKSs) to spectrally broaden and temporally shorten the pulses from your picosecond or femtosecond laser. The modules are compatible with nearly all commercial ultrafast lasers. Pulse shortening factors of 5× to 10× are easily reachable in a single stage with over 90% transmission. The core of our technology is nonlinear spectral broadening in multipass cells. For example, MIKS1_S module shortens the input pulses with 200–400 fs pulse duration and 1–200 μJ energy down to < 50 fs with extremely high transmission of over 90%. This module can be shipped to you and easily installed remotely.
Thorlabs manufactures a suite of options for dispersion management, including a pre-compensation module, dispersion compensating fiber, chirped mirrors, and low GDD optics. For ultrafast applications where dispersion must be well known and managed, Thorlabs’ portfolio includes a robust benchtop white light interferometer for characterizing reflective and transmissive dispersive properties of optics and coatings. Using two different detectors, the Chromatis™ dispersion measurement system is capable of measurements in the 500–1650 nm range, providing a means for measuring optics used for common femtosecond systems, including Ti:sapphire systems as well as 1 µm and 1550 nm oscillators. The Chromatis complements our ultrafast family of lasers, amplifiers, and specialized optics including nonlinear crystals, chirped mirrors, low GDD mirrors/beamsplitters, and dispersion compensating fiber.
FastLas is an incredible tool to change your standard ultra-short pulse laser into an exceptional very Ultra-Short Pulse (USP) with a pulse duration at the output of the system less than 50 fs.
- compatible with current USP lasers
- nonlinearity, broadening and compression is managed thanks to a gas filled fiber
- water cooling allows to shorten pulses at high energy
- responsive and efficient technical support for installation
The FastLas is equipped with a pre-alignment system to facilitate the installation and the injection of the customer’s laser signal into the FastLas.
Geola offers phase-conjugating cells which are suitable for pulse compressors achieving very high peak intensities. This is useful for applications like material processing, laser machining, and certain scientific experiments.
With our software RP Fiber Power, you can simulate and optimize pulse compression in any detail. For example, optimize an amplifier system together with the final pulse compressor such that optimum compression results.
The APE femtoControl is a compact, motorized dispersion compensation unit for optimization of the duration of femtosecond laser pulses.
femtoControl compensates for material dispersion by applying the inverse amount of dispersion to the pulse. This is generated by a pair of prisms on motorized translation stages allowing continuous adjustment of the pulse length.
The femtoControl product:
- femtoControl Ti:Sa: 680 nm to 1080 nm
- femtoControl NIR: 650 nm to 1320 nm
- femtoControl IR: 1150 nm to 2500 nm
- femtoControl Kit: flexible solution for customized dispersion and wavelength ranges
Bibliography
| [1] | F. Shimizu, “Frequency broadening in liquids by a short light pulse”, Phys. Rev. Lett. 19 (19), 1097 (1967); doi:10.1103/PhysRevLett.19.1097 (first demonstration of self-phase modulation) |
| [2] | R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses”, Phys. Rev. Lett. 24 (11), 592 (1970); doi:10.1103/PhysRevLett.24.592 |
| [3] | R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers”, Phys. Rev. A 17 (4), 1448 (1978); doi:10.1103/PhysRevA.17.1448 |
| [4] | F. X. Kärtner et al., “Stabilization of solitonlike pulses with a slow saturable absorber”, Opt. Lett. 20 (1), 16 (1995); doi:10.1364/OL.20.000016 |
| [5] | S. V. Marchese et al., “Pulse energy scaling to 5 μJ from a femtosecond thin-disk laser”, Opt. Lett. 31 (18), 2728 (2006); doi:10.1364/OL.31.002728 |
| [6] | F. Saltarelli et al., “Self-phase modulation cancellation in a high-power ultrafast thin-disk laser oscillator”, Optica 5 (12), 1603 (2018); doi:10.1364/OPTICA.5.001603 |
| [7] | R. Paschotta, "Effect of self-phase modulation on the pulse bandwidth" |
| [8] | R. Paschotta, tutorial on "Passive Fiber Optics", Part 11: Nonlinearities of Fibers |
(Suggest additional literature!)
Questions and Comments from Users
2021-07-12
How can we account for the ray trajectories of a Gaussian beam for the above-mentioned equation, rather than solely considering on-axis phase delay?
The author's answer:
In the simplest case, we still consider unidirectional propagation, just taking into account the spatially dependent nonlinear phase shifts. This leads to nonlinear self-focusing, as explained in the article.
If you consider ray trajectories in different directions, things become substantially more complicated. Quite sophisticated models are required for accurately describing for example the nonlinear effect on tightly focused light pulses, exhibiting strong beam divergence.
2023-03-16
Pure self-phase modulation leads to symmetrical broadening of the spectrum, but why do I see that self-phase modulation has a higher peak at shortwave in some articles？
The author's answer:
Indeed, you see that effect e.g. in Figure 2 on this page. The curve would be symmetrical when shown on a frequency axis, but if you calculate a power spectral density with wavelength units, the conversion makes it asymmetric. This is basically because the frequency range corresponding to a 1-nm wavelength interval, for example, gets larger as the wavelength gets shorter.
2021-07-08
Is there a reference in which I can find the derivation of the self-phase modulation proportionality constant?
The author's answer:
I don't have a reference for that, but it is a fairly simple calculation which you can quickly do yourself.