Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240

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1 Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240 John D. Williams, Ph.D. Department of Electrical and Computer Engineering 406 Optics Building - UAHuntsville, Huntsville, AL Ph. (256) Office Hours: Tues/Thurs 2-3PM JDW, ECE Summer 2010

2 Laser Gain Saturation Chapter 8: Laser Oscillations Laser Beam Growth beyond the Saturation Intensity Optimization of Laser Output Power Energy Exchange between Upper Laser Level Population and Laser Photons Laser Output Fluxuations Laser Amplifiers Above Threshold Chapter 8 Homework: 1, 2, 3, 4, 11, 8, 10 Cambridge University Press, 2004 ISBN-13: All figures presented from this point on were taken directly from (unless otherwise cited): W.T. Silfvast, laser Fundamentals 2 nd ed., Cambridge University Press, 2004.

3 Laser Gain Saturation

4 Population Densities of N u and N l with Beam Present Adding the two equations for upper and lower population states allows one to develop a relation for the lower state in terms of fluxes divided by the lower state decay rate The upper state can be rewritten as: Yielding:

5 Small Signal Gain Coefficient One can define the small- signal gain coefficient as the gain at resonance, when no beam is present as: Where: This value represents the capability of the gain medium to produce a laser. Thus in order to lase, the small-signal gain coefficient must be greater than the threshold gain coefficient Thus the gain: is the amount of gain that would be measured if a lowintensity, I o, beam were directed through a gain medium of length L to generate the high power output intensity, I

6 Saturation of the Laser Gain above Threshold We shall now consider a beam of high intensity, I, with a population difference: The saturated gain coefficient is then calculated using:

7 Saturation of the Laser Gain above Threshold We shall now consider a beam of high intensity, I, with a population difference: Note that the population density above does not depend on the lower level. This can only occur if the additional population flux l u due to stimulated emission is equal to the population flux of absorption ( u l). Thus: This also requires that above threshold that the gain coefficient goes to zero. This is indeed the solution for an ideal system However, in real systems only the net gains goes to zero. Mirror transmission losses, absorption losses of the media, and scattering losses in the system always require that the upper level population density is slight larger than the gain ratio multiplied by the lower state: N u g > g u l N l

8 Lasers Above Threshold: Exponential to Linear Growth

9 Lasers Above Threshold: Exponential to Linear Growth For a two mirror system above threshold: In steady state: Note: Losses associated with the geometry or material properties of the cavity may prevent laser excitation in the center of the homogeneous broadened mode. Thus one must always take the cavity into account when determining wavelength. Such rules will be discussed in Chapter 11

10 Optimizing Laser Output: Optimum Output of Mirror Transmission Lasers can be operated under many different combinations of mirror reflectivities. The key is to provide sufficient gain to overcome all losses associated with the mirror losses. Examples: assume g 0 =20%. Then any combination of reflective losses up to 20% will produce laser output. One chooses a mirror with as little loss as possible to maximize output intensity while maintaining some means by which the laser light is output from the cavity. Let us examine a case in which the mirror has sufficient loss to equal the gain generated. Under these conditions the gain per pass, g th L = a + t, where t is the mirror transmission, and a are the absorption and scattering losses of the mirror. The laser traveling through the cavity within the cavity has an average intensity of I SS /2 Assuming both mirrors have an equal transmittance: The relative intensity maximum can be obtained by differentiation: Which allows one to solve for the optimum transmission, t opt :

11 Examples Example: Semiconductor Lasers: Cleaved ends of the device are used as mirrors. The large change in refractive index between the semiconductor and air generates a Fresnel reflection of approximately 30% at both ends. Thus transmission is 70%. Antireflection coatings are sometimes used to increase transmission, allowing for the use of external mirrors to increase the effective cavity gain. Other times, metallic coatings are placed at one end of the laser to provide increased gain and mono-directional transmission. Example: HeCd laser has optimal transmission between 3-5% Example: HeNe laser Gain coefficient, g0 = 0.1/m at nm Absorption and scattering losses, a =0.5% per pas Discharge length, L = 0.2 m What is the optimum transmission for R = 99%? Based on these calc, t =0.5%, but we want only one exit port for laser, so 2xt = 1% The calculation appears easy, but measuring the constant for a is difficult for small values

12 Optimum Laser Output Intensity Continuing from our previous discussion, the optimized output intensity is: However one must take into account that only one side of the cavity should emit: One should note that this analysis was completed for and is valid for homogeneously broadened laser transmission. It can also be used for Doppler broadening IF the Doppler broadened gain bandwidth is of the order of the homogeneous bandwidth. NOTE: It will be shown later that if a 2-mirror cavity is placed around a gain medium where Doppler broadening is dominant, then the one will observe several longitudinal laser modes operating at slightly different frequencies. Each mode will have a slightly different gain profile. Thus commonly used laser systems with large Doppler broadening effects may be mode locked to emit only a single operating mode.

13 Optimum Laser Output Power The power transmitted by the laser can be written as: Optimization simply requires the optimized intensity: Where one should recall that: Note : Multiply by 2 for transmission through a single mirror in a two mirror system

14 Energy Exchange Between Upper Laser Level Population and Laser Photons So far in this chapter, we have established equations for the steady state output of a laser These equations do not therefore accurately describe time dependent factors, such as output stability, start-up, laser, gain fluctuations inside the medium, etc. In real system, laser output various range from small fluctuations to large spiking effects in laser output. Thus a time dependent solution should also be considered to address more complex phenomenon. Let us revisit the differential equations that govern the system.

15 Laser Decay Time in Optical Cavities Let s examine energy decay from first principles and then add the various decay components as needed Where t c is the effective decay time of the optical cavity resulting in the loss which can be expressed as the distance between the mirrors, over the velocity of light and a single pass length of the cavity, L f Thus one can determine the fractional loss of energy per pass in a mirror system as Note that to account for mirror reflectance, one can make use of the round trip steady state equation for a laser in terms of reflectance

16 Laser Decay Time in Optical Cavities First let us consider losses, so we will ignore the gain term If we express the loss term, α, in terms of the single pass length, then This allows us to define single pass length as in a 2 mirror cavity as: Which in tern allows us to write an explicit term for the energy decay rate of the cavity (or the resonator decay rate) as

17

18 Basic Laser Cavity Rate Equations Take the volume of a laser beam as the cross sectional area, A, times the length of the gain region, L, as V g The volume between the beam and the entire cavity region is V c = Axd V g The gain in any laser is where: Using this description, M is defined as the total number of laser species in each level We will also define a pumping flux, P u, as the number of electrons excited to level u where R u in all of our previous discussions was the number of electrons exited per unit volume We can now solve for the change in intensity per unit time as the laser passes along z as:

19 Basic Laser Cavity Rate Equations Now one can write a simple relation for di/dt for lasers where Vc Vg Next, the entire equation can be related to changes in the number of photons per unit time and compared to photon losses in the medium using the transmission time constant Applying upper and lower states to the amplifier φ/t c = light transmitted through the mirrors Where one uses K to reduce the equation into a very simplistic form One can also include spontaneous emission into the equation as such: t c = resonator decay rate

20 Alternate description

21 Basic Laser Cavity Rate Equations One can also include spontaneous emission into the equation as such: Assuming: must be valid in the laser system to obtain gain And that quantum theory requires: The photons gained in the system per unit time is: Which can be written in terms of pumping flux as:

22 Using Steady State Relations to Obtain Photon Number One can obtain steady state equations from: (Gain -1)* state For low photon output, population grows linearly with flux so: Providing a solution for pumping flux as: Leading to a steady state pumping flux of: Above Threshold: Where: yields: Allowing one to obtain the general solution:

23 Laser Fluctuations: Laser Spiking Solution of cavity based rate equation: Cannot be performed analytically, but can be solved numerically to provide an accurate description of multiple peak modes present in a laser cavity Continuous spiking Damped spiking oscillations

24 Relaxation Oscillations Consider a case where small time dependent perturbations exist in the stead state equation Damped spiking oscillations For the case:

25 Amplifiers and Pulse Lasers

26 Laser Amplifiers Amplifiers can be used to obtain exponential enhancements of ordinarily weak beams without modifying beam shape Think of a light source entering a laser cavity with emission of the same wavelength. The intensity of the source would be magnified exponentially to some value less than the saturation intensity of the laser We can solve for the upper limit of the incident light source required to achieve saturation as The minimum value for I o required to be effective amplification must be times that greater than the spontaneous emission of the amplifier. If not, then the amplified spontaneous emission (ASE) of the amplifier will appear as part of the emerging beam and not have the desired characteristics of the input beam. (the output beam will be that of an independent laser function and not one driven to amplify the input signal) For amplified beams, we do NOT want spontaneous emission in the amplifier to contribute to the beam output. In order to add energy to an already intense beam, then it is desirable to have the input intensity above I sat This allows the beam to transfer all of the stored energy from upper level states to stimulated emission In cases where multiple amplifiers are used, it may be necessary to increase the diameter of the beam in order to take full advantage of the cross sectional area of the amplifier, as I sat can be maintained over much larger cross sections

27 Propagation of High Power, Short Duration Pulses in Laser Amplifiers In short pulse lasers, the upper level lifetime is significantly shorter than the pulse duration. Thus one cannot consider a simple steady state solution We will therefore evaluate short pulse lasers and their amplification in terms of energy fluence, F Recalling our photon population equations for the change of states over time If we apply the chain rule to account for the time dependence of the flux, then Here we have dropped the term for spontaneous emission because we are considering a relatively high power beam where spontaneous emission would be small

28 Propagation of High Power, Short Duration Pulses in Laser Amplifiers Next, we use the condition that we are pumping light into a cavity (laser or amplifier) in which the time constant is much shorter than the pulse duration, thus Let us provide some simplification by converting energy density into intensity We can now write

29 Propagation of High Power, Short Duration Pulses in Laser Amplifiers Allowing one to determine a time dependent differential equation for flux as a function of intensity Where the gain coefficient is of course The solution to this differential equation is expressed as: Where the pulse energy fluence, F, is the energy put unit area up to a time t is solved for as In the case where no pumping or relaxation terms are included, one can find F sat as

30 Propagation of High Power, Short Duration Pulses in Laser Amplifiers In most solid state lasers however, the lower level empties rapidly, requiring Furthermore, one can use the relationship between fluence, and saturated fluence along with To provide the intensity solution in terms of state densities as

31 Propagation of High Power, Short Duration Pulses in Laser Amplifiers If one passes a sufficient number of pulses through the gain medium, then F = Fsat and the gain will be reduced by 1/e. yielding The change in pulse energy fluence does not depend on the shape of I because Therefore

32 Propagation of High Power, Short Duration Pulses in Laser Amplifiers All of these efforts have lead so a simple reduced form Providing a solution of When then When then Which shows that fluence increases linearly with length after saturation is reached.

33 Common Geometrical Configurations of Amplified Laser Systems

34 Fluence and Intensity vs Thermal Damage in Laser Media

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