Overall Ingle and Crouch, Spectrochemical Analysis

For an ideal black body, the rate of absorption and emission must be balanced: B ij U n i = A ji n j + B ji U n j Rearrange:

Population of Energy States Classical Boltzmann Population: n i = n 0 e -  i /kT The higher the energy state of interest, the greater the value of  i, and the fewer atoms there will be in that state. To find the ratio of populations between two states: n j = n i e -(E j -E i )/kT = n i e -h ji /kT Thus, we can predict the excited state population that is eligible for emission.

Are you getting the concept? Determine the population ratio for atoms/molecules in two energy states spaced by 1 eV at T = 300 K: Recall: h = 6.63 x Js k = 1.38 x J/K 1 eV = 1.6 x J njnj nini

We know: Set equal and solve for U v : Looks similar to Planck’s Radiation Law: Spectral Energy Density

Population Inversion Goal: More atoms or molecules in the upper energy level than the lower energy level. Heating the lasing medium will not work: n j = n i e -(E j -E i )/kT Must selectively excite atoms/molecules to particular energy levels. Most common approaches: *light*electricity

Optical Pumping Intense light source at h  (e.g. flash lamp) Excites to a metastable state to achieve population inversion With fast flashing, initial photons start chain reaction Eugene Hecht, Optics, Addison-Wesley, Reading, MA, 1998.

Electrical Discharge Accelerated e - and ions excite atoms/molecules into higher energy states Common in gas lasers Ingle and Crouch, Spectrochemical Analysis

Three - Level System No saturation Not very efficient Better for pulsed mode operation Ingle and Crouch, Spectrochemical Analysis

The ruby laser is a three – level laser Eugene Hecht, Optics, Addison-Wesley, Reading, MA, Commercial ruby laser operates with efficiency ~ 1%

Four - Level System More efficient than 3-level Laser transition does not involve ground state or most highly excited state Easier to achieve population inversion Ingle and Crouch, Spectrochemical Analysis

The He – Ne laser is a four – level laser He* + Ne → He + Ne* + ΔE

Resonance Cavity and Gain Ingle and Crouch, Spectrochemical Analysis Gain = degree of amplification based on positive feedback

Gain Gain (G) = e  (n j -n i )b  = transition cross-section b = length of active medium Oscillation begins when: gain in medium = losses of system  1  2 G 2 = 1 Threshold population inversion: Ingle and Crouch, Spectrochemical Analysis

Eugene Hecht, Optics, Addison-Wesley, Reading, MA, Light Amplification in Resonance Cavity Highly collimated beam Typically ~mm beam width, ~mrad divergence A typical photon travels about 50 times forward and backward within the cavity

Mirror Arrangements Eugene Hecht, Optics, Addison-Wesley, Reading, MA, 1998.

Are you getting the concept? Knowing that the purpose of the resonance cavity is to direct the majority of the photons back through the active medium, what cavity characteristics will be most important?

Achieving Resonance Stimulated emission is coherent (all light waves in phase) If the cavity is an integer multiple of the wavelength, each wave will be at the same phase when it reflects from one of the cavity mirrors (recall that a photon make many round trips in a laser cavity before it is emitted). This allows constructive interference between all photons. Want: m = 2nL Other wavelengths will not be strongly amplified, and thus, will die out. In practice, laser transitions have gain over a range of wavelengths – the gain bandwidth… so that resonance cavity lengths are not impossible to achieve.

Achieving Resonance Goal: Laser cavity where L = m /2 This condition is not as strict as it sounds because: 1.Laser transitions have gain over a range of wavelengths 2.Any integer multiple (longitudinal mode) of will work Amp = (1+Gain) L Estimate amplification factor:

Longitudinal Modes Eugene Hecht, Optics, Addison-Wesley, Reading, MA, Actual is the convolution of the transition bandwidth and the of the longitudinal modes.

Transverse Modes and Transverse modes determine the pattern of intensity distribution across the width of the beam. TEM 00 has a Gaussian distribution and is the most commonly used. The resonator geometry of many commercial lasers is designed to obtain “single transverse mode” operation.

Coherence Factors that compromise coherence: 1. thermal fluctuations 2. vibrational fluctuations 3. emission of multiple wavelengths 4. multiple longitudinal modes Temporal Coherence – How long do the light waves remain in phase as they travel? Coherence Length = 2 /n 

Coherence Spatial Coherence – Over what area does the light remain in phase?

Are you getting the concept Calculate the coherence length for the sources below using n air = 1.00: (a)light bulb emitting from nm (b)semiconductor laser emitting from – nm (c)He-Ne laser emitting from – nm

Laser Wavelengths Factors influencing monochromaticity of laser light: 1. transitions responsible for emission 2. nature of transition determines bandwidth 3. resonance cavity characteristics Doppler bandwidth:  = [5.545 kT/Mc 2 ] ½ where M is the mass of the atom/molecule

Limiting Emitted s with a Fabry-Perot Etalon Insert a pair of reflective surfaces that form a resonant cavity tilted at an angle to the axis of the laser medium. Transmitted  depends on: 1.the angle the light travels through the etalon (  ) 2.the thickness of the etalon (l) 3.the refractive index of the material between the 2 surfaces (n)

Emission Mode Lasers can emit light in continuous wave (cw) mode or they can produce pulses. Heisenberg’s Uncertainty Principle places the limitations: Bandwidth (Hz) = 0.441/Pulse Length (s) Bandwidth (Hz) = 0.441/Pulse Length (s)  E  t ≥ ħ/2 Consequences: Long pulse – narrow bandwidth Short pulse – broad bandwidth Long pulse – high resolution Short pulse – low resolution

Are you getting the concept? Calculate the minimum pulse length for a laser with a 1-nm emission bandwidth at a center wavelength of 500 nm.

Are you getting the concept? Calculate the best spectral resolution (in cm -1 ) that can be achieved with a pulse length of 368 fsec. Recall: ħ = x Js

Pulsed Laser Power Considerations Consider a Gaussian beam profile: Peak Power FWHM Rise Time Fall Time Power Time If power was constant: E = Pt In this case, E = ∫P(t)dt Average Power = ΣE/t or Peak Power x Duty Cycle Duty cycle = Pulse Length x Repetition Rate

Output Power Output power will depend on: 1.variations in power level with time 2.efficiency of converting excitation energy into laser energy 3.excitation method 4.laser size What is wall-plug efficiency? A practical measurement of how much energy put into the laser system (from the wall plug) comes out in the laser beam. Active Medium power supply

Accessible Wavelengths Lasers have also been prepared for the vacuum UV (VUV, nm) and XUV (eXtreme UltraViolet; also called the ultrasoft X-ray region; <100 nm). The shortest wavelength laser produced so far emits at 3.5 nm. Projects to extend this range to 0.1 nm by 2011 are in progress. Why x-ray lasers are so difficult to build: A ji /B ij = 8  h 3 / c 3 Intensity of stimulated emission (B) is proportional to intensity of spontaneous emission (A). Tallents, G.J. J. Phys. D 2003, 36, R259. Dattoli, G.; Renieri, A. Nucl. Inst. Meth. Phys. Res. A 2003, 507,