MEFT / Quantum Optics and Lasers Suggested problems Set 4 Gonçalo Figueira, spring 05 Note: some problems are taken or adapted from Fundamentals of Photonics, in which case the corresponding number is indicated. At the end of the list you may find the solutions to selected problems. Lasers (5.-) Threshold of a Ruby Laser a) At the line center of the λ 0 = 694.3 nm transition, the absorption coefficient of ruby in thermal equilibrium (i.e., without pumping) at T = 300 K is α( ν 0 ) γ ( ν 0 ) 0. cm -. If the concentration of Cr 3+ ions responsible for the transition is N a =.58 0 9 cm -3, determine the transition cross section σ 0 = σ( ν 0 ). b) A ruby laser makes use of a 0 cm Iong ruby rod (refractive index n =. 76) of crosssectional area cm and operates on this transition at λ 0 = 694.3 nm. Both of its ends are polished and coated so that each has a reflectance of 80%. Assuming that there are no scattering or other extraneous losses, determine the resonator loss coefficient α r and the resonator photon lifetime τ p. c) As the laser is pumped, γ(ν 0 ) increases from its initial thermal equilibrium value of 0. cm - and changes sign, thereby providing gain. Determine the threshold population difference N t for laser oscillation. Number of Modes in a Gas Laser (based on 5.-) The spectral distribution of the light generated inside a laser is determined both by the gain profile and by the resonator modes. In order that a laser works, its gain coefficient must be greater than the loss coefficient of the resonator [ γ 0 ( ν) > α r ]. Since the gain changes with frequency, this condition is normally verified only inside a finite spectral band of width B centered about the atomic resonance frequency ν 0 (see figure below). The bandwidth B increases with the atomic linewidth Δν and the ratio γ 0 ( ν 0 )/ α r ; the precise relation depends on the shape of the function γ 0 ( ν). It is clear that only a finite number of oscillation frequencies ( ν,ν ν M ) are possible. The number of possible laser oscillation modes is therefore M B ν F, where ν F = c/d is the approximate spacing between adjacent modes. Consider a Doppler-broadened gas laser having a gain coefficient with a Gaussian spectral profile (see Sec. 3.3D and Exercise 3.3-) given by γ 0 ( ν) = γ 0 ν 0 exp ν ν 0 σ D, FWHM = Δν D = 8ln σ D
a) Derive an expression for the allowed oscillation band B as a function of Δν D and the ratio γ 0 ( ν 0 )/ α r where α r is the resonator loss coefficient. b) A He-Ne laser has a Doppler linewidth Δν D =.5 GHz and a midband gain coefficient γ 0 ( ν 0 ) = 0 3 cm -. The length of the laser resonator is d = 00 cm, and the reflectances of the mirrors are 00% and 97% (all other resonator losses are negligible, i.e. α s 0). Assuming that the refractive index n =, determine the number of laser modes M. (5.-) Number of Longitudinal Modes An Ar + -ion laser has a resonator of length 00 cm. The refractive index n =. a) Determine the frequency spacing ν F between the resonator modes. b) Determine the number of longitudinal modes that the laser can sustain if the FWHM Doppler-broadened linewidth is Δν D = 3.5 GHz and the loss coefficient is half the peak small-signal gain coefficient. c) What would the resonator length d have to be to achieve operation on a single longitudinal mode? What would that length be for a CO laser that has a much smaller Doppler linewidth Δν D = 60 MHz under the same conditions? (5.3-) Operation of an Ytterbium-Doped YAG Laser Yb 3+ :YAG is a rare-earth-doped dielectric material that lases at λ 0 =.030 µm on the F 5/ F 7/ transition (see Tables 3.-, 4.3-, 5.3-, and the figure below). This threelevel laser is usually optically pumped with an InGaAs laser diode. a) The pump band (level 3) has a central energy of. 395 ev and a width of 0.0475 ev. Determine the wavelength of the desired laser-diode pump and the width of the absorption band in nm.
Energy levels pertinent to the ytterbium-doped YAG laser transition at λ 0 =.030 µm. b) At the central frequency of the laser transition ν 0, the peak transition cross section σ 0 σ( ν 0 ) = 0 0 cm. Given that the Yb 3+ ion doping density is set at N a =.4 0 0 cm -3, determine the absorption and gain coefficients of the material at the center of the line, α( ν 0 ) γ ( ν 0 ). Assume that the material is in thermal equilibrium at T = 300 K (i.e., there is no pumping). c) Consider a laser rod constructed from this material with a length of 6 cm and a diameter of mm. One of its ends is polished to a reflectance of 80% ( = 0.8) while the other is polished to unity reflectance ( =.0). Assuming that there is no scattering, and that there are no other extraneous losses, determine the resonator loss coefficient α r and the resonator photon lifetime τ p. d) As the laser is pumped, the gain coefficient γ ( ν 0 ) increases from its initial negative value at thermal equilibrium and changes sign, thereby providing gain. Determine the threshold population difference N t for laser oscillation. e) Why is it advantageous to have the energy of level 3 close to that of level? Pulse mode-locking in different laser media Consider a mode-locked laser consisting of a d =.5 m long cavity and with Nd:glass as the gain medium, operating at λ 0 =.053 µm. The transition linewidth, according to Table 4.3-, is Δν = 7 THz. a) Calculate the spacing between consecutive resonator modes ν F, the corresponding period T F and the repetition rate (number of pulses per second). b) What is the number of allowed modes contributing to the generated pulses? c) Determine the mode-locked pulse duration and their spatial length. d) For an average output intensity of 50 mw, what is the peak power of each pulse? What is the energy carried by each pulse? e) Repeat (a) (e) for a Ti:sapphire laser with λ 0 = 0.8 µm and Δν = 00 THz. 3
The Structure of Laser Pulses (.-) The Hyperbolic-Secant Pulse A pulse has a complex envelope sech(t / τ), where sech(x) = / cosh(x) and τ is a time constant. Show that the width of the intensity function τ FWHM =.76 τ, the spectral intensity S( ν) = sech ( π τν), and the FWHM spectral width ν = 0.35/τ. Compare to the Gaussian pulse. Temporal broadening of a 0 fs Gaussian pulse in BK7 A 0 fs FWHM transform-limited Gaussian laser pulse at λ 0 = 800 nm propagates through a mm plate of BK7, a transparent optical material. This optical material can be considered as a chirp filter with a chirp coefficient b = GDD L, where GDD = 44 fs /mm is called the group delay dispersion and L is the material thickness. a) Calculate the /e pulse duration τ and FWHM bandwidth Δν FWHM of this pulse. b) Calculate the (empty space) wavelengths λ + and λ - corresponding to the limits of the FWHM spectral bandwidth. c) Determine the percentage of pulse temporal broadening after propagation through mm of BK7. What is the new FWHM duration? d) Show that for a large chirp coefficient such that b >> τ the formula relating the broadened to the intial pulse duration can be simplified to τ = b τ. e) Calculate the pulse duration after propagation through cm of BK7. f) What is the temporal separation between the wavelengths λ + and λ - in the resulting chirped pulse? Compare the result with the broadened pulse duration. 4
Selected Solutions (5.-) Threshold of a Ruby Laser a) At 300 K: We have N σ ν 0 γ 0 ( ν 0 ) = N 0 σ( ν 0 ) N 0 = N N N = exp ΔE kt = exp hc λkt e 69 9 0 3 N a = N + N N N 0 = N N N a = γ 0 ( ν 0 ) N 0 = α ν 0 N a.66 0 0 cm b) Loss coefficient: α r = α s + d ln R R = 0 + 0 ln 0.03 cm - 0.64 Photon lifetime: c) τ p = α r c = c 0 / n N t = α r.63 ns α r σ( ν) 0.03.66 0 0.76 0 8 cm -3 Number of Modes in a Gas Laser a) For simplicity let us define γ 0 ( ν 0 )/ α r = ρ. The threshold gain condition is γ 0 ( ν) = ρexp ν ν 0 α r σ D > The definition of B (see figure) is such that values of the frequency inside the range [ ν 0 B,ν 0 + B ] obey the expression above. At the limiting frequencies the expression becomes equal to (gain = loss) so we may write ρexp B σ D From this we arrive at B = Δν D = lnρ ln b) The number of modes M can be estimated fromm B Δν F. Δν F = c d = ( c 0 / n ) = 50 MHz d α r = d ln = 0.05 m - R R ρ = 0. 0.05 3.3 B = Δν D lnρ ln.97δν D.9 GHz M = B Δν F = 9 5
Number of Longitudinal Modes a) Δν F = c d = ( c 0 / n ) = 50 MHz d b) For this particular case we have ρ = γ 0 ( ν 0 )/ α r = B = Δν D The number of modes is therefore M = B Δν F = Δν D Δν F = 3 c) The requirement for a single-mode is equivalent to the following condition: M = Developing this, B Δν F = B Δν F < B Δν F = Δν D d c < d < c 0 / n Δν D For a CO laser: 8.57 cm We have c) N N = exp( ΔE kt ) e 46.6 5.73 0 N a = N + N N N 0 = N N N a α( ν 0 ) = γ 0 ( ν 0 ) = N 0 σ( ν 0 ) =.8 cm - α r = α s + d ln R R = 0 + ln 0.086 cm - 0.8 τ p = α r c = α r c 0 n 3.6 ns [Note that n(yb:yag) =.8 cf. Table 4.3-] d < c 0 / n Δν D 5 m d) N t = α r σ( ν) 9.3 07 cm -3 (5.3-) Operation of an Ytterbium- Doped YAG Laser a) λ = hc ΔE Δν = ΔE h Δλ = λ c 940 nm 5.985 THz Δν 7.63 nm b) [Cf. 5.- (a)] γ 0 ν 0 = N 0 σ ( ν 0 ) N 0 = N N ΔE = hc λ 0 =.93 0 9 J.06 ev At 300 K: (.-) The Hyperbolic-Secant Pulse A( t ) = sech(t τ) = I ( t ) = sech (t τ) = cosh(t τ) cosh (t τ) To calculate the temporal FWHM we write I ( t ) = cosh( t τ) = ± t = ±τ cosh ±0.88τ t FWHM = t + t =.76 τ The Fourier transform is A( ν) sech π ν I ( ν) sech ( π ν) = sech π ν 6
To calculate the spectral FWHM we write Finally: = ± ( ) I ( ν) = cosh π τν ν = ± cosh π τ ± 0.0893 τ ν FWHM = ν + ν = 0.786 τ t FWHM ν FWHM 0.35 Temporal broadening of a 0 fs Gaussian pulse in BK7 a) τ FWHM τ = 8.47 fs ln Δν FWHM 0.44 τ FWHM = 44 THz b) Given the relatively large bandwidth (compared to the central frequency) we calculate each limiting frequency individually: ν 0 = c 0 λ 0 = 375 THz ν + = ν 0 + Δν FWHM ν - = ν 0 Δν FWHM = 397 THz = 353 THz e) For a 0 mm long plate of BK7: b = 44 0 = 880 fs τ b 04 fs τ () τ FWHM = lnτ fs f) A chirp filter applied on a transform-limited Gaussian pulse introduces a chirp coefficient: a = b τ = 880 8.47.3 This chirped pulse will have the instantaneous frequencies linearly distributed over its duration such that ν i t = ν 0 + a πτ t We now need to find the time delay between the two frequencies defined in (b) Δν FWHM ν + ν = a πτ Δt Δt = πτ Δν a FWHM fs This results for the wavelengths: λ + = c 0 ν + = 756 nm λ = c 0 ν = 850 nm Δλ = 94 nm c) τ = τ + b () τ FWHM τ 4.58τ.58τ FWHM = 5.8 fs d) τ = τ + b τ 4 τ b τ 4 = b τ 7