The Photonics Spotlight
The Photonics Spotlight – associated with the Encyclopedia of Laser Physics and Technology – is a “blog” (web log) with the purpose of highlighting interesting news and useful information in the area of photonics, particularly laser technology and applications. The content can be related to particularly interesting scientific papers or to other forms of publications, reporting for example cute new techniques, special achievements, or useful hints.
Note that the Spotlight articles (as well as those of the Encyclopedia) are citable. Permanent links are given for each article.
This blog is operated by Dr. Rüdiger Paschotta of RP Photonics Consulting. Comments and suggestions are welcome. The news items are definitely not available for advertising, but advertisers can order banners on the right column of this page.
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Gain Saturation with Pulses
Posted on 2015-08-08 as a part of the Photonics Spotlight.
Permanent link: https://www.rp-photonics.com/spotlight_2015_08_08.html
It is well-known that the saturation of the optical gain of a laser amplifier operated with constant pump and signal input power can be described with a formula like g(Ps) = g0 / (1 + Ps / Psat), where g0 is the small-signal gain, Ps is the signal input power and Psat is the saturation power. (The gain should be small, as otherwise the signal power rises within the gain medium; otherwise, one may have to numerically split the gain medium into several sections.) What, however, happens when we apply a short signal pulse? That question is relevant in many important cases, for example involving pulsed amplifiers and Q-switched lasers.
Calculating Gain Saturation
Some people believe that saturation of a laser amplifier becomes substantial once the power of a signal pulse approximately reaches the level of the saturation power. That guess can be quite wrong, however. The formula given above applies only in the steady state, and that is typically reached after a few times the upper-state lifetime of the laser medium. For most solid-state laser gain media, the upper-state lifetime is well above 1 μs, so that a nanosecond or even picosecond pulse is by far not long enough to bring the gain medium to its steady state.
For pulse durations far below the upper-state lifetime, it is quite clear that spontaneous emission has a negligible impact on the upper state population during the time of such a pulse. Therefore, the value of the upper-state lifetime cannot be relevant for the saturation behavior, and the same holds for the saturation power, which depends on the upper-state lifetime. What is relevant here is the saturation energy Esat, which is essentially determined by the emission cross section at the signal wavelength and is not influenced by the upper-state lifetime. (For example, it is not affected by quenching processes reducing that lifetime.) Again, a simple formula can be used to calculate the gain after the pulse: gf = gi exp(−Es / Esat), where gi is the initial gain and Es is the signal pulse energy.
Example: Saturation in a Q-switched Laser
When the resonator of a Q-switched solid-state laser is switched to its high-Q state, the signal power rises rapidly, starting from a very low level. When it reaches the saturation power, gain saturation still stays very weak, as explained above. Only once the temporally integrated laser power reaches the order of the saturation energy, gain saturation sets in. Therefore, the peak power occurring within such a laser (and also the output peak power) can be orders of magnitude above the saturation power of the gain medium. (For a continuous-wave output power, that is usually impossible.)
Gain Saturation Caused by an Ultrashort Pulse
A difficult issue arises when we want to calculate the saturation of a laser gain medium in an ultrafast amplifier caused by an optical pulse with a duration far below 1 ps. Such a pulse has a substantial optical bandwidth – at least the Fourier-limited bandwidth or even more in case of chirped pulses. One would naturally describe the spectral properties in the frequency domain, but that is clearly unsuitable for calculating the time-dependent gain. Obviously, neither the time domain nor the frequency domain is suitable for fully treating what happens here.
The interaction of single atoms or ions with pulsed light can be described with Bloch equations, which are differential equations in the time domain. For a complicated ensemble of laser-active ions interacting with a host crystal, however, this approach is hardly applicable. The first challenge which one meets here is that one will normally not get all the detailed data to describe the behavior of ions in such a situation.
How can one obtain at least approximate results? As long as the pulse energy is well below the saturation energy, one can easily calculate the pulse amplification in the frequency domain. The weak effect of gain saturation can be simulated afterwords by reducing the gain according to the amount of energy the pulse has extracted – which is simply the difference of output and input pulse energy, assuming that additional effects such as parasitic losses do not occur.
For the case that the pulse energy reaches or exceeds the level of the saturation energy, the described approach can be extended. One may simulate the interaction of the pulse with the gain medium in multiple steps, where each time only a fraction of the gain is applied to the pulse, and afterwords the gain is reduced according to the extracted energy. This is an approximate method, however, which cannot precisely describe the details of such an amplification process.
Self-phase Modulation Causes Spectral Broadening – Does it Really?
Posted on 2015-07-01 as a part of the Photonics Spotlight.
Permanent link: https://www.rp-photonics.com/spotlight_2015_07_01.html
For many, it seems to be common wisdom that the effect of self-phase modulation (SPM), which results from the Kerr nonlinearity, always increases the optical bandwidth of an ultrashort pulse. After all, it creates a so-called chirp, i.e., a temporal variation of the instantaneous frequency, which then runs through a wider range of frequencies. This effect is utilized, for example, in a method of temporal pulse compression, where one first broadens the bandwidth using SPM and then temporally compresses the pulse by applying an appropriate amount of chromatic dispersion which removes the created chirp.
However, it should then be irritating that various example cases quite clearly contradict the mentioned belief:
- When fundamental soliton pulses propagate through an optical fiber, these continuously experience self-phase modulation. Nevertheless, their optical bandwidth does not change at all! Well, here we also have chromatic dispersion acting on the pulse, but that linear effect is known not to affect the optical bandwidth. Therefore, how could the chromatic dispersion remove the bandwidth-broadening effect of the fiber nonlinearity?
- If one injects a higher-order soliton pulse into such a fiber, its optical bandwidth changes periodically, i.e., it regularly expands and contracts again.
- There are indeed cases where e.g. a broadband down-chirped pulse enters a fiber and leaves it with much reduced optical bandwidth and reduced chirp.
All this can be resolved by considering more carefully the generation of additional frequency components by the Kerr nonlinearity. The essential point is to realize that the Kerr nonlinearity adds certain complex amplitudes to other frequency components. (The typically used differential equations for light propagation in fibers clearly show that.) How the intensities of these frequency components change, depends on the relative signs of existing and added complex amplitudes:
- If the pulse is unchirped (i.e., it has a constant instantaneous frequency), the complex amplitudes added in the frequency domain are 90° out of phase with the already existing frequency components. As a result, the magnitude of these frequency components does not change to first order. This explains, for example, how a fundamental soliton pulse can have a constant optical bandwidth: the chirp introduced by the nonlinearity is constantly removed by the (anomalous) chromatic dispersion, and unchirped pulses are effectively not broadened by SPM.
- Without chromatic dispersion, the situation is different: the pulse can acquire a growing chirp, and here the magnitude of the generated frequency components can indeed grow, so that the bandwidth increases.
- If, however, a pulse initially has a down-chirp, the outer frequency components are indeed reduced by SPM, because the added amplitudes are out of phase with the existing ones. In that situation, SPM leads to spectral compression rather than broadening.
We see that the sign of the chirp of the pulse is essential for the nonlinear effects on the pulse spectrum, as is also illustrated in the following two diagrams:
It is also instructive to consider a soliton mode-locked laser where the pulse bandwidth is constantly reduced by the finite gain bandwidth (→ gain narrowing). In the steady state of the laser, there must be an effect to compensate for this. Apart from a modulator or a saturable absorber, SPM can take over that function. The pulse then must develop a positive chirp, as spectral broadening is possible only with that. Indeed one can observe in computer simulations that pulses in soliton mode-locked lasers exhibit a slight up-chirp, depending on the magnitude of bandwidth-reducing effects.
The presented thoughts demonstrate that one can learn a lot by thinking about basic effects e.g. in ultrafast optics or laser physics a little more closely. Many people are too quickly satisfied with inaccurate descriptions of effects, which are in contradiction even with quite common observations.
The probably most effective way for detecting and subsequently revising inaccurate beliefs is to deal with numerical models. Here, existing beliefs are continuously put to test. Seeing quite easily what exactly goes on in various systems (e.g., in a transparent laser realized in the form of a computer model), one quite quickly realizes that certain thoughts cannot be correct. At the same time, this is one of the greatest opportunities to obtain new ideas.
Alignment Sensitivity of Laser Resonators – an Important Design Criterion
Posted on 2015-05-14 as a part of the Photonics Spotlight.
Permanent link: https://www.rp-photonics.com/spotlight_2015_05_14.html
Everyone in the field knows that an essential design criterion for laser resonators is to have appropriate mode radii, particularly within a laser crystal. For example, the mode radius in the laser crystal should approximately match the mode radius of the pump beam if one wants to achieve transverse single-mode operation, which results in a high beam quality.
What is much less known is the importance of the alignment sensitivity of laser resonators, and that it can be greatly influenced by the resonator design. The easy part is to understand why a low alignment sensitivity is very desirable; obviously, you do not want a resonator which needs to be aligned with extreme care, and it needs to be realigned when ever some optomechanical parts (e.g. mirror holders) are slightly affected e.g. by thermal expansion, or when the thermal lens is modified by changes of the pump beam profile. Particularly for an industrial product, an excessive alignment sensitivity cannot be tolerated.
In extreme cases, one would not even achieve any reasonable performance, even when working hard on a fine alignment, if thermal effects in the laser crystal are strong. Interestingly, the alignment affects the power and position of the circulating laser beam, which can also lead to modifications of thermal lensing, and that again affects the beam position and power; if such mutual influences are strong enough, you can get a very strange behavior of the laser which prevents any reasonable performance.
Calculating Alignment Sensitivities and Applying that Knowledge
Experts in the field of laser resonator design know very well that the alignment sensitivity of a laser resonator, e.g. concerning angular positions of laser mirrors, can be calculated with suitable software, which does not use a simple ABCD matrix algorithm but rather an extended algorithm using 3×3 matrices. The alignment sensitivity can then be used as a criterion for the quality of the laser design; it can (and often should) even be included in a figure of merit within an automatic optimization procedure. Ignoring this important aspect in laser design can easily lead to designs with unnecessarily high alignment sensitivity which do not work well in practice.
Mode Areas are Important – What Else?
It is also known that lasers with large mode areas tend to have a higher alignment sensitivity. (This is the essential reason why high-power lasers are often much more delicate to align; the wide-spread believe that this results simply from the large size of the resonator itself is wrong, as discussed in an earlier posting.) However, there is no fixed relation between mode size and alignment sensitivity; one can have two different laser resonator designs with the same mode area in the laser crystal which differ very much (e.g. by a factor larger than 5) concerning alignment sensitivity. In case of linear resonators (standing-wave resonators), one has two different stability zones in terms of the focusing power of the thermal lens, and these can have very different alignment sensitivities. Unfortunately, one cannot always use the less sensitive zone, because that involves limitations in other aspects.
The issue of alignment sensitivities even at the heart of an often encountered trade-off: high-power laser can often be designed either for a high power conversion efficiency and robustness or for highest beam quality, but not both at the same time. Obviously, one can hardly find optimized designs without understanding these issues very well.
How to Find a Suitable Resonator Design?
Even in seemingly simple cases, it is very desirable that the person developing a laser resonator design understands the matter well. The required knowledge goes far beyond a basic understanding of resonator modes; it should definitely include a precise knowledge on alignment sensitivity issues and substantial experience concerning various typical trade-offs. Simply having a heavy textbook in the office, or even having read it, will often not be sufficient.
A suitable laser resonator design software (such as our product RP Resonator) must definitely be able to calculate alignment sensitivities and to take them into account in optimizations. However, no software in this area can replace a decent technical understanding of the person using it; I think it is not possible to make it such that it takes into account all important issues without bothering the user with it. For example, a software can hardly “know” the importance of various resonator properties for the particular application, i.e., it could not put appropriate weights on certain factors in the trade-offs which are necessary. At least, however, software from a good source comes with very helpful technical support, giving you crucial pieces of advice.
If you need a proper laser resonator design, you basically have two different options:
- You can try to acquire all the required expertise (which is certainly not easy and will require substantial time) and also get a good resonator design software.
- You can try to find an experienced expert who can do that job for you. That person would first closely analyze the concrete requirements in a dialogue with you, then translate that into appropriate resonator properties such as mode sizes and maximum alignment sensitivities, and finally work out a suitable design using proper design software.
If you quite often need resonator designs, it will probably try to get into the position of doing it yourself. If that is not the case, however, it will often be much more economical (and also lead to better results) to have an external expert doing it.
In any case, I warmly recommend to take the question of resonator design very serious, because this very much contributes to an efficient product development, avoiding a lot of possible problems causing delays and cost overruns.
By the way, the fact that a very simple resonator has been used so far for certain lasers does in no way prove that more elaborate laser design considerations would be wasting resources. After all, how can you know that the simple type of resonator is doing its job well and could not be improved? Also, even seemingly simple resonators are not so easy to understand. Finally, a few hours of good work by a competent expert cost far less than what you might well pay for wrong decisions on such matters.
2015-02-05: Attenuating Laser Beams – not That Easy
2014-10-03: Fiber Optics Tutorials
2014-07-28: How to Define the Mode Radius of a Fiber?
2014-05-16: 10-Year Anniversary of RP Photonics
2014-01-17: Mediation in Disputes on Laser Technology
2013-12-13: Avoiding Trouble with Laser Specifications
2013-11-12: Beam Quality Limit for Multimode Fibers
2013-08-26: Frequency Doubling and the Reverse Process
2013-06-13: Two New Photonics Newsletters
2012-08-06: The New RP Photonics Buyer's Guide
2012-03-12: New Raman Lasers
2012-03-03: Conflicting Definitions of s and p Polarization
2011-12-23: Kerr-lens Mode-locked Thin-disk Laser
2011-06-10: Are Compact Resonators More Stable?
2010-07-12: Laser Development: Get an Expert Early on!
2010-06-09: Poor Man's Isolator
2010-04-26: Resolution and Accuracy of Measurements
2010-04-08: Creating a Top-hat Laser Beam Focus
2010-03-22: All-in-one Concepts versus Modular Concepts
2010-03-09: Nonlinearities in Fiber Amplifier Modeling
2010-01-29: Far From Maturity: The Photonics Industry
2010-01-22: Pumping Fiber Lasers with Fiber Lasers
2010-01-11: Beams of Laser Pointers: Visible in Air?
2009-12-31: Tilt Tuning of Etalons
2009-12-13: Johnson–Nyquist Noise in Photodiode Circuits
2009-11-18: Articles and a Quiz on Photonics Issues
2009-11-13: Photodetection: Optical and Electrical Powers
2009-11-03: Coherent Light from a Bulb?
2009-10-03: Peak Intensity of Gaussian Beam
2009-09-27: Lasers with Short Upper-state Lifetime
2009-09-19: Are Laser Resonators Power Scalable?
2009-09-01: Fresnel Reflections from Double Interfaces
2009-08-14: Progress on Green Laser Diodes
2009-08-12: What is an Optical Transistor?
2009-07-29: No Beat Note for Orthogonal Modes
2009-07-21: Signal-to-Noise Ratio and Measurement Bandwidth
2009-07-09: Gain-guiding Index-antiguiding Fibers
2009-06-29: Doing Things Properly: It's the Economy, Stupid!
2009-06-23: Coherence – a Black-or-White Issue?
2009-06-08: Prizes of the European Physical Society
2009-06-02: 5 Years of RP Photonics Consulting
2009-05-13: The Minimum Time–Bandwidth Product
2009-04-28: SPIE Field Guides
2009-04-05: Stability of Resonators – an Ambiguous Term
2009-03-02: User Interfaces for Simulation Software
2009-01-12: Chaotic Lasing Generates Random Numbers
2009-01-05: Extremely Long Mode-locked Fiber Laser
2008-12-16: Why Fiber Amplifiers, not Fiber Lasers?
2008-11-25: The Gouy Phase Shift Speeds up Light
2008-11-08: Validating Numerical Simulation Software
2008-09-24: Decoupling Pulse Duration and Pulse Energy
2008-09-10: Unpolarized Single-Frequency Output
2008-07-26: Beat Signals with Zero Linewidth
2008-07-02: Stronger Focusing Avoids SESAM Damage
2008-06-20: All-in-One Ultrafast Laser Systems
2008-06-06: Fiber Lasers Which Are No Fiber Lasers
2008-05-25: Einstein and the Laser
2008-05-05: Length of a Photon
2008-04-28: Different Kinds of Polarization
2008-04-22: Abused Photonics Terms: Coherence
2008-04-15: Abused Photonics Terms: Modes
2008-03-10: Automatic Phase Matching
2008-03-04: What is a “High” Laser Beam Quality?
2008-02-14: How Laser Development Can Go Wrong
2008-02-03: Quantifying the Chirp of Ultrashort Pulses
2008-01-27: Beam Quality in Second-Harmonic Generation
2008-01-14: Frequency Doubling: Long Pulses Cause Trouble
2007-12-18: The Role of Laser Safety Goggles
2007-12-03: New Paper on Power Scaling of Lasers
2007-11-26: Solving Laser Problems Step by Step
2007-11-10: Retirement of Prof. David C. Hanna
2007-11-02: Ultrafast Laser Kills Viruses
2007-10-31: Thermal Equilibrium in Laser Crystals
2007-10-25: The Gain Bandwidth of Laser Crystals and Glasses
2007-10-17: Why the Second-Harmonic Beam is Smaller
2007-10-11: Understanding Fourier Spectra
2007-09-21: Optimum Crystal Length for Frequency Doubling
2007-09-07: Power Scaling in Downward Direction
2007-08-27: Distant Healing of Lasers
2007-08-23: An OPO Without Resonator Mirrors
2007-08-15: Light = Electromagnetic Waves?
2007-07-06: Promoting Dangerous Practices in Laser Labs
2007-07-01: Nonsensical Regulations Undermine Laser Safety
2007-06-24: The Plague of a Narrow Emission Linewidth
2007-06-11: Beam Quality Measurements Can Easily Go Wrong
2007-06-01: Characterize Your Pump Beam!
2007-05-19: Why Strong Birefringence in Fibers Helps
2007-04-16: Questions and Answers on Shot Noise
2007-03-23: Explaining the Nature of Photons to Lay Persons
2007-03-11: Divided-Pulse Amplification
2007-03-09: The Trouble with Crystal and Coating Damage
2007-02-26: No Laser, no Result?
2007-02-22: Lossy Laser Cavities
2007-02-16: The Science of Biophotons
2007-02-09: Papers Reporting Yet Another Laser Crystal
2007-02-04: Continuing Struggle for Larger Fiber Mode Areas
2007-01-27: Noise Figure of Amplifiers
2007-01-21: Operation Far Above Threshold
2007-01-15: Origins of Heating in Laser Crystals
2007-01-09: The Myth of Fiber-Optic Polar Bears
2006-12-31: Peak Position of an Optical Spectrum
2006-12-16: Dangerous Green Laser Pointers
2006-12-09: The Laser Industry - High Tech or Low Tech?
2006-12-03: Diffraction in Optical Fibers
2006-11-28: The Role of Diffraction in Optical Resonators
2006-11-21: The Resonator Mystery
2006-11-16: Laser Models - not Always Useful
2006-11-02: Reflection Spectrum of Tilted Dielectric Mirrors
2006-10-22: Lasers Attract Dust to Cavity Mirrors
2006-10-01: Stability Zones of Laser Resonators
2006-09-22: Coherence Length of Ultrashort Pulses
2006-09-16: Q-switched Lasers: YAG versus Vanadate
2006-09-01: Test Yourself with the Photonics Quiz
2006-08-20: Lower Noise from Longer Lasers
2006-08-12: Understanding Quasi-Three-Level Lasers
2006-08-10: Single-Mode Fibers with Large Mode Areas
2006-08-01: Lasers Disturbed by Vacuum?
2006-07-24: Beam Distortions in Laser Cavities
2006-07-23: Single-Atom Lasers
2006-07-22: No Magnetic Field on the Axis of a Coil?
2006-07-16: Spontaneous Emission and Amplifier Noise
2006-07-14: Lasers Like it Cool
2006-07-10: Strength of Thermal Lensing Effects
2006-07-01: Characterizing a Cavity with a Frequency Comb
2006-07-01: With Wavelength Combs to Picometer Resolution