Parametric Amplification

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

Parametric amplification is a phenomenon where a signal can be amplified using a parametric nonlinearity and a pump wave. This articles focuses on optical amplification, even though there are also electronic parametric amplifiers, used e.g. for microwaves. For optical parametric amplifiers, either the <$\chi^{(2)}$> nonlinearity of certain nonlinear crystal materials or the <$\chi^{(3)}$> nonlinearity e.g. of an optical fiber [7] can be utilized.

This article discusses the physical mechanism of parametric amplification, which is based on some optical nonlinearity. It is different from laser amplification via stimulated emission by excited atoms, ions or molecules, or by excited carriers in a semiconductor laser; parametric amplification does not involve the excitation of media to higher-lying energy levels (even though some frequently used explanations involve some virtual excited states). It is also important to distinguish between degenerate and nondegenerate parametric amplification because these lead to very different features.

Nondegenerate Parametric Amplification

Here, we consider parametric amplification based on a <$\chi^{(2)}$> nonlinearity in a nonlinear crystal material such as LiNbO₃ or lithium triborate (LBO).

In the nondegenerate case, there is an interaction between three distinct light waves, the angular frequencies of which are <$\omega_1$>, <$\omega_2$>, and <$\omega_3$> (with the indices in the order of the frequency values):

• The pump wave has the frequency <$\omega_3$>.
• The signal wave has the frequency <$\omega_2$>.
• A so-called idler wave with the frequency <$\omega_1$> is generated in the interaction.

For reasons of energy conservation, the relation <$\omega_3 = \omega_1 + \omega_2$> must hold. The idler frequency <$\omega_1$> is often below the signal frequency <$\omega_2$>, but it can also be higher than that; in any case, both signal and idler frequency are below the pump frequency.

Signal and idler generally have different optical frequencies, but even the case of equal frequencies may be considered as degenerate if the two waves can still be distinguished through different polarization directions.

Essentially, the amplification process implies that some of the pump photons are converted to signal and idler photons. More precisely, for each disappearing pump photon, one signal photon and one idler photon is generated. Although the idler photons leaving the nonlinear crystal are often not used, there are essential in the amplification process: for a material with strong absorption of the idler wave, the amplifier performance can be strongly degraded.

The local growth rate for the signal amplitude is proportional not only to the pump amplitude, but also to the already present signal amplitude. Therefore, the process can be interpreted as an amplification process – unlike the process of frequency doubling, for example.

For a simple case with plane waves or with weakly focused collinear Gaussian beams, the nonlinear interaction of the waves with complex amplitudes <$A_1$> to <$A_3$> can be described with the equations

where

is the phase mismatch (calculated from the wavenumbers), <$z$> is the coordinate of the propagation direction, and <$\kappa$> is a coupling constant which is proportional to the effective nonlinear coefficient of the material. A more symmetrical set of equations is sometimes used, which can be obtained from the equations above by replacing the amplitude <$A_3$> with <$i A_3$>.

The simplest situation is that with a zero phase mismatch (<$\Delta k = 0$>). When there is no idler input (<$A_1 = 0$> at the beginning of the crystal), in the first order there is no signal amplification nor any pump depletion, but a buildup of an idler wave. If the initial <$A_3$> and <$A_2$> are real, a real amplitude of the idler will be built up. (This process may be considered as difference frequency generation.) In the following, this leads to a growth of <$A_2$>, i.e., to signal amplification, and later to the depletion of the pump wave.

It is instructive to consider that the optical phase of the initial signal is now rotated by e.g. 30°. This will lead to a rotation of the idler phase by −30°. Consequently, the field contributions added to the signal are also rotated by +30°, so that the signal is amplified as before. This shows that the amplification is phase insensitive (i.e. independent of the signal phase) in this nondegenerate case (with a separate signal and idler).

In another case, the pump phase is rotated by 30°. This will also rotate the idler by 30°. Both phase changes cancel in the contribution added to the signal, so the signal is again amplified, and only the idler phase is changed.

When there is a phase mismatch (i.e. a nonzero <$\Delta k$>), the relative phases of the waves will change during propagation, so that after some distance the power in signal and idler may be converted back towards the pump. Therefore, phase matching, which can be achieved only in a limited frequency range, is important for efficient amplification. Under certain special conditions, a very large phase-matching bandwidth may be achieved.

The equations also show that the signal amplification is reduced in a medium with strong absorption for the idler wave, which keeps the idler amplitude small. Therefore, the transparency range of the nonlinear crystal has to be large enough to keep the idler wave in the low-loss window, even if one is not interested in the idler wave itself. However, some level of idler absorption may even be beneficial in certain cases [9].

The equations used are based on classical physics. In a quantum-mechanical picture, there are processes where pump photons are converted into signal–idler photon pairs. These are strongly correlated, leading to nonclassical statistics.

Degenerate Parametric Amplification

In the rarely used degenerate case, signal and idler waves are identical not only in frequency but also in polarization, i.e., they are indistinguishable. There is therefore only a signal amplitude <$A_1$> and a pump amplitude <$A_2$>, and no idler. The signal wavelength is then exactly twice the pump wavelength. The interaction is described with the equations

with

Here, the amplification is phase sensitive. For example, signal amplification occurs (for zero phase mismatch) if the signal and pump amplitudes are real and positive, or there is signal deamplification when the sign of the pump amplitude is changed. The direction of energy transfer is governed by the complex phase of the term <$A_1^2 A_2^*$>.

Phase-sensitive amplification provides a mechanism for producing so-called squeezed states of light (e.g. a squeezed vacuum), and also in principle it allows one to avoid excess amplifier noise, i.e., to keep the noise figure near 1. However, the need to maintain a fixed phase relationship between pump and signal makes this kind of optical amplification too inconvenient e.g. for use in optical fiber communications.

Parametric Amplification in Fibers

Due to the centrosymmetric nature of the material, glass fibers do not exhibit a <$\chi^{(2)}$> nonlinearity (unless under certain circumstances, for example when fibers are “poled” with a strong electric field). However, parametric amplification can also occur as a result of the <$\chi^{(3)}$> nonlinearity. In that case, four different frequencies can be involved: two pump frequencies, a signal frequency, and an idler frequency. A frequent case is that of partial degeneracy, where one has only a single pump wave.

The interaction is somewhat complicated because the optical phases of signal and idler are influenced both via cross-phase modulation (XPM) and the chromatic dispersion of the fiber. (For strong signals, self-phase modulation occurs in addition.) Parametric amplification is obtained only within some wavelength range around the pump wavelength. That range can be fairly wide when the chromatic dispersion is weak, and its width depends on the pump power.

Figure 1 shows the gain spectra for an example case. The highest gain occurs for a signal wavelength where phase matching is obtained by mutual cancellation of XPM and dispersion effects. This is possible only for anomalous dispersion.

The required fiber length can be minimized by using a highly nonlinear fiber. That also has the advantage that phase matching is easier to achieve and less sensitive to external influences such as temperature differences.

Parametric gain is mostly relevant in cases where short light pulses propagate in a fiber. Numerical pulse propagation modeling can simulate such situations, where a variety of effects can play a role. Examples are the partial temporal overlap of pulses due to group velocity mismatch and soliton effects. The interactions are particularly complicated in multimode fibers because the phase matching is different for different combinations of propagation modes.

More to Learn

Encyclopedia articles:

• optical parametric amplifiers
• optical parametric chirped-pulse amplification
• optical parametric oscillators
• optical parametric generators
• optical amplifiers

Blog articles:

• The Photonics Spotlight 2007-12-11: “The Idler Wave – Essential for Parametric Amplification and Oscillation”

Bibliography

(Suggest additional literature!)

Suppliers

The RP Photonics Buyer's Guide contains 23 suppliers for optical parametric amplifiers. Among them:

Thorlabs

Amidst our extensive selection of femtosecond lasers and related products for control and characterization, Thorlabs manufactures a compact, tunable repetition rate Ytterbia OPA with integrated microjoule-class pump laser. Complimenting our ultrafast laser systems is a suite of nonlinear crystals, chirped mirrors, low GDD optics, and related products for pulse measurement, pre-compensation, and dispersion measurement.

RPMC Lasers

Serving North America, RPMC Lasers offers a single-box optical parametric amplifier that converts ultrafast pulses from Ytterbium-based lasers into broadly tunable pulses with wavelengths ranging from 210 nm to > 10 µm. With intuitive, user-friendly PC software, an integrated mini spectrometer enables automatic tuning from 315 to 2600 nm. Designed to operate from a single pulse up to 300kHz rep. rate with pump pulse energy from 30 µJ to 200 µJ, Harmony is compatible with all Ytterbium-based femtosecond lasers, but it works best with the Jasper series: the Jasper X0 & Jasper Flex lasers featuring exceptional beam-pointing stability. Standard and custom options available. Let RPMC help you find the right laser today!

Covesion

Researchers have used MgO:PPLN crystals to build a compact and dependable, tunable, CW seeded synchronization-free OPA with a robust, commercial pico-second pump laser.

Utilizing our knowledge and experience in poling techniques, Covesion can offer a variety of stock and custom free space bulk crystal solutions for amplifying low power near-IR CW seed lasers for applications such as spectroscopy and environmental monitoring.

Our custom design capabilities include:

• multiple grating, chirped or fan-out designs
• tailored AR coatings
• custom grating periods and apertures
• compatibility with both CW and pulsed lasers

EKSPLA

For researchers demanding a wide tuning range, high conversion efficiency and narrow line-width, EKSPLA optical parametric oscillators/amplifiers are an excellent choice. All models feature hands-free wavelength tuning, a protection system for valuable optical components as well as a wide range of accessories and extension units.

EKSPLA's long-term experience and close cooperation with scientific institutions made it possible to create a range of models, offering probably the widest tuning range: from 193 nm to 16000 nm. Versions offering near transform-limited linewidth as well as operating at kilohertz repetition rates are available.

Class 5 Photonics

Class 5 Photonics delivers ultrafast, high-power laser technology at outstanding performance to advance demanding applications from bio-imaging to ultrafast material science and attosecond science. Our robust optical parametric chirped pulse amplifiers (OPCPA) provide high-power, tunable femtosecond pulses at user-friendly operation.

Features of the White Dwarf OPCPA 5 W:

• compact and user-friendly
• CEP stability available
• pumped by Coherent Monaco industrial femtosecond laser

White Dwarf HE OPCPA 30 W:

• high-performance, ultrafast OPCPA
• pump-probe configuration
• pumped by Yb-based laser up to 300 W and 3 mJ

Supernova OPCPA 100 W:

• our award-winning flagship product
• highest average power OPCPA for demanding applications
• pumped by kW-class Yb:YAG Innoslab amplifiers or thin-disk lasers

Light Conversion

The highlight of our tunable wavelength sources is the ORPHEUS series of femtosecond optical parametric amplifiers (OPAs) for pumping with Yb-doped lasers. The ORPHEUS series enables operation at high repetition rates while maintaining the best properties of the TOPAS series OPAs, such as a wide tuning range from deep-UV to mid-IR and high output stability. When combined with the PHAROS or CARBIDE femtosecond lasers, it becomes an invaluable source for ultrafast spectroscopy, nonlinear microscopy, and a variety of other applications. The list of tunable wavelength sources has been extended by the addition of I-OPA, a compact industrial-grade OPA, and CRONUS-3P, an OPA-based ultrafast source with GDD control for advanced nonlinear microscopy.

Stuttgart Instruments

The Stuttgart Instruments Alpha is an ultrafast and fully wavelength-tunable frequency conversion system in an ultra-compact and completely passively stable system based on revolutionary parametric oscillator design which guarantees outstanding stability, reproducibility and shot-noise limited performance.

The revolutionary design of Stuttgart Instruments Alpha, characterized by outstanding low noise and passive long-term stability, is based on the fiber-feedback optical parametric oscillator (FFOPO) technology and results in outstanding performance and high flexibility at the same time.

The Alpha covers a gap-free rapid tunable spectral range from 700 nm to 20 µm wavelengths, while maintaining high output power up to the Watt-level with femto- or picosecond pulses at several MHz pulse repetition rates. It provides multiple simultaneously tunable outputs with a selectable bandwidth from a few to 100 cm-1. Shot-noise limited performance above 300 kHz, passive spectral stability (< 0.02% rms) and wavelength-independent stable beam pointing (< 30 µrad) enable excellent sensitivity. In addition, each Alpha is equipped with a user-friendly ethernet and Wi-Fi interface and a matching graphical user interface (GUI) as well as easy to access API interfaces for e.g. LabView, Python, C++.

Typically, the Alpha is pumped by an ultra-low-noise Primus pump laser, which provides more than 8 W average output power at 1040 nm wavelength and 450 fs pulse duration at 42 MHz repetition rate. In addition, the Alpha can be operated with other pump lasers around 1 µm wavelength and enough power.

Due to our modular platform, the Alpha can be adapted and optimized for various applications and is particularly suited for spectroscopic applications requiring a robust and reliable tunable radiation with low noise.

Fluence

Find harmony in four perfectly synchronised and precisely tunable wavelength outputs produced by the Fluence Harmony Optical Parametric Amplifier (OPA).

Fluence Harmony is a device that precisely converts ultrafast pulses of one band (e.g. 1030-nm pulses coming from Fluence Jasper) into a set of pulsed laser beams tunable in an extremely broad spectrum of wavelengths, ranging from 210 nm up to 2600 nm. All automated, all software-controlled.

A growing number of ultrafast spectroscopy techniques has prompted the need for a robust and reliable, self-diagnostic device like Harmony. The optical parametric amplifier is fully compatible with all femtosecond lasers and provides automated tuning across the basic tuning range.

Active Fiber Systems

AFS offers customized OPCPA/OPA add-ons to supplement our high-power beamlines.

APE

AVUS is the very latest Optical Parametric Amplifier (OPA) providing widely tunable high-energy pulses. It is ideal for use with 1-μm femtosecond lasers and opens doors for up to 50 W pump power. The user-friendly and maintenance-free unit is air-cooled and constructed with a monolithic case design for long-term thermal stability, even at maximum pump power. Versions with pulse durations <200 fs or alternatively <70 fs are available.

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