1 Photon optics! Photons and modes Photon properties Photon streams and statistics

Transcription

1 1 Photons and modes Photon properties Photon optics! Photon streams and statistics

2 2 Photon optics! Photons and modes Photon properties Energy, polarization, position, momentum, interference, time Photon streams Mean photon flux Randomness of photon flow Photon-number statistics References: Fundamentals of Photonics, Ch. 12

3 3 From ray optics to quantum optics! Ray optics Wave optics Electromagnetic optics Quantum optics!

4 In quantum optics, light is described by photons 4 Ray optics!! Wave optics!! EM optics!! Quantum optics Rays Scalar wave function u(r,t) Vectorial wavefunction E(r,t), H(r,t) Photon! Carries EM energy, momentum and angular momentum (spin) Can carry orbital angular momentum Zero rest mass: travels at c 0 (or c in matter) Particle and wavelike character

5 5 EM optics theory of light in a resonator! Light EM field consisting in a superposition of discrete orthogonal modes! A combination of frequency spatial distribution polarization One possibility for the expansion functions are monochromatic waves! E ( r,t ) = Re[E ( r,t )] ( ) = A q U q r E r,t q ( )exp( i 2πν q t )ê q The (complex) spatial distribution function is normalized: V U q ( r) 2 = 1 polarization!

6 6 Light inside a cubic resonator! A convenient choice for the U q (r) is a set of standing waves: 3 U ( q r ) =! 2 $ 2! π # d & sin # qx % d x $! π &sin# q y % d y $! π &sin# q z % d z $ & % q = ( q x,q y,q z ), q i = 1,2,3 The energy density W of each mode and the total energy contained in the mode are: W 1 2 εe 2 = 1 2 ε A q E q = 1 2 ε V A q 2 Uq ( r) 2 2 Uq ( r) 2 dr = 1 ε A 2 q 2 In classic EM any positive value of E q is possible.!

7 7 Photon optics theory of light in a resonator! Mode energy is restricted to discrete values separated by a fixed energy. Only multiples of this unit are allowed: the energy is quantized. Each unit of energy is carried by one photon. Light Set of modes, each containing an integral number of identical photons. Mode frequency, spatial distribution, direction of propagation and polarization are assigned to the photon.

8 8 The photon is the energy unit of an electromagnetic mode! Energy of a photon in a mode of frequency ν: (only added or taken in multiples of hν ) Energy of mode with n photons: E = hν = ω E n = ( n + 1 2)hν, n = 0, 1, 2, Example: Photon with λ = 1 μm E = hν = hc λ = J 1.24 ev zero-point energy:! E 0 = 1 2 hν (in most experiments, only energy differences are measured)

9 9 The particle/wave nature of a photon depends on its frequency! photonics electronics particlelike nature (higher frequency) wavelike nature (lower frequency)

10 10 A photon may have linear or circular polarization! For each monochromatic wave traveling in some direction, there are always two polarization modes: x and y directions x and y directions, etc right- and left circular the photon polarization is that of the mode. How to describe the polarization of a photon?! In classical EM theory, polarization is described by a Jones vector (A x, A y ): i LP (linear pol.) along x-axis LP along 45 º from x-axis RCP (right hand circular pol.)

11 11 The components of the Jones vector become probabilities! A x, A y A x 2, A y 2 complex probability amplitudes probability that the photon is observed in the x, y polarization modes

12 12 The position of a photon is also a probabilistic function! A photon of frequency ν is associated with a wavefunction U(r)exp(i2πνt) of the mode and extends over the entire mode. But when it hits a detector of area da, it is either detected or not detected. da The probability p(r)da of observing a photon at a point r within da, at any time, is proportional to the local optical intensity I(r): p(r)da I(r)dA, I(r) U(r) 2 photon! position rule!

13 13 Photons behave as extended AND localized entities! Example: standing wave in cavity of length d I(x,y,z) sin 2 ( πz d ) We apply the photon position rule to find out the probability of finding the photon 1 0,75 0,5 0,25 0 max probability 0 0 0,25 0,5 0,75 d 1 zero probability Waves! Particles! Photons! extended in space localized in space wavelike (probability) + particlelike (when detected) = wave-particle duality!

14 14 Transmission of a single photon through a beamsplitter! T R = 1 T intensity transmittance intensity reflectance I t =T I I r = RI = ( 1 T )I (classical expressions for waves) What is the probability of finding a transmitted/ reflected photon (it can only choose one direction)? Applying the photon position rule: p t = I t I =T p r = I r I = 1 T

15 15 A photon carries linear and spin angular momentum! An EM plane wave carries a linear momentum density (per unit volume): (check Fund. Photon., 5.4) A photon carries a linear momentum: ( W c) ˆk p = ( E c) ˆk E = ω = ck Linear momentum associated with a photon in a plane-wave mode k: Magnitude: Photon spin: p = k p = p = k = ω c = E c = h λ S = ±

16 16 A single photon can exhibit interference effects! Performing Young s experiment with a single photon: ( ) 2I 0 $ 1+ cos 2πθx I x # λ % ' & max prob. zero prob. G. I. Taylor (1909) did this experiment and obtained the classical interference pattern by acquiring many individual photons. The extended nature of each photon allows it to pass through both holes at the same time, but be detected individually.

17 17 Describing photons using non-monochromatic modes! In monochromatic (single frequency) modes, with constant intensity, a photon is likely to be detected at any time. More generally, light can be expanded in polychromatic modes, of time-varying intensity. The probability p(r) dadt of observing a photon at a point r within an area da and a time dt, is proportional to the optical intensity I(r, t): p(r,t )dadt I(r,t )dadt, I(r,t ) U(r,t ) 2 photon position and time!

18 18 Time-energy uncertainty! Energy Time Monochromatic mode!!! Polychromatic mode!! U ( t ) = V ( ν)exp( i 2πνt )dν hv σ v I ( ν) V ( ν) 2 (uncertain) σ t I ( t ) U ( t ) 2 A photon in a wavepacket described by U(t) with a duration σ t has an uncertainty in its frequency (and energy) given by σ v such that σ ν σ t 1 4π σ ω σ t 1 2 σ E σ t 2 Time-energy uncertainty!

19 19 What happens when there is a large number of photons?! The number of photons in each mode is generally random It is described by a probability distribution that depends on the quantum state of light Temporal pattern:! A detector that integrates spatially registers photons at random times Spatial pattern:! A detector that integrates temporally registers photons at random positions

20 20 Mean variables for photon streams! Monochromatic light mean photon-flux density φ(r) = I(r) hν (photons/s cm 2 ) Quasi-monochromatic light mean photon-flux density φ(r) I(r) hν (photons/s cm 2 )

21 21 Mean variables for photon streams! mean photon-flux Φ(r) = P = φ( r) = P hν, A I(r) da A (photons/s) (J/s = W) Example: P = µm E J = 6.24 ev Φ 10-9 / photons/s = 1 photon/ns mean photon number detected in an area A and in a time T n = ΦT = E hν E = PT (J)

22 22 Randomness applied to photon streams! single photons! photon streams! ( ) p ( r,t ) ( ) φ ( r,t ) I r,t I r,t probability at (r,t) mean photon flux density at (r,t) constant intensity, but random detection times (depends on source) density of photon detections follows P(t)

23 23 Photon number statistics! source! detector! Example: P = µm E J = 1.25 ev! Φ 10-9 / = 5 photon/ns or ps -1 on average Question: If we divide 100 ns in 10 5 intervals of T = 1 ps each how many will have n = 0 photons? how many will have n = 1 photon? how many will have n = 2 photons? etc photon number statistics

24 24 The statistical distribution of the photons depends on the source! Coherent light!!!!! Thermal light! Photon arrivals are independent Poisson distribution! Photon arrivals are not independent Boltzmann distribution! ( ) = n n exp( n ) p n p E n n! # ( ) exp E n % $, n = 0, 1, 2, & (, k = J/K kt '

25 25 Photon statistics for coherent light! Constant optical power but random arrival times: Poisson distribution p(n) for photon number n!

26 26 Photon statistics for coherent light! The mean value, variance and signal-to-noise ratio are defined as n = n=0 np ( n) σ 2 n = n n n=0 ( ) 2 p ( n) SNR = n 2 2 σ n For the Poisson distribution: σ n 2 = n = SNR i.e. the SNR improves with the number of photons. This is very important for the elimination of errors in optical communications.

27 27 Photon statistics for thermal light! Photon number depends on their energy and the temperature. At 300 K, kt = ev Boltzmann! distrib. p(e n )! Using E n = ( n + 1 2)hν p ( n) = 1 n=0 the probability of finding n photons is p ( n) = ( ) ( ) 1 exp hν kt exp nhν kt

28 28 Photon statistics for thermal light! This probability distribution may be written more simply as Where the mean value is p ( n) = 1 n =! n $ # & n +1 n +1% 1 ( ) 1 exp hν kt n Bose-Einstein distribution! (or geometric distribution) B-E distribution p(n) for photon number n!

29 29 Photon statistics for thermal light! For the Bose-Einstein distribution: σ n 2 = n + n 2 SNR = n n +1 Thermal light has a larger variance than coherent light (more uncertainty) The SNR is always <1, no matter the power This form of light is too noisy for high-data-rate information transmission