Statistical Optics Lecture 5

Transcription

1 s Spring Quarter 2018 ECE244a - Spring

2 Noise distributions s Start w/quantum optics description of noise; signal consists of photons Let p n (n) be the distribution of photons in a single mode Goal: Determine distribution p n (n) that maximizes randomness Solve in two parts: First: determine p n (n) that maximizes the randomness within a single mode. Second: determine how this distribution is generated by the interaction of the mode with an external thermal energy source. ECE244a - Spring

3 Measuring randomness - s Random fluctuations, (physical or from information sources), are quantified using the concept of entropy H H = k N p n (n) ln p n (n) n=0 N represents the number of discrete states that occur with probability p n (n) k is s constant (not used for information sources) Units of entropy are J/K. First step: Find the distribution p n (n) that maximizes H for a single mode. ECE244a - Spring

4 Maximizing the s Constrain maximization so that the mean number of photons n = n=0 np n (n) and thus mean energy E = n hf within any single mode is finite. Equivalent to constraining average power, P = E B, as long as the bandwidth of the system is finite. To determine p n (n), use a variational principle. If H is maximized, then a small functional variation ɛf(n) will not affect H, the mean n or normalization Form the sum of H, the mean, n, and the normalization S = p n (n) ln p n (n) + C 1 np n (n) + C 2 n=0 n=0 n=0 where C 1 and C 2 are unknown. p n (n) ECE244a - Spring

5 Calculus of Variations s Now vary S about the maximum value by replacing p n (n) with p n (n) + ɛf(n) Group terms in powers of ɛ. Order 0 term is S If p n (n) is to be optimal, then the order ɛ term must sum to zero [ ] ln pn (n) C 1 n + C 2 f(n) = 0 n=0 If sum is zero, term in brackets must be zero. Solve for p n (n) p n (n) = Ku n where K = e (1+C 2) and u = e C 1. ECE244a - Spring

6 s Constants can then be determined using the constraint on the finite mean, and normalization (Homework). The distribution for p n (n) can then be written as p n (n) = 1 ( n ) n 1 + n 1 + n where n is the mean number of photons in a mode. This distribution is the distribution It is the fundamental distribution for statistical optics. All other distributions (including Gaussians) can be derived from it. ECE244a - Spring

7 from the s Given the distribution, the entropy is H m = k = k p n (n) ln p n (n) n=0 [ ( ) ] n n ln + ln (1 + n ) 1 + n Form of entropy is valid for a single mode Step two: Determine specific form of the distribution when single mode is coupled to an external energy source is thermal equilibrium ECE244a - Spring

8 Discrete distribution in thermal equilibrium s Assume that initially no photons are in the mode Mode is coupled to an external environment at a temperature T 0. Systems in thermal equilibrium maximize the total entropy of the system H Total system: sum of entropy in mode H m + entropy added from the external environment H. Form sum H(x) = H m (x) + H(x) ECE244a - Spring

9 s Take derivative dh(x) dx and set to zero (Homework). Final distribution may be written as [ ( )] hf p n (n) = (1 exp[ hf/kt 0 ]) exp n }{{} kt 0 normalization }{{} Geo. distribution Maximum entropy distribution for a system of discrete energy states in thermal equilibrium. is geometric because n is discrete Probably scales as ratio of hf (photon energy) to kt 0 (ave. thermal energy) ECE244a - Spring

10 Photon Matter s Assume a two level system w/levels, E 1 and E 2, separated by hf so E 2 E 1 = hf. In thermal eq. ratio of density p n (n) so [ N 2 p(n + 1) = exp hf ] N 1 p(n) kt 0 ECE244a - Spring

11 Types of s A photon is absorbed. A photon stimulates the creation of an identical second photon. This process is called stimulated emission and is the fundamental quantum-level amplification process. A photon is emitted spontaneously. This spontaneous emission has no classical counterpart and the emitted photons are noise. ECE244a - Spring

12 s If there are n photons in the mode, then the energy in the mode is ( E(n) = hf n + 1 ). 2 The factor of 1 2 is the energy in the mode when no photons are present (n = 0) Associated with fundamental momentum and position fluctuations in the field. This vacuum or zero-point energy interacts with matter producing spontaneous emission. ECE244a - Spring

13 Mean Energy () in a Mode s Mean energy per mode is the product of dis. and the energy per mode summed over all modes n E = N 0 (f) = E(n)p n (n) n=0 1 1 = hf + exp[hf/(kt 0 )] 1 2 }{{}}{{} Thermal Quantum = hf [ ] hf 2 coth (W/Hz), 2kT 0 Mean energy typically expressed as a power density (W/Hz) ECE244a - Spring

14 Thermal Spectrum s Scaled Noise N (f) E Wavelength in microns Photon Energy E in units of kt 30 Frequency in hertz Thermal noise regime 1 2 E coth (E /2) E e E 1 Quantum noise regime 100 ECE244a - Spring

15 Limiting Cases: Thermal of Light s When hf kt 0 : hf/2 is insignificant (cannot see discrete effects) The spacing between the energy levels in the discrete Boltz. distribution becomes extremely small The energy distribution f E (E) becomes essentially continuous. ECE244a - Spring

16 Derivation of Cont. when hf kt 0 s Cont. distribution is obtained by setting E = nhf, where n is now treated as continuous Using f E (E)dE = f n (n)dn where f n (n) is, f E (E) is f E (E) = f n (n) dn de = 1 ( [ 1 exp hf ]) [ exp E ] hf kt 0 kt 0 Taking the limit as E = hf 0 produces a continuous exponential distribution f E (E) = 1 N 0 exp[ E/N 0 ]. The energy in a single mode is exponentially distributed with a mean energy per mode N 0 = kt 0. ECE244a - Spring

17 Thermal Source Spectrum s To find power density spectrum, expand exp[hf/(kt 0 )] 1 + hf/(kt 0 ) and neglect hf/2 with respect to kt 0. The limiting form for the power density spectrum is then a constant N 0 = kt 0. (W/Hz) The power density spectrum in a single mode contains an equal amount of power per unit bandwidth independent of frequency. This is the low-frequency limit when hf kt 0. ECE244a - Spring

18 Optical thermal sources s At room temperature (300K), hf = kt f = 6.25 THz. Thermal noise w/constant power spectrum only true for f 6.25THz. At temperature of sun (6000K), hf = kt THz. (2.4 microns) Thermally generated photons in near IR likely Basically see thermal noise for optical thermal sources Other light sources that do not produce light via thermal process called pseudo-thermal sources if they exhibit same statistics as thermal sources Example: Spontaneous emission from an optical amplifier ECE244a - Spring

19 s Other extreme is when hf kt 0 (optical room temperature) Energy of a single photon hf is much larger than the average thermal energy kt 0. Thermal noise fluctuations at lightwave frequencies at room temperature are extremely small. exp[hf/kt 0 ] for λ = 1.5 µm and T 0 = 300K The photon distribution p n (n) in the quantum noise limit is still a distribution. Discuss photoelectron noise generated from quantum noise later in the class. ECE244a - Spring