Mirrors, Interferometers and Thin-Film Structures

2.3 Mirrors, Interferometers and Thin-Film Structures One of the most striking wave phenomena is interference. Many optical devices are based on the concept of interfering waves, such as low loss dielectric mirrors and interferometers and other thin-film optical coatings. After having a quick look into the phenomenon of interference, we will develope a powerful matrix formalism that enables us to evaluate efficiently many optical (also microwave) systems based on interference. 2.3.1 Interference and Coherence Interference Interference of waves is a consequence of the linearity of the wave equation (2.13). If we have two individual solutions of the wave equation E1 (r, t) = E1 cos(ω 1 t − k1 · r + ϕ1 ) e1 , E2 (r, t) = E2 cos(ω 2 t − k1 · r + ϕ2 ) e2 , (2.137) (2.138) with arbitrary amplitudes, wave vectors and polarizations, the sum of the two fields (superposition) is again a solution of the wave equation E(r, t) = E1 (r, t) + E2 (r, t). (2.139) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES63 If we look at the intensity, wich is proportional to the amplitude square of the total field ³ ´2 E(r, t)2 = E1 (r, t) + E2 (r, t) , (2.140) we find E(r, t)2 = E1 (r, t)2 + E2 (r, t)2 + 2E1 (r, t) · E2 (r, t) with E12 E1 (r, t) = 2 E2 E2 (r, t)2 = 2 2 2 ³ ´ 1 + cos 2(ω 1 t − k1 · r + ϕ1 ) , ³ ´ 1 + cos 2(ω 2 t − k2 · r + ϕ2 ) , (2.141) (2.142) (2.143) E1 (r, t) · E2 (r, t) = (e1 · e2 ) E1 E2 cos(ω 1 t − k1 · r + ϕ1 ) · (2.144) · cos(ω2 t − k2 · r + ϕ2 ) 1 E1 (r, t) · E2 (r, t) = (e1 · e2 ) E1 E2 · 2 ³ ´ ´ ⎤ ⎡ ³ cos (ω1 − ω2 ) t − k1 − k2 · r + (ϕ1 − ϕ2 ) ´ ³ ´ ⎦ ³ ·⎣ + cos (ω 1 + ω2 ) t − k1 + k2 · r + (ϕ1 + ϕ2 ) (2.145) (2.146) Since at optical frequencies neither our eyes nor photo detectors, can ever follow the optical frequency itself and certainly not twice as large frequencies, we drop the rapidly oscillating terms. Or in other words we look only on the cycle-averaged intensity, which we denote by a bar E(r, t)2 = E12 E22 + + (e1 · e2 ) E1 E2 · 2 ³ 2 ´ ³ ´ · cos (ω 1 − ω 2 ) t − k1 − k2 · r + (ϕ1 − ϕ2 ) (2.147) Depending on the frequencies ω 1 and ω 2 and the deterministic and stochastic properties of the phases ϕ1 and ϕ2 , we can detect this periodically varying intensity pattern called interference pattern. Interference of waves can be best visuallized with water waves, see Figure 2.38. Note, however, that water waves are a scalar field, whereas the EM-waves are vector waves. Therefore, the interference phenomena of EM-waves are much richer in nature than 64 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS for water waves. Notice, from Eq.(2.147), it follows immedicatly that the interference vanishes in the case of orthogonally polarized EM-waves, because of the scalar product involved. Also, if the frequencies of the waves are not identical, the interference pattern will not be stationary in time. Figure 2.38: Interference of water waves from two point sources in a ripple tank [1] p. 276. If the frequencies are identical, the interference pattern depends on the wave vectors, see Figure 2.39. The interference pattern which has itself a wavevector given by k1 − k2 (2.148) shows a period of 2π ¯. Λ = ¯¯ ¯ ¯k1 − k2 ¯ (2.149) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES65 Wavefronts k1 Λ Lines of constant differential phase k2 Figure 2.39: Interference pattern generated by two monochromatic plane waves. Coherence The ability of waves to generate an interference pattern is called coherence. Coherence can be quantified both temporally or spatially. For example, if we are at a certain position r in the interference pattern described by Eq.(2.147), we will only have stationary conditions over a time interval Tcoh << 2π . ω1 − ω2 Thus the spectral width of the waves determines the temporal coherence. However, it depends very often on the expermental arrangement whether a given situation can still lead to interference or not. Even so the interfering light may be perfectly temporally coherent, i.e. perfectly monochromatic, ω 1 = ω 2 ,yet the wave vectors may not be stable over time and the spatial inteference pattern may wash out, i.e. there is insufficient spatial coherence. So for stable and maximum interference three conditions must be fulfilled: • stable and identical polarization • small change in the relative phase between the beams involved over the observation time, temporal coherence, often achieved by using narrrow linewidth light • stable beam propagation or guiding of light to achieve spatial coherence. 66 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS It is by no means trivial to arrive at a light source and an experimental setup that enables good coherence and strong interference of the beams involved. Interference of beams can be used to measure relative phase shifts between them which may be proportional to a physical quantity that needs to be measured. Such phase shifts between two beams can also be used to modulate the light output at a given position in space via interference. In 6.013, we have already encountered interference effects between forward and backward traveling waves on transmission lines. This is very closely related to what we use in optics, therefore, we quickly relate the TEM-wave progagation to the transmission line formalism developed in Chapter 5 of 6.013. 2.3.2 TEM-Waves and TEM-Transmission Lines The motion of voltage V and current I along a TEM transmission line with an inductance L0 and a capacitance C0 per unit length is satisfies ∂I(t, z) ∂V (t, z) = −L0 (2.150) ∂z ∂t ∂I(t, z) ∂V (t, z) = −C 0 (2.151) ∂z ∂t Substitution of these equations into each other results in wave equations for either the voltage or the current ∂ 2 V (t, z) 1 ∂ 2 V (t, z) − = 0, ∂z 2 c2 ∂t2 ∂ 2 I(t, z) 1 ∂ 2 I(t, z) − = 0, ∂z 2 c2 ∂t2 (2.152) (2.153) √ where c = 1/ L0 C 0 is the speed of wave propagation on the transmission line. The ratio between voltage pand current for monochomatic waves is the characteristic impedance Z = L0 /C 0 . The equations of motion for the electric and magnetic field of a x-polarized TEM wave according to Figure 2.1, with E−field along the x-axis and Hfields along the y- axis follow directly from Faraday’s and Ampere’s law ∂E(t, z) ∂H(t, z) = −μ , ∂z ∂t ∂H(t, z) ∂E(t, z) = −ε , ∂z ∂t (2.154) (2.155) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES67 which are identical to the transmission line equations (2.150) and (2.151). Substitution of these equations into each other results again in wave equations √ for electric and magnetic fields propagating p at the speed of light c = 1/ με and with characteristic impedance ZF = μ/ε. The solutions of the wave equation are forward and backward traveling waves, which can be decoupled by transforming the fields to the forward and backward traveling waves r Aef f a(t, z) = (E(t, z) + ZF o H(t, z)) , (2.156) 2ZF r Aef f (E(t, z) − ZF o H(t, z)) , (2.157) b(t, z) = 2ZF which fulfill the equations µ ¶ 1∂ ∂ + a(t, z) = 0, ∂z c ∂t ¶ µ 1∂ ∂ − b(t, z) = 0. ∂z c ∂t (2.158) (2.159) Note, we introduced that cross section Aef f such that |a|2 is proportional to the total power carried by the wave. Clearly, the solutions are a(t, z) = f (t − z/c0 ), b(t, z) = g(t + z/c0 ), (2.160) (2.161) which resembles the D’Alembert solutions of the wave equations for the electric and magnetic field s ZF o (a(t, z) + b(t, z)) , (2.162) E(t, z) = 2Aef f s 1 (a(t, z) − b(t, z)) . (2.163) H(t, z) = 2ZF o Aef f Here, the forward and backward propagating fields are already normalized such that the Poynting vector multiplied with the effective area gives already the total power transported by the fields in the effective cross section Aef f P = S · (Aef f ez ) = Aef f E(t, z)H(t, z) = |a(t, z)|2 − |b(t, z)|2 . (2.164) 68 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS In 6.013, it was shown that the relation between sinusoidal current and voltage waves © ª © ª V (t, z) = Re V (z)ejωt and I(t, z) = Re I(z)ejωt (2.165) along the transmission line or corresponding electric and magnetic fields in one dimensional wave propagation is described by a generalized complex impedance Z(z) that obey’s certain transformation rules, see Figure 2.40 (a). Figure 2.40: (a) Transformation of generalized impedance along transmission lines, (b) Transformation of generalized impedance accross free space sections with different characterisitc wave impedances in each section. Along the first transmission line, which is terminated by a load impedance, the generalized impedance transforms according to Z 1 (z) = Z 1 · Z 0 − jZ 1 tan (k1 z) Z 1 − jZ 0 tan (k1 z) (2.166) with k1 = k0 n1 and along the second transmission line the same rule applies as an example Z (−L1 ) − jZ 2 tan (k2 z) Z 2 (z) = Z 2 · 1 (2.167) Z 2 − jZ 1 (−L1 ) tan (k2 z) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES69 with k2 = k0 n2 . Note, that the media can also be lossy, then the characteristic impedances of the transmission lines and the propagation constants are already themselves complex numbers. The same formalism can be used to solve corresponding one dimensional EM-wave propagation problems. Antireflection Coating The task of an antireflection (AR-)coating, analogous to load matching in transmission line theory, is to avoid reflections between the interface of two media with different optical properties. One method of course could be to place the interface at Brewster’s angle. However, this is not always possible. Let’s assume we want to put a medium with index n into a beam under normal incidence, without having reflections on the air/medium interface. The medium can be for example a lens. This is exactly the situation shown in Figure 2.40 (b). Z 2 describes the refractive index of the lense material, e.g. n2 = 3.5 for a silicon lense, we can deposit on the lens a thin layer of material with index n1 corresponding to Z 1 and this layer should match to the free space index n0 = 1 or impedance Z 0 = 377Ω. Using (2.166) we obtain Z − jZ 1 tan (−k1 L1 ) Z 2 = Z 1 (−L1 ) = Z 1 0 (2.168) Z 1 − jZ 0 tan (−k1 L1 ) If we choose a quarter wave thick matching layer k1 L1 = π/2, this simplifies to the famous result Z2 or n1 Z 21 = , Z0 √ λ = n2 n0 and L1 = . 4n1 (2.169) (2.170) Thus a quarter wave AR-coating needs a material which has an index corresponding to the geometric mean of√the two media to be matched. In the current example this would be n2 = 3.5 ≈ 1.87 2.3.3 Scattering and Transfer Matrix Another formalism to analyze optical systems (or microwave circuits) can be formulated using the forward and backward propagating waves, which transform much simpler along a homogenous transmission line than the total fields, i.e. the sum of forward adn backward waves. However, at interfaces 70 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS scattering of these waves occurs whereas the total fields are continuous. For monochromatic forward and backward propagating waves a(t, z) = a(z)ejωt and b(t, z) = b(z)ejωt (2.171) propagating in z-direction over a distance z with a propagation constant k, we find from Eqs.(2.158) and (2.159) µ a(z) b(z) ¶ = µ e−jkz 0 0 ejkz ¶µ a(0) b(0) ¶ . (2.172) A piece of transmission line is a two port. The matrix transforming the amplitudes of the waves at the input port (1) to those of the output port (2) is called the transfer matrix, see Figure 2.41 T Figure 2.41: Definition of the wave amplitudes for the transfer matrix T. For example, from Eq.(2.172) follows that the transfer matrix for free space propagation is ¶ µ −jkz e 0 . (2.173) T= 0 ejkz This formalism can be expanded to arbitrary multiports. Because of its mathematical properties the scattering matrix that describes the transformation between the incoming and outgoing wave amplitudes of a multiport is often used, see Figure 2.42. 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES71 S Figure 2.42: Scattering matrix and its port definition. The scattering matrix defines a linear transformation from the incoming to the outgoing waves b = Sa, with a = (a1, a2 , ...)T , b = (b1, b2 , ...)T . (2.174) Note, that the meaning between forward and backward waves no longer coincides with a and b, a connection, which is difficult to maintain if several ports come in from many different directions. The transfer matrix T has advantages, if many two ports are connected in series with each other. Then the total transfer matrix is the product of the individual transfer matrices. 2.3.4 Properties of the Scattering Matrix Physial properties of the system reflect itself in the mathematical properties of the scattering matrix. Reciprocity A system with constant scalar dielectric and magnetic properties must have a symmetric scattering matrix (without proof) S = ST . (2.175) 72 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS Losslessness In a lossless system the total power flowing into the system must be equal to the power flowing out of the system in steady state ¯ ¯2 ¯ ¯ |a|2 = ¯b¯ , i.e. S+ S = 1 or S−1 =S+ . (2.176) (2.177) The scattering matrix of a lossless system must be unitary. Time Reversal To find the scattering matrix of the time reversed system, we realize that incoming waves become outgoing waves under time reversal and the other way around, i.e. the meaning of a and b is exchanged and on top of it the waves become negative frequency waves. aej(ωt−kz) time reversal → aej(−ωt−kz) . (2.178) To obtain the complex amplitude of the corresponding positive frequency wave, we need to take the complex conjugate value. So to obtain the equations for the time reversed system we have to perform the following substitutions Original system Time reversed system ¡ ¢∗ . ∗ (2.179) a∗ = Sb → b = S−1 a b = Sa 2.3.5 Beamsplitter As an example, we look at the scattering matrix for a partially transmitting mirror, which could be simply formed by the interface between two media with different refractive index, which we analyzed in the previous section, see Figure 2.43. (Note, for brevity we neglect the reflections at the normal surface input to the media, or we put an AR-coating on them.) In principle, this device has four ports and should be described by a 4x4 matrix. However, most often only one of the waves is used at each port, as shown in Figure 2.43. 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES73 Figure 2.43: Port definitions for the beam splitter The scattering matrix is determined by b = Sa, with a = (a1, a2 )T , b = (b3 , b4 )T and S= µ r jt jt r ¶ , with r2 + t2 = 1. (2.180) (2.181) The matrix S was obtained using using the S-matrix properties described above. From Eqs.(2.113) we could immediately identify r as a function of the refractive indices, angle of incidence and the polarization used. Note, that the off-diagonal elements of S are identical, which is a consequence of reciprocity. That the main diagonal elements are identical is√ a consequence of unitarity for a lossless beamsplitter and furthermore t = 1 − r2 . For a given frequency r and t can always be made real by choosing proper reference planes at the input and the output of the beam splitter. Beamsplitters can be made in many ways, see for example Figure 2.37. 2.3.6 Interferometers Having a valid description of a beamsplitter at hand, we can build and analyze various types of interferometers, see Figure 2.44. 74 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS Figure 2.44: Different types of interferometers: (a) Mach-Zehnder Interferometer; (b) Michelson Interferometer; (c) Sagnac Interferometer [6] p. 66. Each of these structures has advantages and disadvantages depending on the technology they are realized. The interferometer in Figure 2.44 (a) is called Mach-Zehnder interferometer, the one in Figure 2.44 (b) is called Michelson Interferometer. In the Sagnac interferometer , Figure 2.44 (c) both beams see identical beam path and therefore errors in the beam path can be balance out and only differential changes due to external influences lead to an output signal, for example rotation, see problem set 3. To understand the light transmission through an interferometer we analyze as an example the Mach-Zehnder interferometer shown in Figure 2.45. If we excite input port 1 with a wave with complex amplitude a0 and no input at port 2 and assume 50/50 beamsplitters, the first beam splitter will 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES75 φ3 φ4 Figure 2.45: Mach-Zehnder Interferometer produce two waves with complex amplitudes b3 = √12 a0 b4 = j √12 a0 (2.182) During propagation through the interferometer arms, both waves pick up a phase delay φ3 = kL3 and φ4 = kL4 , respectively a5 = √12 a0 e−jφ3 , a6 = j √12 a0 e−jφ4 . (2.183) After the second beam splitter with the same scattering matrix as the first one, we obtain ¡ ¢ b7 = 12 a0 ¡e−jφ3 − e−jφ4 ¢, (2.184) b8 = j 12 a0 e−jφ3 + e−jφ4 . The transmitted power to the output ports is |b7 |2 = |b8 |2 = |a0 |2 4 |a0 |2 4 ¯ ¯ ¯1 − e−j(φ3 −φ4 ) ¯2 = ¯ ¯ ¯1 + e−j(φ3 −φ4 ) ¯2 = |a0 |2 2 |a0 |2 2 [1 − cos (φ3 − φ4 )] , [1 + cos (φ3 − φ4 )] . (2.185) The total output power is equal to the input power, as it must be for a lossless system. However, depending on the phase difference ∆φ = φ3 − φ4 between both arms, the power is split differently between the two output ports, see Figure 2.46.With proper biasing, i.e. φ3 − φ4 = π/2 + ∆φ, the difference in 76 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS 1.0 2 |b8| Output power 0.8 0.6 0.4 2 |b7| 0.2 0.0 -3 -2 -1 0 Δφ 1 2 3 Figure 2.46: Output power from the two arms of an interfereometer as a function of phase difference. output power between the two arms can be made directly proportional to the phase difference ∆φ. Opening up the beam size in the interferometer and placing optics into the beam enables to visualize beam distortions due to imperfect optical components, see Figures 2.47 and 2.48. Figure 2.47: Twyman-Green Interferometer to test optics quality [1] p. 324. 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES77 Figure 2.48: Interference pattern with a hot iron placed in one arm of the interferometer ([1], p. 395). 2.3.7 Fabry-Perot Resonator Interferometers can act as filters. The phase difference between the interferometer arms depends on frequency, therefore, the transmission from input to output depends on frequency, see Figure 2.46. However, the filter function is not very sharp. The reason for this is that only a two beam interference is used. Much more narrowband filters can be constructed by multipass interferences such as in a Fabry-Perot Resonator, see Figure 2.49. The simplest Fabry Perot is described by a sequence of three layers where at least the middle layer has an index different from the other two layers, such that reflections occur on these interfaces. 78 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS n1 n2 n3 2 1 S1 S2 Figure 2.49: Multiple intereferences in a Fabry Perot resonator. In the simplest implementation a Fabry Perot only consists of a sequence of three layers with different refractive index so that two reflections occur with multiple interferences. Each of this discontinuites can be described by a scattering matrix. Any kind of device that has reflections at two parallel interfaces may act as a Fabry Perot such as two semitransparent mirrors. A thin layer of material against air can act as a Fabry-Perot and is often called etalon. Given the reflection and transmission coefficients at the interfaces 1 and 2, we can write down the scattering matrices for both interfaces according to Eqs.(2.180) and (2.181). µ ¶ µ ¶µ ¶ µ ¶ µ ¶µ ¶ r1 jt1 ã1 r2 jt2 ã3 b̃1 b̃3 = and = . jt1 r1 ã2 jt2 r2 ã4 b̃2 b̃4 (2.186) If we excite the Fabry-Perot with a wave from the right with amplitude. ã1 6= 0, then a fraction of that wave will be transmitted to the interface into the Fabry-Perot as wave b̃2 and part will be already reflected into b̃1 , (0) b̃1 = r1 ã1 . (2.187) The transmitted wave will then propagate and pick up a phase factor e−jφ/2 , with φ = 2k2 L and k2 = 2π n, λ 2 ã3 =jtã1 e−jφ/2 . (2.188) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES79 After propagation it will be reflected off from the second interface which has a reflection coefficient ¯ b̃3 ¯¯ Γ2 = ¯ = r2 . (2.189) ã3 ¯ a4 =0 Then the reflected wave b̃3 propagates back to interface 1, picking up another (1) phase factor e−jφ/2 resulting in an incoming wave after one roundtrip of ã2 = jt1 r2 e−jφ ã1 . Upon reflection on interface 1, part of this wave is transmitted leading to an output (1) (2.190) b̃1 =jt1 jt1 r2 e−jφ ã1 . (1) The partial wave a2 is reflected again and after another roundtrip it arrives (2) at interface 1 as ã2 = (r1 r2 ) e−jφ · jt1 r2 e−jφ ã1 . Part of this wave is transmitted and part of it is reflected back to go through another cycle. Thus in total if we sum up all partial waves that contribute to the output at port 1 of the Fabry-Perot filter, we obtain b̃1 = ∞ X (n) b̃1 n= = Ã0 r1 − t21 r2 e−jφ µ = r1 − t21 r2 = ∞ X n=0 −jφ r1 r2 e−jφ ã1 e 1 − r1 r2 e−jφ r1 − r2 e−jφ ã 1 − r1 r2 e−jφ 1 ! ¶ ã1 (2.191) Note, that the coefficient in front of Eq.(2.191) is the coefficient S11 of the scattering matrix of the Fabry-Perot. In a similar manner, we obtain µ ¶ µ ¶ b̃3 ã1 =S (2.192) ã2 b̃4 and 1 S= 1 − r1 r2 e−jφ µ r1 − r2 e−jφ −t1 t2 e−jφ/2 −t1 t2 e−jφ/2 r2 − r1 e−jφ ¶ (2.193) In the following, we want to analyze the properties of the Fabry-Perot for the case of symmetric reflectors, i.e. r1 = r2 and t1 = t2 . Then we obtain for 80 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS the power transmission coefficient of the Fabry-Perot, |S21 |2 in terms of the power reflectivity of the interfaces R = r2 ¯ ¯ ¯ 1 − R ¯2 (1 − R)2 2 ¯ = |S21 | = ¯¯ (2.194) 1 − Re−jφ ¯ (1 − R)2 + 4R sin2 (φ/2) Fabry-Perot Transmission, |S21| 2 Figure 2.50 shows the transmission |S21 |2 of the Fabry-Perot interferometer for equal reflectivities |r1 |2 = |r2 |2 = R. 1.0 FSR R=0.1 0.8 0.6 ΔfFWHM 0.4 R=0.5 R=0.7 0.2 R=0.9 0.0 0.0 0.5 1.0 1.5 2.0 (f -fm)/ FSR 2.5 3.0 Figure 2.50: Transmission of a lossless Fabry-Perot interferometer with |r1 |2 = |r2 |2 = R At very low reflectivity R of the mirror the transmission is almost everywhere 1, there is only a slight sinusoidal modulation due to the first order interferences which are periodically in phase and out of phase, leading to 100% transmission or small reflection. For large reflectivity R, due to the then multiple interference operation of the Fabry-Perot Interferometer, very narrow transmission resonances emerge at frequencies, where the roundtrip phase in the resonator is equal to a multiple of 2π φ= 2πf n2 2L = 2πm, c0 (2.195) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES81 Fabry-Perot Transmission, |S21| 2 which occurs at a comb of frequencies, see Figure 2.51 c0 . fm = m 2n2 L (2.196) 1.0 am am-1 0.8 am+1 am+2 0.6 0.4 0.2 0.0 2fm fm 1fm-1 f3m+1 fm+2 4 frequency 5 Figure 2.51: Developement of a set of discrete resonances in a dimensionsal resonator. one- On a large frequency scale, a set of discrete frequencies, resonances or modes arise. The frequency range between resonances is called free spectral range (FSR) of the Fabry-Perot Interferometer F SR = c0 1 = , 2n2 L TR (2.197) which is the inverse roundtrip time TR of the light in the one-dimensonal cavity or resonator formed by the mirrors. The filter characteristic of each resonance can be approximately described by a Lorentzian line derived from Eq.(2.194) by substituting f = fm + ∆f with ∆f ¿ F SR, |S21 |2 (1 − R)2 ¡£ ¤ ¢ (1 − R)2 + 4R sin2 m2π + 2π F∆f /2 SR 1 ≈ ³ √ ´2 , 2π R ∆f 1 + 1−R F SR = ≈ 1+ ³ 1 ∆f ∆fF W HM /2 ´2 , (2.198) (2.199) 82 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS where we introduced the FWHM of the transmission filter ∆fF W HM = F SR , F with the finesse of the interferometer defined as √ π R π F = ≈ . 1−R T (2.200) (2.201) The last simplification is valid for a highly reflecting mirror R ≈ 1 and T is the mirror transmission. From this relation it is immediately clear that the finesse has the additional physical meaning of the optical power enhancement inside the Fabry-Perot at resonance besides the factor of π, since the power inside the cavity must be larger by 1/T , if the transmission through the Fabry-Perot is unity. 2.3.8 Quality Factor of Fabry-Perot Resonances Another quantity often used to characterize a resonator or a resonance is its quality factor Q, which is defined as the ratio between the resonance frequency and the decay rate for the energy stored in the resonator, which is also often called inverse photon lifetime, τ −1 ph Q = τ ph fm . (2.202) Lets assume, energy is stored in one of the resonator modes which occupies a range of frequencies [fm − F SR/2, fm + F SR/2] as indicated in Figure 2.52. Then the fourier integral am (t) = Z +F SR/2 b̃2 (f )ej2π(f −fm )t df, (2.203) −F SR/2 ¯2 ¯ ¯ ¯ where ¯b̃2 (f )¯ is normalized such that it describes the power spectral density of the forward traveling wave in the resonator gives the mode amplitude of the m-th mode and its magnitude square is the energy stored in the mode. Note, that we could have taken any of the internal waves ã2 , b̃2 , ã3 , and b̃3 . The time dependent field we create corresponds to the field of the forward or backward traveling wave at the corresponding reference plane in the resonator. 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES83 Transmission, |S21| 2 1.0 am-1(t) 0.8 am+1(t) am(t) 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 (f -fm)/ FSR 2.5 3.0 Figure 2.52: Integration over all frequency components within the frequency range [fm − F SR/2, fm + F SR/2] defines a mode amplitude a(t) with a slow time dependence We now make a "Gedanken-Experiment". We switch on the incoming waves ã1 (ω) and ã4 (ω) to load the cavity with energy and evaluate the internal wave b̃2 (ω). Instead of summing up all the multiple reflections like we did in constructing the scattering matrix (2.192), we exploit our skills in analyzing feedback systems, which the Fabry-Perot filter is. The scattering equations set force by the two scattering matrices characterizing the resonator mirrors in the Fabry-Perot can be visuallized by the signal flow diagram in Figure 2.53 a~1 _ r b~1 _ + jt + S1 r jt ~ b2 _ a~2 _ e-jφ/2 -jφ/2 e a~3 _ b~3 _ jt + r S2 r + jt ~ b4 _ a~4 _ Figure 2.53: Representation of Fabry-Perot resonator by a signal flow diagram 84 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS For the task to find the relationship between the internal waves feed by the incoming wave only the dashed part of the signal flow is important. The internal feedback loop can be clearly recognized with a closed loop transfer function r2 e−jφ , which leads to the resonance denominator 1 − r2 e−jφ in every element of the Fabry-Perot scattering matrix (2.192). Using Blacks formula from 6.003 and the superposition principle we immediately find for the internal wave b̃2 = ¡ ¢ jt ã1 + re−jφ/2 ã4 . 2 −jφ 1−r e (2.204) Close to one of the resonance frequencies, Ω = 2πfm + ω, using t = 1 − r2 , (2.204) can be approximated by j b̃2 (ω) ≈ ¢ ¡ ã1 (ω) + r(−1)m e−jωTR /2 ã4 (ω) , R 1 + j 1−R ωTR ¡ ¢ j ≈ ã1 (ω) + r(−1)m e−jωTR /2 ã4 (ω) 1 + jωTR /T (2.205) (2.206) for high reflectivity R. Multiplication of this equation with the resonant denominator ¡ ¢ (1 + jωTR /T ) b̃2 (ω) ≈ j ã1 (ω) + r(−1)m e−jωTR /2 ã4 (ω) (2.207) and inverse Fourier-Transform in the time domain, while recognizing that the internal fields vanish far off resonance, i.e. Z +π·F SR Z +∞ jωt am (t) = (2.208) b̃2 (ω)e dω = b̃2 (ω)ejωt dω, −π·F SR −∞ we obtain the following differential equation for the mode amplitude slowly varying in time TR d am (t) = −T (am (t) + ja1 (t) + j(−1)m a4 (t − TR /2)) dt (2.209) 2.3. MIRRORS, INTERFEROMETERS AND THIN-FILM STRUCTURES85 with the input fields a1/4 (t) = Z +π·F SR ã1/4 (ω)ejωt dω. (2.210) −π·F SR Despite the pain to derive this equation the physical interpretation is remarkably simple and far reaching as we will see when we apply this equation later on to many different situations. Lets assume, we switch off the loading of the cavity at some point, i.e. a1/4 (t) = 0, then Eq.(2.209) results in am (t) = am (0)e−t/(TR /T ) (2.211) And the power decays accordingly |am (t)|2 = |am (0)|2 e−t/(TR /2T ) (2.212) twice as fast as the amplitude. The energy decay time of the cavity is often called the cavity energy decay time, or photon lifetime, τ ph , which is here τ ph = TR . 2T Note, the factor of two comes from the fact that each mirror of the FabryPerot has a transmission T per roundtrip time. For exampl a L = 1.5m long cavity with mirrors of 0.5% transmission, i.e. TR = 10ns and 2T = 0.01 has a photon lifetime of 1μs. It needs hundred bounces on the mirror for a photon to be essentially lost from the cavity. Highest quality dielectric mirrors may have a reflection loss of only 10−5...−6 , this is not really transmission but rather scattering loss in the mirror. Such high reflectivity mirrors may lead to the construction of cavities with photon lifetimes on the order of milliseconds. Now, that we have an expression for the energy decay time in the cavity, we can evaluate the quality factor of the resonator Q = fm · τ ph = m . 2T (2.213) Again for a resonator with the same parameters as before and at optical frequencies of 300THz corresponding to 1μm wavelength, we obtain Q = 2 · 108 . 86 CHAPTER 2. CLASSICAL ELECTROMAGNETISM AND OPTICS 2.3.9 Thin-Film Filters Transfer matrix formlism is an efficient method to analyze the reflection and transmission properties of layered dielectric media, such as the one shown in Figure 2.54. Using the transfer matrix method, it is an easy task to compute the transmission and reflection coefficients of a structure composed of layers with arbitrary indices and thicknesses. A prominent example of a thin-film filter are Bragg mirrors. These are made of a periodic arrangement of two layers with low and high index n1 and n2 , respectively. For maximum reflection bandwidth, the layer thicknesses are chosen to be quarter wave for the wavelength maximum reflection occures, n1 d1 =λ0 /4 and n2 d2 =λ0 /4 ~ b2 _ ~ a _ 1 n1 n2 n1 n2 n1 n2 .... d1 d2 d1 d2 d1 d2 b~1 _ ~ a _ 2 Figure 2.54: Thin-Film dielectric mirror composed of alternating high and low index layers. As an example Figure 2.55 shows the reflection from a Bragg mirror with n1 = 1.45, n2 = 2.4 for a center wavelength of λ0 = 800nm. The layer thicknesses are then d1 =134nm and d2 =83nm. 1 0.9 0.8 0.7 reflectivity 0.6 0.5 0.4 0.3 0.2 0.1 0 500 600 700 800 900 lambda 1000 1100 1200 1300 Figure 2.55: Reflectivity of an 8 pair quarter wave Bragg mirror with n1 = 1.45 and n2 = 2.4 designed for a center wavelength of 800nm. The mirror is embedded in the same low index material.