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Education and Training in Optics and Photonics

Understanding the characteristics of gain saturation for homogeneously broadened laser medium from the point of view of the bandwidth

Geguo Du

Published Jan 1, 2007• 0 citations
Publisher

Abstract

It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δ ν = 1 + I ν 1 / I s Δ ν H for condition 2). Here, I v 1 and Is are the intensity of the large signal and the saturation intensity at the center frequency Vo, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.

DOI
10.1364/etop.2007.esb4

Related papers

Understanding the characteristics of gain saturation for homogeneously broadened laser medium from the point of view of the bandwidth

Geguo Du

It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δν = √1+Iν1 / IsΔνH for condition 2). Here, Iν1 and Is are the intensity of the large signal and the saturation intensity at the center frequency ν0, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 3, 2015•0 citations

Strong-Field Saturation Effects in Laser Media

Donald H. Close

Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the $m\mathrm{th}$ sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.

Physical Review•Jan 10, 1967•81 citations

Strong field saturation effects in laser media

Donald H. Close

Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the $m\mathrm{th}$ sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.

IEEE Journal of Quantum Electronics•Apr 1, 1966•2 citations

Effect of Spectral Hole-Burning and Cross Relaxation on the Gain Saturation of Laser Amplifiers

Amado Y. Cabezas, Richard P. Treat

A theory is presented for the effect of cross relaxation (CR) on the saturation with driving-signal intensity of the steady-state gain of a laser amplifier with an inhomogeneously broadened linewidth. The condition of steady-state gain results when the amplifier excitation level W and the driving-signal intensity are constant for about 3τ or longer, where τ is the fluorescence lifetime of the upper laser level. The driving signal is derived from an oscillator operated in the normal mode and which does not saturate the amplifier inversion density over times short compared with τ. We assume that CR, at a rate F, restores to thermal equilibrium any distortion of the spectral inversion density caused by the driving signal. Formulas are derived for steady-state gain saturation with arbitrary F for two cases: the spectral width of the driving signal is (1) narrow and (2) broad relative to the ionic or homogeneous line. The ionic linewidth Δvi is assumed to be much less than the inhomogeneous linewidth Δva and refers to the minimum frequency interval for which infinitely fast CR is operative. The theory also yields the CR coefficients f=(Δvi/Δva) [F/(τ−1+W)] for the narrow-band case and f*=(Δvs/Δva) [F/(τ−1+W)] for the broadband case, where Δvs is the driving-signal bandwidth. These coefficients significantly parametrize the CR effect: for f, f*>1, hole-burning is not severe, whereas for f, f*<1, considerable hole-burning takes place and the gain saturates faster. Finally, the theory is applied to experimental results obtained for Nd3+:glass broadband and narrow-band amplification; an approximate lower bound of 4×105 sec−1 is determined for F.

Journal of Applied Physics•Aug 1, 1966•58 citations

Population Inversion, Saturation, and Laser Systems

Surendra Singh

This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.

CRC Press eBooks•Sep 23, 2026•0 citations

Optical frequency amplifiers

Christophér C. Davis

Introduction When an electromagnetic wave propagates through amedium stimulated emissions increase the intensity of the wave, while absorptions diminish it. The overall intensity will increase if the number of stimulated emissions can be made larger than the number of absorptions. If we can create such a situation then we have built an amplifier that operates through the mechanism of stimulated emission. This laser amplifier , in common with electronic amplifiers, has useful gain only over a particular frequency bandwidth. Its operating frequency range will be determined by the lineshape of the transition, and we expect the frequency width of its useful operating range to be of the same order as the width of the lineshape. It is very important to consider how this frequencywidth is related to the various mechanisms by which transitions between different energy states of a particle are smeared out over a range of frequencies. This line broadening affects in a fundamental way not only the frequency bandwidth of the amplifier, but also its gain. A laser amplifier can be turned into an oscillator by supplying an appropriate amount of positive feedback. The level of oscillation will stabilize because the amplifier saturates. Laser amplifiers fall into two categories, which saturate in different ways. A homogeneously broadened amplifier consists of a number of amplifying particles that are essentially equivalent, whereas an inhomogeneously broadened amplifier contains amplifying particles with a distribution of amplification characteristics. Homogeneous line broadening All energy states, except the lowest energy state of a particle (the ground state) cover a range of possible energies.

Cambridge University Press eBooks•Mar 20, 2014•0 citations
It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δ ν = 1 + I ν 1 / I s Δ ν H for condition 2). Here, I v 1 and Is are the intensity of the large signal and the saturation intensity at the center frequency Vo, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.
It is very important to understand the characteristics of gain saturation for homogeneously broadened laser medium. Assuming a Lorentzian lineshape function with a linewidth of ΔνH, we have derived the analytical functions of the gain coefficient under three different conditions: 1) a small and 2) a large signal incident on the medium respectively, and 3) a small signal accompanied by a large signal simultaneously incident on the medium. We have found that the bandwidth Δν of the gain coefficient is equal to the linewidth ΔνH for condition 1) and 3) while it is Δν = √1+Iν1 / IsΔνH for condition 2). Here, Iν1 and Is are the intensity of the large signal and the saturation intensity at the center frequency ν0, respectively. The reasons are also presented for such results. For condition 2), gain saturation effect is strongly dependent on the frequency of the large signal: the more the frequency deviates from the center frequency, the weaker the gain saturation effect. This results an increase of the bandwidth. For condition 3), the intensity of the large signal only changes the distribution of populations on the upper and lower energy levels. Hence, the shape of the gain coefficient does not change with the gain reduction of the small signal, thus, its bandwidth remains the same.
Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the mthm\mathrm{th}mth sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.
Using Lamb's model, we analyze the effects of gain saturation by strong traveling-wave fields in dilute laser media. Using approximate solutions of an integral equation for the population inversion density (PID), the index of refraction and incremental gain are studied for arbitrarily strong fields. Effects of atomic motion are included for a Maxwellian velocity distribution, but pressure effects are neglected. The case of a monochromatic field leads to the saturation results of Gordon, White, and Rigden, which are studied as a function of frequency. For a small ratio of natural to Doppler linewidths, there is a transition from inhomogeneous to homogeneous broadening for sufficiently strong fields. An effect of particular interest is the generation of waves at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, $2{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$, and higher order sidebands by two strong input signals at ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$. The source of the parametric gain at these frequencies is the time-dependent gain saturation due to the presence of multiple strong fields. For ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ small compared to the decay rates ${\ensuremath{\gamma}}_{a}$ and ${\ensuremath{\gamma}}_{b}$ of the laser levels, the gain at these intermodulation sidebands is computed as a function of the field strengths. The limiting cases of homogeneous and inhomogeneous (due to atomic motion) broadening are studied in detail. These two cases give essentially the same results, for a given unsaturated gain. Numerical results indicate that a first-order side-band intensity at least 10% as large as that of the inducing fields can be easily observed in practice. The integral equation for the PID is converted into an infinite set of linear algebraic equations for a typical solid-state laser. The conditions under which this set of equations can be limited to a finite number is discussed, and the dependence of the sideband gain on ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ is calculated. For large ${\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$, the gain at the mthm\mathrm{th}mth sideband decreases as ${({\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2})}^{\ensuremath{-}m}$.
A theory is presented for the effect of cross relaxation (CR) on the saturation with driving-signal intensity of the steady-state gain of a laser amplifier with an inhomogeneously broadened linewidth. The condition of steady-state gain results when the amplifier excitation level W and the driving-signal intensity are constant for about 3τ or longer, where τ is the fluorescence lifetime of the upper laser level. The driving signal is derived from an oscillator operated in the normal mode and which does not saturate the amplifier inversion density over times short compared with τ. We assume that CR, at a rate F, restores to thermal equilibrium any distortion of the spectral inversion density caused by the driving signal. Formulas are derived for steady-state gain saturation with arbitrary F for two cases: the spectral width of the driving signal is (1) narrow and (2) broad relative to the ionic or homogeneous line. The ionic linewidth Δvi is assumed to be much less than the inhomogeneous linewidth Δva and refers to the minimum frequency interval for which infinitely fast CR is operative. The theory also yields the CR coefficients f=(Δvi/Δva) [F/(τ−1+W)] for the narrow-band case and f*=(Δvs/Δva) [F/(τ−1+W)] for the broadband case, where Δvs is the driving-signal bandwidth. These coefficients significantly parametrize the CR effect: for f, f*>1, hole-burning is not severe, whereas for f, f*<1, considerable hole-burning takes place and the gain saturates faster. Finally, the theory is applied to experimental results obtained for Nd3+:glass broadband and narrow-band amplification; an approximate lower bound of 4×105 sec−1 is determined for F.
This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.
Introduction When an electromagnetic wave propagates through amedium stimulated emissions increase the intensity of the wave, while absorptions diminish it. The overall intensity will increase if the number of stimulated emissions can be made larger than the number of absorptions. If we can create such a situation then we have built an amplifier that operates through the mechanism of stimulated emission. This laser amplifier , in common with electronic amplifiers, has useful gain only over a particular frequency bandwidth. Its operating frequency range will be determined by the lineshape of the transition, and we expect the frequency width of its useful operating range to be of the same order as the width of the lineshape. It is very important to consider how this frequencywidth is related to the various mechanisms by which transitions between different energy states of a particle are smeared out over a range of frequencies. This line broadening affects in a fundamental way not only the frequency bandwidth of the amplifier, but also its gain. A laser amplifier can be turned into an oscillator by supplying an appropriate amount of positive feedback. The level of oscillation will stabilize because the amplifier saturates. Laser amplifiers fall into two categories, which saturate in different ways. A homogeneously broadened amplifier consists of a number of amplifying particles that are essentially equivalent, whereas an inhomogeneously broadened amplifier contains amplifying particles with a distribution of amplification characteristics. Homogeneous line broadening All energy states, except the lowest energy state of a particle (the ground state) cover a range of possible energies.