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 35899 Ph. (256) 824-2898 email: [email protected] Office Hours: Tues/Thurs 2-3PM
JDW, ECE Summer 2010 Chapter 11: Laser Cavity Modes
• Longitudinal Cavity Modes – Fabry-Perot resonator – Fabry-Perot cavity modes – Longitudinal modes – Mode number • Transverse Laser Cavity Modes – Diffraction integral – Plane parallel mirrors – Curved mirrors – Spatial distributions – Gaussian shamed modes • Properties of Laser modes – Mode characteristics – Effects of mode on gain medium profile Cambridge University Press, 2004 Chapter 11 Homework: 2,4,5,6,8,11,12 ISBN-13: 9780521541053
All figures presented from this point on were taken directly from (unless otherwise cited): W.T. Silfvast, laser Fundamentals 2nd ed., Cambridge University Press, 2004. Fabry-Perot Resonator
• Consider two partially reflective mirrors parallel to one another. • The mirrors may be nearly entirely reflective, but will always have some transmission value • Let us consider light incident on the mirror at an angle θ • Light reflecting back and forth between the mirrors is depicted here
Mirror plane 1 Mirror plane 2 Fabry-Perot Resonator
• Let us consider light incident on the mirror at an angle θ • Light reflecting back and forth between the mirrors can be modeled at an angle 2θ • The reflected light follows an additional path length of a+b
Mirror plane 1 Mirror plane 2 Fabry-Perot Resonator
• The additional path length generates a phase change in the propagating wave. • This can be shown by taking an incident plane wave, eikz , where k is the wave vector and z is the incident propagation direction • One can express the phase factor as
• Furthermore, one can sum all of the transmitted amplitudes as
Fabry-Perot Resonator
• The transmitted intensity of the wave is
• Where the exponential term represents the phase change upon two reflections
• If we define the reflectivity of the mirror, R, and transmission, T, then one can write
Airy Function Fabry-Perot Resonator Intensity Fabry-Perot Resonator
• Each of the Airy function peaks is identical in shape for each value of Φmax • Thus, we can solve for the FWHM by considering the peak at n = 0 • For values of R > 0.6, we can approximate the sin(Φ/2) as Φ/2. • The value of Φ at which the Airy function reduces to its half max is referred to as Φ’
Fabry-Perot Resonator
• This value, F, is called the finesse and is solved in a two mirror system as
• Let us consider the simple case where the incident angle , θ, is zero and the phase change, φr, =0
Frequency difference between allowed modes Fabry-Perot Resonator Intensity
Note: use this value for ν if the gain medium is placed with a gap distance d-L between the medium and the mirrors Fabry-Perot Resonator
• The frequency spread of the FWHM can be solved as:
• For mirrors with unequal reflectivity, the function is slightly more complex
• The quality factor, Q
Fabry-Perot Cavity Modes
• For a cavity with refractive index = 1,
• Consider the n = 1 mode of a 0.1 m cavity with Reflectivity 0.99
14 • For visible light, ν is approx 5x10 /s so nmax is something like 35,000 • In order to reflect with a high surface finish, the surface quality must be better than λ/10 • Furthermore, the reflective intensity of the cavity at the operating frequency should be 99 times greater than [1-0.01]/0.01, thus the total cavity intensity at the desired wavelength s 199 times greater than the light transmitted through the mirror • Therefore light energy is essentially stored within the cavity at wavelengths with high reflectivity
Longitudinal Cavity Modes Longitudinal Cavity Modes Longitudinal Cavity Modes Transverse Cavity Modes
• So far we have considered a plane wave traveling through a Fabry-Perot cavity • However, we don’t have a plane wave. • We have a wave propagating and being generated in a medium that is most accurately described as a Gaussian beam • Therefore, we must consider longitudinal and transverse effects acting on the wave as it propagates
Fresnel Kirchhoff Diffraction Integral
• Let us begin by first examining the properties of a Gaussian beam. We can develop these properties by extracting the near field diffraction of a plane wave exiting a small aperture
• Let Up be the wave function exiting the aperture
• Let Uo be the wave front incident
Fresnel Kirchhoff Diffraction Integral in Fabry-Perot Cavities
• Consider two circular mirrors
Fresnel Kirchhoff Diffraction Integral in Fabry-Perot Cavities
• Equivalence of two passes
Fresnel Kirchhoff Diffraction Integral in Fabry-Perot Cavities
• Equivalence of two passes
Transverse Modes Using Curved Mirrors
• The same method can be applied to two curved mirrors to evaluate the diffractive losses in the system. • The Fresnel number is a geometric parameter that allows one to quickly evaluate the fractional losses per pass in the mirror system based on the integral previously described
Example 2: TEM mode distributions Example 2: TEM Modes 00 and 01 Transverse Mode Frequencies
• Typical transverse modes are slightly off axis of the longitudinal modes • Thus, there is significantly greater differences between longitudinal and transverse modes • However, at least one transverse mode will always exist in a cavity of one or more longitudinal modes
Transverse Mode Patterns in Circular Symmetric Cavities Transverse Mode Patterns in XY Symmetric Cavities Brewster Angle within a Cavity
• In the x-y symmetric mode distribution, the modes demonstrated a preferred orientation due to the angle of incidence with respect to the mirror • This is determined by radially nonsymmetric loss within the cavity • One can use the Brewster angle window arrangement shown below to minimize the reflective losses associated with beam propagation between the gain medium (or amplifier) and the mirrors • At the Brewster’s angle there is no loss of reflectivity for polarized light parallel to the slide as shown in the figure • Furthermore, there is only 15% loss of all transverse light perpendicular to the slide Mode Properties
• Spatial Dependence – Each mode in a typical two mirror cavity is associated with a mode number, n – Each transverse mode must be associated with a standing longitudinal wave and therefore is a specific longitudinal mode number, n, as well as a transverse mode numbers l and m • Frequency Dependence – Each mode has a slightly different frequency – Transverse modes will have the same n number but slightly different frequency due to l and m mode numbers – Longitudinal mode numbers, n, typically have a larger frequency difference than l and m • Mode Competition – For homogeneous broadening, the waves associated with different modes are all competing for the same upper laser level species – Thus each mode is attempting to reach saturation – The mode at the center of the gain profile will reach saturation first causing other modes to decrease in amplitude – In homogeneous broadening, it is common to saturate one longitudinal mode yet leave transverse modes associated with that mode gaining b/c of their distinctly different spatial regions Mode Properties
• Effects of Modes on Gain Medium Profile – Modes that reach gain saturation intensity generate significant increases in gain over the short spectral width of the mode – Two effects of mode competition can have a significant impact on this condition • Spectral Hole Burning • Spatial Hole Burning Spectral Hole Burning
• Spectral Hole Burning: mode competition can lead to multiple laser output modes, causing the amplifier gain to be decreased dramatically at each of the competing frequency bands • This gain decrease is observed in Doppler broadened emissions where the width of the Doppler broadened beam gives rise to the output spectral density of the laser Spatial Hole Burning
• Consider a single standing wave longitudinal mode developed in a 2 mirror homogenously broadened laser cavity • The intensity of the incident light will be zero at every half wavelength interval within the cavity • There is therefore no stimulated emission present at those notes where the electric field is zero. • The gain profile therefore has a periodic spatial variation within the gain medium that is 90 degrees out of phase with the laser intensity profile within the gain medium • It is possible that higher order transverse modes might take advantage of this periodicity to generate strong secondary modes
• Ring lasers were developed in attempt to recycle the light back into the system with a slight phase error in attempt to generate more gain in the primary mode