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Theory off chaotic light and coherence Rotem Manor ID:036235976 It is important to distinguish between tow types of light source: • Chaotic source: gas discharge lamp, where the different atoms are excited by an electrical discharge and emit their radiation independently of one another. emission line determined by statistical spread in atomic velocities and random occurrence of collisions. • Laser: high intensity ,coherence There are three mechanisms witch are mainly responsible for the spread in frequency of emitted light. The decay process it self: leads to broadening of an absorption line of Lorentzian frequency distribution (with a width Γ). Collision broadening: form the atomic motion. Spread in atomic velocities: leads to Doppler spread in emitted frequencies. Note: it is convenient to neglect the other linewidth contributions while treating a particular broadening process. Doppler broadening Atom in exited energy level E2 , Has velocity v2 . Emits photon of energy and falls to a lower energy level E1 . The photon has a momentum k where k / c and is emission causes a recoil of the atom to a new velocity v1 Momentum : Mv2 Mv1 k 1 1 2 2 E2 Mv2 E1 Mv1 Energy: 2 2 Let 0 be the frequency of light which would be emitted if the atom had zero velocity before and after the emission. 0 E2 E1 It gives us: 2k 2 0 v2 k 2M We’ll take k to be on the z-axis v2z h 2 0 c 2Mc 2 Typical orders of magnitude: 9 v 10 ; 10 2 Mc 2 5 z 2 Hence 0 1 v z 2 0 (1 v2z c ) c There for the frequency of emitted light suffers a Doppler shift. The Maxellian velocity distribution define the frequency distribution 2 ( ) 1 1 c z 2 z 2 0 exp( M (v ) )dv exp( Mc ) d 2 2 2 0 0 where Probability function: FG ( ) where 1 2 2 e 1 k BT ( 0 ) 2 2 2 0 2 Mc 2 Collision broadening We focus attention on the same pair of atomic states that we have used before, but we now ignore the other linewidth( assumption: the radiative lifetime of an atom is long compared to time between collisions). Consider a particular exited atom radiating light of frequency. When it suffers a collision his energy levels are shifted by the force of interaction between the two colliding atoms. After the collision the wave frequency 0 is resumed with all the other characteristics, except from the phase of the wave, that is unrelated to the one before the collision. If the duration of the collision is sufficiently brief, it is possible to ignore any radiation emitted during the collision while 0 is shifted. The collision broadening effect can then be adequately represented by a model in witch each exited atom always radiates at frequency 0 , but with random changes in the phase of radiated wave each time a collision occurs. The wave train radiated by single atom. Vertical line represent a collision. The Phase of The wave train radiation According to the kinetic theory of gases, the probability that an atom has period of free flight lasting a length of time between t and t+dt is t p(t )dt Hence 0 1 0 V 4d 2 N e 0 dt M If we consider one period of free flight of an atom, The field can be written in a complex form: E (t ) E0 exp( i0t i ) (t0 t t0 ) And represent as a Fourier integral 1 E ( ) 2 t0 E 0 exp( i0t i it ) t0 E0 exp{i ( 0 ) } 1 exp{i ( 0 )t0 i} 2 i ( 0 ) Thus the cycle averaged intensity is 2 E sin {(0 ) / 2} 2 0 I ( ) | E ( ) | 2 ( ) 0 2 At any instant of time the total intensity of radiation is made up of contribution from large number of exited atoms. The probability of the free flight time is given( in slide 8 ), so it’s necessary to integrate the intensity with the probability over time for getting the total intensity. 2 sin {(0 ) / 2} I total ( ) | E ( ) |2 exp( / 0 )d 2 0 0 (0 ) 1 1/ 2 (0 ) 2 (1 / 0 ) 2 realistic values for gas in room temperature and pressure 105 Pa . 0 3 10 11 s 0 3 1015 Hz 0 0 9 10 4 So average atom doing about 15,000 periods of oscillations before a collision, And the collision linewidth is about 100 times the natural radiative linewidth( Г ) Composite emission lineshape It is interesting to note that the Doppler width( slide 6 ) is approximately equal to the collision linewidth for this parameter values. In this case it is necessary to determine the composite lineshape of all the dominant processes. A combination of two line broadening mechanisms with individually generate normalized lineshape functions f(ω) and g(ω) is F ( ) f (w) g ( 0 w)dw Where 0 is the common central frequency component of the distributions. Clearly if the two sources of broadening are Lorentzians And for Gaussians total f g total f g 2 2 2 Time dependence of the chaotic light beam It is clear from the above discussions that the frequency spread of the emitted light is governed by the same physical parameters as the time dependence of emitting source atom. The wave train emitted by a single atom( number 1 ) is E1 (t ) E0 exp( i0t i1 ) The total emitted wave is represented by a sum of this terms E (t ) En E0 exp( i0t ) exp( in ) n n E0 exp( i0t )a(t ) exp( i (t )) For simplicity we will assume that the observed light has a fixed polarization so that the electric field can be added algebraically The amplitude a(t) is illustrated in this diagram Im[ a(t )] Re[ a(t )] a(t ) and (t ) are different and different instate of time. How ever it is not possible in practice to resolve the oscillation in E(t) witch occur at the frequency of the carrier wave. A good experimental resolving time is of the order 10 9 (six order magnitude too long to detect the oscillation of the carrier wave). The real electric field has zero cycle average and the beam intensity in free space is 1 1 2 2 2 I (t ) 0 c | E (t ) | 0 cE0 a(t ) 2 2 I (t ) 0 This is a computer simulation of a collision-broadened light source. It is seen that substantial changes in intensity and phase can occur over time span 0 , but these quantities are reasonably constant over (t ) time intervals t 0 . It gives us two new parameters • c - coherence time, • c c c - coherence length. 0 Intensity fluctuation of chaotic light Intensity fluctuation can be measured experimentally only with a detector whose response time is short compared to the coherence time c . So a normal detector is measuring averages. a(t ) 2 | exp( in ) |2 n (the bar denotes a long tome average for a time long compared to ) c So the long time average intensity is 1 2 I (t ) 0 cE0 2 It is seen that in any instant of time the movement of a(t) is in depend on stepsin random direction like in the ‘random walk’ problem, so let p[a(t)] be the probability of the end point of a(t). a(t ) 2 p[a(t )] exp 1 Probability distribution for the amplitude and the phase of electric field of chaotic light I (t ) p[ I (t )] exp I 1 If we’ll calculate: I (t ) n 1 I (t ) n exp( I (t ) / I )dI (t ) n! I n I 0 It’s means that the fluctuation are I 2 I 2 I . Despite the very large fluctuation compared to the intensity, the coherence time c is usually so short that it is difficult to observe the fluctuation in experimentally and their influence is often small. Young’s interface fringes The model experiment ignores complications arising from the finite source diameter and consequent lack of parallelism in the beam witch illuminates the first screen. Let E(r,t) be total electric field of radiation at position r on the observation screen at time t. E(r, t ) u1E(r1t1 ) u2 E(r2t2 ) Where t1 t (s1 / c), t2 t (s2 / c), And u1 ,u2 are inversely proportional to s1 , s2 respectively, and depend on the geometry of the experiment. The intensity of the light in the position r averaged on over a cycle of oscillation is 1 I (rt ) 0 c | E (rt ) |2 2 1 0 c{| u1 |2 | E (r1t1 ) |2 | u2 |2 | E (r2t 2 ) |2 2 * * 2u1 u2 Re E (r1t1 ) E (r2t 2 ) } The fringes in Young’s interface experiment are normally can be observed by the naked eye. In this case the recording time is long compared to c and there for it is necessary to average I (rt ) . I I (rt ) T 1 1 0c dt{| u1 |2 | E (r1t ) |2 | u2 |2 | E (r2t t21 ) |2 2 T0 2u u Re E * (r1t ) E (r2t t 21 ) } * 1 2 Where t21 t2 t1 The first two terms represents the intensities caused by each pinholes in the absence of others. The third one is called first-order correlation function of the field, and it’s clearly depend on t 21 (the deference between the times at witch the fields are measured). Evaluation of the first order correlation function Let’s calculate the correlation function of a Lorentzian distribution light source. The light witch strikes the first screen in Young experiment is assumed to be propagating in the z direction, the optical cavity then is one dimensional case and the spacing between the wave vector is L – length of cavity k / L We’ll use Fourier sum of the normal-modes contribution to the light source Where E ( zt ) Ek exp( ikz ik t ) k ck The number of different normal-modes contributes significantly to the electric field is depend on the length L and the coherence length. L c If we’ll take c / L On the other hand linewidth of the assumed Lorentzian emission line can be taken as 2 where 1 / c / It that the only significant mode is k0 0 / c 2I | Ek0 | oc 2 Now will take L c / It mean that the distribution is | Ek | (0 ) 2 2 2 - constant of proportionality 1 1 2 I 0c | E ( zt ) | 0c | Ek |2 2 2 k Since 0 0 ( L / ) dk ( L / c) d If we’ll take the sum to be k 0 L 1 I dk 0 L 2 2 2 0 (0 k ) 2 We’ll get the intensity Witch give us the constant 2I | Ek | 0 L (0 k ) 2 2 2 Now we’ll continue the calculation of the correlation function E * ( z1t1 ) E ( z2t2 ) Ek* Ek ' exp{i(kz1 k t1 k ' z2 k 't2 )} k ,k ' | Ek |2 exp( ik ) k Where t1 t2 ( z1 z2 ) / c exp( ik ) 2I * E ( z1t2 ) E ( z2t2 ) dk 2 2 0c 0 (0 k ) For narrow emission line the lower limit on the integral can be replaced by without significant change in its value. exp( ik ) dk exp( i0 | |) 2 2 (0 k ) 2I E ( z1t2 ) E ( z2t2 ) exp( i0t | |) 0c * Note: We should remember that: •We used the independent randomness of different normal modes. •The large number of cavity modes, is a necessary condition for the replacement of the sum with an integral. Fringes intensity and first order coherence Since: I (r ) I {| u1 |2 | u2 |2 2u1*u2 exp( | | cos 0t ) Where (s1 s2 ) / c The fringes visibility at the position r on the second screen is defined by: I (r ) max I (r ) min 2u1*u2 exp( | s1 s 2 | / c) I max I min | u1 |2 | u2 |2 It’s easy to see that when s1 s2 and u1 u2 the fringes visibility is unity, and it’s less than unity otherwise. The degree of first order coherence between the light fields at the space-time points (r1 , t1 ) and (r2 , t2 ) is denoted by g (1) (r1t1 , r2t2 ) . g (1) (r1t1 , r2t 2 ) g12(1) | E * (r1t1 ) E (r2t 2 ) | | E (r t ) | 11 2 | E (r2t 2 ) |2 1 2 The angle brackets indicate that ensemble average mast be taken when the field E(rt) is define statistically. Coherent - g12 1 (1) (1) Incoherent - g12 0 Example: Lorentzian frequency distribution in Young experiment: g (1) (r1t1 , r2t2 ) exp( | |) (1) The form of dependence of g12 is illustrated below while t1 t2 ( z1 z2 ) / c c Since the corresponding coherence length is about 102 it is possible. Note: this calculation is for a collision model Motivation for higher order coherence The classical stable wave E1 (t ) E0 exp( ikz i0t i ) provides another example of coherence properties. The first order correlation function is determined without any ensemble averaging in this case. 2I E ( z1t2 ) E ( z2t2 ) E0 exp( i0 ) exp( i0 ) 0c * 2 It means that the first order coherent is g12 1 . (1) There are some more examples under witch light have perfect first order (if the beam is single cavity mode, the filed can be specified precisely, with no statistical features). Recent development in optics have gone beyond the domain of classical theory. • The laser has coherence properties which can be varied chaotic sources. •Experiments have been preformed in with the intensity fluctuation of a chaotic source are directly measured. Intensity interface and higher order coherence Hanbury Brown and Twiss experiment (intensity of the fluctuation) •The detectors are symmetrically placed with respect to the mirror. •The half-silvered mirror produces two exactly similar light beams. For now we’ll ignore the finite response-times of the detector. The experiment measures fluctuations in the intensity: ( I ( zt1 ) I )( I ( zt 2 ) I ) I ( zt1 ) I ( zt 2 ) I 2 , Since I ( zt1 ) I ( zt 2 ) I The correlation function in here is: 0c * * I ( z1t1 ) I ( z 2t 2 ) E ( z t ) E ( z 2t 2 ) E ( z 2t 2 ) E ( z1t1 ) 11 2 2 The electric field can be expanded as superposition of plane wave. 2 c I ( z1t1 ) I ( z2t 2 ) 0 Ek*1 Ek*2 Ek3 Ek4 2 k1k2 k3k4 exp{i(k1 z1 1t1 k2 z2 2t2 k3 z3 3t3 k4 z4 4t4 ) Where i cki . Thus equation can simplifies to 0c I ( z1t1 ) I ( z2t2 ) 2 2 | Ek1 |2 | Ek2 |2 [exp{i(1 2 ) } 1] k1k 2 This equation can be written in terms of first order correlation function 0c I ( z1t1 ) I ( z 2t 2 ) 2 2 * E ( z1t1 ) E ( z2t 2 ) 2 I2 I ( z1t1 ) I ( z2t 2 ) I 2 {exp( 2 | |) 1} Hence ( I ( z t ) I )( I ( z t ) I ) I 2 exp( 2 | t t |) 11 2 2 1 2 Note: I ( zt ) 2 2 I 2 since there is two light beams. This result is not realistic, Every detector has minimum response time r . so now we’ll calculate the average intensity in the response time. ( I ( z1t1 ) I )( I ( z2t2 ) I ) 1 r 2 r r dt dt 1 0 2 ( I ( z1t1 ) I )( I ( z2t2 ) I ) 0 ( I 2 / 2 2 r ){exp( 2 r ) 1 2 r } 2 The properties of a light beam witch are relevant to an intensity interface can be expressed in terms of an extension of the coherence concept. Analogous to the definition of the first order coherence we define the second order coherence. g12( 2) E * (r1t1 ) E * (r2t2 ) E (r2t 2 ) E (r1t1 ) | E (r1t1 ) |2 | E (r2t 2 ) |2 The light is said to be second order coherent if simultaneously (1) and g12 1 g12( 2 ) 1 If we’ll use the development of the equations from slides 32-33 we’ll get a new definition for the second-order coherence of chaotic light: ( 2) (1) 2 g12 | g12 | 1 Note: this calculation is good for a collision and Doppler broadening models With those definition it’s not possible for chaotic light to be second-order coherent for any choice of space-time points. Examples for a different frequency distribution For lorentzian : ( 2) g12 exp( 2 | |) 1 For gaussian : ( 2) g12 exp( 2 2 ) 1 Although you can notice that for a classical stable wave: E ( zt ) E0 exp( ikz i0t i ) The correlation of intensity is I ( z1t1 ) I ( z 2t 2 ) I 2 Hence g12( 2) 1 It’s second-order coherence in all space-time points. The degree of first- and second- order coherence is define by the same pattern. This is just two members of a hierarchy of coherence functions. It is possible to envisage a general interface experiment in which the measured result depends on the correlation of electric fields at an arbitrary number of space-time points. The results of this experiment is depends upon the hierarchy of coherence functions with define like this: g (n) (r1t1...rnt n ; rn 1t n 1...r2 nt 2 n ) | E * (r1t1 )...E * (rntn ) E (rn 1tn 1 )...E (r2 nt 2 n ) | * 2 * | E (r1t1 ) | ... | E (rnt n ) | 2 2 | E (rn 1tn 1 ) | ... | E (r2 nt2 n ) | 2 Reference http://people.deas.harvard.edu/~jones/ap216/lectures/l s_3/ls3_u6A/ls3_unit6A.html The quantum theory of light, Lauden, R 1973 Appendix I Quantum coherence For first order coherence for ” Young interference” The filed operators associated with the modes are hence if the pine holes are equal: So it can be written that: So that Thus for a n-photon, single mode incident beam, the intensity at an observation point Q is given by Or Second order coherence for Hanbury Brown interference correlation between photomultiplier currents is proportional to Thus, the degree of second-order coherence is Appendix II Huygens principle: we assert that component fields radiated by each coherent cell are spherical waves of the form . Appendix III Free Space Propagation of Coherence Functions how coherence propagates through space. We start by assuming that the analytic signal representing the field of interest satisfies an wave equation of the form If we multiply through by , we see that Similarly Therefore the coherence function must satisfy the fourthorder equation Appendix IIII Addition calculation for second order coherence To obtain the root-mean-square deviation in the cycle average of the intensity, we first calculate For a collision model it can be written as Calculation of the equation in slide 36 (relation between first and second order coherent in collision model). Again, for the collision- and Doppler-broadening models Hence: