Chapter 11. Laser Oscillation : Power and Frequency Power & Frequency of Single mode continuous wave (cw) laser ? 11.2 Output Intensity : Uniform-Field Approximation Assumptions : (1) Homogeneous broadened gain medium (2) Mean-field (uniform-field) approximation (3) All loss processes are independent of the cavity intensity (4) Steady-state (temporal) or CW operation 1) Gain (10.5.8) , => : The gain is clamped on the threshold gain for cw(steady-state) oscillation.

2) Output intensity (10.12.10) or (10.11.14) => Output intensity : If => : A given medium, laser intensity depends on how the laser cavity (t or s) is chosen. =>

11.3 Optimal Output Coupling 1) Optimal mirror transmission coefficient 2) Maximum output intensity : Lasing is possible when : scattering loss coefficient If : theoretical upper limit of laser intensity

2) Maximum output intensity – another approach (10.12.8) => : uniform field approx. (11.2.5) => If : maximum intensity per unit volume Maximum intensity extracted from the medium of length l ; : (11.3.5a)

3) Input-to-output power conversion efficiency 3-level system, (10.7.12) => (10.11.12) => (10.11.7) => In the case of strongly saturated case, : quantum efficiency

11.4 Effect of Spatial Hole Burning Standing wave inside the cavity, (10.2.9) => (11.3.6) => : Power per unit volume, at the point z, extracted from the medium by stimulated emission. The rate at which the field gains energy should equal the rate at which it losses energy ; For

Put, Output intensity : : Disired solution is the one with minus sign since should be equal to 1 when g0/gt=1. Output intensity : : The effect of spatial hole burning is to reduce the output intensity

11.5 Large Output Coupling Our analysis of output power thus far has assume that the output coupling is small( ), and we have also assumed time averaged intensity I(+) and I(-) are independent of z. We will now allow arbitrary output coupling and therefore allow the possibility I(+) and I(-) may vary with z. Ignoring the spatial hole burning, (11.2.4) => (10.4.3) =>

=> =>

=> Output intensity : When r1=1, t1=s1=0, r=r2, t=t2, s=s2, and t+s<<1 (11.2.11) : small output coupling

<Total two-way intensity> (11.5.10) => (11.5.12) => (11.5.13) => Total intensities are comparable at the two mirrors for reflectivity as low as 50%.

11.6 Measuring Small-Signal Gain and Optimal Output Coupling Eq. (11.2.11) for output intensity or its generalization (11.5. 18) has been shown experimentally to be quite accurate, because the spatial hole burning effect is usually negligible in gas lasers. In general the small signal gain and the saturation intensity are difficult to calculate accurately, because the puping and decay rates of the relavant atomic levels may not be well known. => Experimentally measured !

<Maximal loss method to measure g0> - The cavity loss is varied by inserting a reflecting knife-edge into the cavity - The cavity loss is increased until the laser oscillation ceases. - (11.2.4) => g0(=gt) : loss just when the laser oscillation ceases. <Simultaneous measurement method> - Scattering coefficient, s=Pin/P+ - Effective output coupling : t+s - t=Pout/P+ : known => s=tPin/Pout - Ptotal=Pin+Pout : total output power - Determine s-value for which Ptotal is maximum => topt=sopt + t., Ptotal=(Pin+Pout)topt - Small signal gain : s-value at which laser oscillation stops.

11.7 Inhomogeneously Broadened Laser Media In an inhomogeneously broadened gain medium the different active atoms have different central transition frequencies n21. # Small signal gain : Doppler broadened lineshape The gain coefficient is obtained by integrating the contributions from the different frequency components, each of which saturates to a differnet degree depending on its detuning from the cavity mode frequency n.

cf) homogeneous medium, (11.2.9) where, The gain saturation set in more slowly as the intensity I is increased in the case of inhomogeneous broadening medium. Output intensity : cf) homogeneous medium, (11.2.9)

11.8 Spectral Hole Burning and the Lamb Dip Spectral packets : The atom group with the central transition frequency of The gain for spectral packets with frequency n21~n(field frequency) is saturated more strongly than others : spectral packets with frequency detuned from n by much more than the homogeneous linewidth, i.e., |n21-n|>>dn21, are hardly saturated at all. Spectral hole burning (Bennet hole)

<Lamb dip> Suppose the cavity mode frequency n the center frequency of the Doppler gain profile i) The traveling-wave field propagating in the +z direction will strongly saturate the spectral packet of atoms with Doppler-shifted frequencies n’21=n. : The Doppler effect has brought these atoms into resonance with the wave. Therefore, those atoms have the z component of velocity given by ii) Similarly, the traveling-wave field propagating in the -z direction will strongly saturate those atoms with the z component of velocity given by

=> The standing wave cavity field will burn two holes in the Doppler line profile. When the mode frequency is exactly at the center of the Doppler line, the two holes merge together. => The field can now strongly saturate only those atoms having no z component of velocity. => The output power exactly at resonance will be lower than slightly off resonance. : Lamb Dip.

11.9 Cavity Frequency and Frequency Pulling Cavity mode frequency : In general, where, l : gain medium length, n: refractive index. or, where, : bare(passive) cavity mode frequency (3.3.22), (3.3.25) => for homogeneous broadening medium put, : cavity bandwidth : frequency pulling

<Frequency pulling and gain> In most lasers, (homogeneous broadening) (inhomogeneous broadening) : The larger the threshold gain gt, the greater the frequency pulling for fixed gain linewidth (dn21 or dnD).

<Mode spacing> : The effect of frequency pulling is to reduce the mode spacing from c/2L.

11.11 Laser Power at Threshold Laser power near the threshold ? => spontaneous emission. (10.5.7) (Mean-field approx.) (10.11.12) Including the spontaneous emission :

Steady-state solution : (11.11.1) => <1 : below the threshold >1 : above the threshold define, (far above threshold) : (11.2.5), (11.2.9) : (11.2.5), (11.2.9) (near threshold) (10.11.8), (10.11.10), P>>G21

In many lasers, PV~103s-1, f~1, dn21~10GHz => qsat~1010 (very low power) cf) (ex) 6328 A He-Ne laser, <The rate of change of qn with y> Extremely rapid rise in the cavity photon number at the point y=1 (threshold).

11.12 Obtaining Single-Mode Oscillation 1) Short cavity length Ex) Dng~1500 MHz (He-Ne laser) => c/2L>1500MHz => L<10 cm (low power) 2) Homogeneous broadening medium

3) Fabry-Perot etalon / Grating / Prism => Selective transmission Ex) Fabry-Perot etalon - resonance frequency :