{ } 1 2. c 2 + p2. m r

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1 5. Optics, n-level systems, and lasers In section, we saw how the refractive index n, absorption coefficient α and nonlinear susceptibility χ of a material arises through interaction with the electromagnetic field. An alternative derivation of the absorption coefficient α was also given appendix B (time dependent perturbation theory and the Raman cross section for any substance was discussed in section 0 as part of second quantization and the formulation of electromagnetics in terms of photons instead of electric fields. Here we will assume these material properties as given, and discuss how they affect the propagation of light, and how interaction of light with molecules of n> levels can lead to light amplification by stimulated emission (lasing. 5. Photons and Snell's Law The refractive index of a substance affects the propagation of electromagnetic radiation through that substance because the wavevector k=ωn/c. The energy of a relativistic particle is given by E = (m r c + (pc, where m r is the rest mass. For slow particles, if p<<m r c { } m r c + p E = m r c 4 + p c m r c m r c + p m r +, the usual result of rest mass energy and kinetic energy of a free particle. But for photons, m r m r E = pc = hν or p = hν c = k. Now consider a flat boundary between two substances, one with refractive index and the other with refractive index n t. To conserve momentum along the direction of the boundary (x axisduring transmission from to n t. p ix = p tx p i = p t sinθ t k i = k t sinθ t. Using k=ωn/c it follows that also known as Snell s law. sinθ t = n t, 5. Fermat's principle The above result, as well as any propagation of photons through media with index n(x,y,z can also be obtained from a much more general principle, without the aid of relativity. This derivatios based on a Lagrangian (in this case the time t required for the photon to travel from A to B, to which we apply a least action principle. (See calculaus of variations in appendix A for a detailed derivation.

2 The least action principle in this case is Fermat s law: Photons travel along stationary (usually minimum time optical paths." "Stationary" means there are no first order variations in the optical path Σ L i, where L i are segments of the optical path, and are their refractive indices. Mathematically stated, photon motion minimizes t[x(u] = u f ( x / u c / n(x(u du, u i the time it takes a photon to travel the path x(u given parametrically by u. The numerator is the arc length traveled along the path, the denominator the local velocity. By varying x(u and finding the minimum t, the correct path x(u is determined. Consider three examples: ex: propagation within a medium of constant index n from point A to point B u f In that case, t[x(u] = c ( x / u du n. But the shortest distance between point A (u = u i u i and B (u=u f is a straight line. So photons in constant index media travel in straight lines. ex: transmission through a flat boundary from A (index to B (index n t In that case, it has already been settled that within the first and second media, the photon will travel in a straight line. Thus the path must consist of two straight line segments connected at the boundary from to n t. For simplicity let A, B and the connection be in the same plane. Then c(t + t = + L A { } + { + (L L A } l where L is the horizontal distance between A and B, and L A is the horizontal distance of A to the connection. To minimize this function l (called the optical path length because each path segment is weighted by its refractive index, set the derivative with respect to L A (the only variable equal to zero: l L A = n L i A + L n t (L L A A + (L L A. The two ratios are related to the angles between the vertical through the connector and points A and B, or l L A n t sinθ t This is again Snell s law for refraction. Note that you could do a more general derivation assuming the connection point is not in the same plane as A and B. Then you have to minimize with respect to two projected distances L xa and L xb, but the final result will be the same.

3 ex: reflection of the photon by a flat surface from point A to point B in the same medium Now the pathlength is (assuming again a connector in the same plane as A and B { } + { + (L L A }. l = + L A This cammediately be rewritten terms of incidence and reflection angles as sinθ r or θ i = θ r (Euclid' s Law. Similarly, we can derive the path of photons under any geometry of refractive index n(x,y,z, no matter how complicated, explaining propagation through prisms, lenses, gratings, etc. 5.3 Relation of Fermat s principle and wave equations Of course, one should be able to derive Fermat s principle from the electromagnetic wave equations discussed in Section. We saw that the solution for a monochromatic electric field is E(r = e(re ikir iωt We can rewrite this in terms of the arc-length traveled as E(r = e(re ik0l(r iωt, where k 0 = ω/c is the vacuum wave vector, and l(r = n(x(t path x (t is the length traveled along the propagation direction. (Compare this with t[x] given earlier. Inserting this into the wave equation we derived in section, E + n c t E x + y + z e 0l(r i (reik n ω e c i (re ik 0l (r e i 0l x eik + e i x (ik l 0l l 0 x eik + ik 0 x e 0l i eik l k 0 e x i e ik 0 l + n k 0 e i (re ik 0 l(r Now we make an assumption that makes our next result less general than full electromagnetics: If e(r varies slowly compared to the wavelength λ = π / k 0 (i.e. the electric field looks locally like a plane wave traveling in a straight line along k, only the highest powers of k 0 (last two terms survive as k 0 (λ 0 l x + l y + l z = n (x, y, z. This is called the eikonal equation. It relates the square of the gradient of the optical path arc length l to the square of the refractive index: ( [ l] = n δ path dl The equivalence of the differential equation on the left to the minimum principle on the right (Fermat s principle that the optical path is minimized follows directly from dx path

4 variational calculus, in the same way that Lagrange s equation of motion follows from the minimum action principle. We now see the assumption of geometrical optics: the rays cannot be focused to a size approaching λ, or e is no longer locally a plane wave, and the first three terms in the wave equation cannot be neglected. 5.4 Gaussian beams In order to increase signals from light beams interacting with a substance, or to operate a laser, the beam may have to be focused, violating Fermat s principle. In that case the electromagnetic wave must be treated without the eikonal approximation. Many complex electromagnetic fields are possible, but we will just consider the simplest case: focusing to a uniform Gaussian beam. If the wave equatios rewritten cyclindrical geometry, we have E + n c t E = r + r r + r ϕ + E + n z c t E. We now assume the field is propagating along the z (symmetry axis, and does not depend on ϕ, so E=E(z,r. Such a wave with cylindrical symmetry is known as a TEM 00 wave, or Gaussian mode. The solution of the equation is, w E(z,r = ε 0 0 w(z e w(z = w 0 + z z R R(z = z + r + r z R z r w e ik(z+ r + z r E + n c t E R(z iφ(z iωt, where is the beam half-waist ( / e intensity point, is the beam radius of curvature, and z φ(z = tan is the phase shift, z R which goes through π as the beam goes through the focus where the beam waist w is smallest (=w 0. The distance z R is the Raleigh length: it equals z R = πw 0 /λ, and is roughly the depth of focus, or the distance over which the beam remains close to w 0 in diameter. At large distances from the focus ( z >>0, the angle made by the Gaussian beam with the z-axis is θ=λ/πw 0. Squaring the expression for the electric field yields the intensity I(r, z = c 4π EE* = c ε 0 w 0 4π w(z e from which the Gaussian beam derives its name. r w(z w 0 = I w(z e r w(z,

5 There are two ways in which a Gaussian focus can be achieved: by a lens, or in a mirror cavity. Consider a lens of diameter D and focal length f (the curvature of the lens surface can be calculated using Fermat s principle. Far from the focus, a collimated beam entering the lens will converge at an angle θ=tan - (D/f D/f if D<<f. Inserting this into the Gaussian beam equation for θ and eliminating θ, w 0 = λ f π D. The smaller the wavelength, the bigger the lens, and the shorter the focal length, the tighter the focus is going to be. Unless the focal length of the lens is much smaller than its diameter (difficult from a practical point of view, the beam waist is not going to be much smaller than the wavelength of light. This is the diffraction limit, the smallest size to which an electromagnetic wave can be focused. For visible light it is about 300 nm, for a GHz NMR radio wave it is about 30 cm. The second way involves a spherical cavity: two spherical mirrors of radius R, separated by a distance z such that R(z=R of a Gaussian wave exactly matches the surface of the two mirrors, will support a Gaussian beam. By Euclid s law, the beam reaching the mirror on the left will be reflected back to the right exactly back otself, go through the focus, and then do the same thing at the mirror on the right. If the mirrors are perfect reflectors, the beam will remain between them. If one mirror is semi-transparent, the wave will leak out and the intensity in the cavity will decay by a fraction -/F at each pass of the beam. For example, a cavity that loses % of the intensity at each pass decays by 0.99, and has a finesse F=00. Note that the electric field at the mirror surface must be equal to zero. For an on-axis separation of z of the two mirrors, this means that mλ = z = L or ν = mc cav m. L cav Only certain frequencies are supported by the cavity, and c/l cav is known as the free spectral range of the cavity or resonator. 5.5 The laser gain If we place at the center of our cavity a two-level system that undergoes stimulated emission at the frequency of the cavity, the loss due to the semi-transparent mirror can be made up, and the cavity can continue to output a Gaussian beam. As proved in the section on time-dependent perturbation theory (Appendix B, as well as sections 0 and by different methods, the stimulated emission rate for a single two-level system in the excited state is given by Γ(s = π 3/ µ iε(z,r L (ω. L is a Lorentzian profile centered at ω = πv m (assuming population decays exponentially from excited state to ground state to reach thermal equilibrium. z and r specify the location of the two-level system in the beam. Now consider a material at the center of the cavity containing many two-level systems either in ground state or excited state. For simplicity, we assume the slug of material is wide enough so the entire Gaussian beam fits within, and the concentration of excited (n and ground state (n two level systems is constant

6 throughout the material. Using the formula that relates electric field and intensity derived above for the Gaussian beam, we then obtain Γ(s π 3/ µ = dv i ˆε 4π I w 0 c w(z e w(z L (ω(n for the total rate of photon emission. Here ˆε is the polarization direction of the electric field. In cylindrical coordinates, dv=dϕ rdr dz. Doing the area integration but not the dz integration yields dγ(s = π 3/ µ i ˆε π I c πw 0L (ω(n dz. Defining the beam area as πw 0 and multiplying both sides by ω to convert the rate into energy/sec, this becomes d ωγ A = π 3/ µ i ˆε I πω c L (ω(n n dz The left hand side is just the change intensity di. Also, by taking all of the small numerical factors and combining them with the average of cos θ, the angle between field and dipole, into a geometrical factor g on the order of unity, we have di I = g ω c µ L (ω(n dz = α(ωdz Note that α is analogous to the absorption coefficient from chapter 0, and proportional to the susceptibility χ from chapter, except if c > c, α is positive because there is stimulated emission. Integrating the equation over the thickness of the material from 0 to d, and assuming it is thin enough so we are within the Raleigh length of the focus, we have simply I(d = I(0e α (ω d = I(0e γ (ω. In this equation, γ is the gain coefficient of the medium, which amplifies the incoming intensity to make up for the loss by the semi-transparent cavity mirror. Note the gain coefficient is proportional to the populationversion Δn=n -n. 5.6 The four-level laser Of course the bad news is that no two level system can achieve a populationversion by heating or steady-state pumping. We would have to apply a coherent pi-pulse (from another laser!. At low frequencies (radio frequencies used in NMR it would be easy to generate such a pi-pulse electrically, but then you already have coherent radiation otherwise simply known as a function generator in the lab. How do we create an optical laser without another pulsed laser for input? The trick to get steady-state lasing is to use a four level system, part of which will be the desired two-level system with a populationversion. Using the same rate equations as for Einstein coefficients, we can calculate Δn so we can calculate γ (ω. The idea is that the two-level system that will lase is nested within another two-level system that s pumped without achieving a populationversion. But if the upper pumped state 3> relaxes fast to, and the lower state > of the lasing two-level system relaxes fast to r

7 0>, then states > and > will have a populationversion. The Einstein equations for this case are dn 3 dn dn = R pump n 3 T 3 = + n 3 T 3 Bu(n n T = n T + Bu(n T 0 dn 0 = R pump + n T 0 For simplicity, we include only the pump, relaxation and emission processes minimally needed for the discussion. In steady-state operation, d /. First consider the gain medium without cavity so u 0. Then n 3 = R T pump n n n 3 T T Δn 0 = R pump (T T 0 0 T = n T 0 As expected, T > T 0 is necessary to obtain a populationversion: the two-level system made out of states > and > must have a lower state that quickly empties into 0> to get populationversion. If a cavity is added, Bu 0 n (Bu + = n (Bu + T T 0 Δn n + T Bu As ρ (power builds up inside the cavity due to stimulated emission, the population inversion eventually falls of to zero. This is called gain saturation. At some point, laser power building up in the cavity will deplete all the excited molecules in state > and the laser cannot have any further gain. Inserting our formula for the populationversion Δn = n into the equation for γ (or α, using I=cu, and writing the Lorentzian line shape for relaxation from > to > out explicitly, we have γ (ω = g µ ω T c (ω ω + T R pump(t T 0 d BI(ω + T c. where B = 4 µ π cm 3 erg 3 ergs and I = ρc in cm s Hz BI(ω The low signal gain at resonance (ω = ω and >> T is given by c

8 g µ T (T T 0 R pump d. c 5.7 Lasing threshold We can now summarize the conditions that have to be met for lasing in the cavity to occur: e γ (ω e αl / ( T (gain exceeds losses k(l d + φ gain d = πm (phase-matched rounrip In, T is the fractional power loss due to transmission (e.g for a 5% output coupler, and αl includes any other losses due to absorption the resonator cavity of length L. is the boundary condition that the emitted mode must match the laser cavity. When lasing starts, Δn decreases from the initial value because I(ω in the cavity increases, leading to gain saturation. The power in the cavity will build up until γ (ω=αl-ln(-t, clamping the gain at the threshold value: I thr (ω = gω µ R pump (T T 0 d c (αl ln( T B T B (at ω = ω cm3, B in s erg This is the intracavity power per unit frequency at the gain maximum. To get the output power, one has to multiply by T. ex: a W laser with a 0% output power has I thr W ; at startup, the power may transiently overshoot this power 5.8 Mode locking Last, we discuss mode-locking. It is often desirable to have a pulsed laser output rather than a continuous wave output. Mode-locking is currently the means to achieve the fastest possible laser pulses. The idea is to have the gain medium able to amplify many cavity modes ν m. By superimposing these cavity modes ~cos(πνt in phase, a pulse results. We can construct the electric field amplitude in the cavity as a sum over longitudinal modes, E(z,t = A m e πi(ν mc 0 + m L t z c m,±,, * where we take the refractive index in the cavity to be n for simplicity. The A m (= A m to make the field real are complex amplitudes largest at the center of the gain profile (at ν 0 and smaller as m increases. In a cw laser, the phrases of these "unlocked" modes m are generally random, resulting in a roughly uniform intensity distribution the cavity with some power fluctuation proportional to / I. Now assume all A m have the same phase and amplitude over some range M of modes: E(z,t = Ae πiν 0 z t c M e m= M πimc L t z c

9 We can do the summation by using which yields M x m = x M + m=-m x M x x sin(mπ t z c E(z, t = Ae ikr iωt c L sin(π t z c c L The result is a coherent train of short pulses spaced L in time (cavity rounrip time. c The locking of phases can be achieved in two ways: Active: place and acousto-optic modulator running at ν = c in the cavity; this L modulates the gain so γ (ω is higher for a pulse train than for cw lasing. Passive: Kerr lensing makes use of the fact that the refractive index is power dependent ( n(i = n 0 + n I, see section. The cavity can be adjusted so higher power is focused better in the gain medium, giving better gain. Thus a noise fluctuation will be preferentially amplified until a locked pulse train results. Since Δν Δt, short pulses require a large gain bandwih, i.e. γ must be large over a wide range of frequencies to support cavity modes of many different frequencies that can be locked to make a pulse. ex: Ti:S gain band from nm Δν = Hz = 75 THz Δt 3 fs,