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Some quantum properties of light Blackbody radiation to lasers Density of Field Modes in a Cavity z Perfectly conducting walls. Tangential component of E must vanish at bondary y L x kx nx ky L nx , n y , nz 0,1,2,3 n y L kx nz L kz Lattice points in the positive octant of k-space kx , k y , kz 0 ky kx Density of States for Radiation k dk k2 2 dk or 2 2 3 d c The number of field modes per unit volume having their wavenumber between k and k + dk or angular frequency between and + d Each mode has total energy of: 1 Ek (nk + ) k 2 nk is the number of photons in a specific mode Quantization of electromagnetic modes produces energy levels just like that of the harmonic oscillator. 1 Note that if nk = 0 the energy is: 2 This is called the “energy of the vacuum. In thermal equilibrium at temperature T the probability Pn that the mode is thermally excited to the nth state is given by the Bolzmann factor Pn exp En k BT exp E n 0 n k BT where k B Bolzmann' s constant 1.38 10 Pn e n k BT 1 e k BT 23 J /K The mean number, of photons excited in a particular mode at temperature T is: n k BT n k BT e n nPn n1 e n 0 n 0 Planck thermal excitation function kT 1 e B 1 What is the mean energy density of the radiation in these modes at temperature T? W d n d 2 1 d k BT 2 3 1 c e 3 d 2 3 k BT c e 1 Planck’s Law for radiative energy density Energy Density of Radiation d 0 W d 0 2c3 e kBT 1 2 k B4 3 3 15c 3 T urad 4 Units of J/m3 Stefan-Bolzmann Law q What is intensity of radiated blackbody light. Maximum of distribution found by taking derivative with respect to , and equate to zero. max T 2.898 10 m K 3 Wien’s Displacement Law Cosmic Microwave Background Einstein A & B Coefficients E2, N2 A21 spontaneous emission B12W absorption B21W stimulated emission Consider N atoms in a cavity where the energy density of the radiation is: W E1, N1 From Planck: 3 1 W 2 3 k BT c e 1 Thermal Equilibrium implies: dN1 dN 2 N 2 A21 + N 2 B21W N1 B12W 0 dt dt A21 A21 W k BT N1 B12 B21 B12 B21 e N2 B12 B21 and 3 B21 A21 2 3 c Note that: 3 1 1 B21 2 3 k BT A21 k BT 1 e 1 c e B21W A21n Thermal-stimulated emission rate is equal to the spontaneous emission rate multiplied by mean number of photons of frequency . Total Emission Rate B21W + A21 A21n + A A21 n + 1 Spontaneous emission, or is it stimulated emission that was “stimulated” by the vacuum!??! Atom in a cavity?!? In general, W() tends to be small. In order to excite atoms one needs a powerful source of light, like a laser. N atoms dN1 dN 2 N 2 A + N 2 N1 BW dt dt Say at t=0 N=N1 (all atoms in ground state) N2 A + BW N t N A + 2BW 1 Lasers Pump R~.95 R~.999 Gain Medium L 2L nc n some integer Standing light wave between mirrors. If 1W exits laser, 20W inside cavity 2 E2 1 E1 For laser to work we need pump to provide “Population Inversion” Normally relative populations are small: N2 e E2 E1 k BT N1 Say DE~2eV, T=300K => N2/N1=3x10-34 Way small! Real laser: pumping process typically something like E3 Pump State fast decay E2 Metastable state with lifetime ts hf0 pump input hf laser output E1 Ground State In order for one to achieve population inversion the lifetime of the metastable state must be greater than time atom spends in ground state or pump state during the pumping process. Pulsed Lasers Gain Medium Periodic Switch Pump gain media but do not let light flow out until you say so. All energy released as a high power pulse when “lasing” is allowed. 1) Put mirror on a rotating shaft 2) Voltage signal to an electro-optic crystal. Optical properties change with applied voltage. Mode Locking Periodic loss Periodic loss modulated at frequency equal to time for pulse to travel distance of 2L. Pulse propagates when losses are minimal.