EE 485 Introduction to Photonics ― Photon Optics and Photon Statistics

1 Historical Origin ― Photo-electric Effect (Einstein, 1905)

Different metals, same slope Clean metal Vstop

Slope = h/q Light I ν = c/λ

λ λ Current flows for < 0, for any intensity of light Apply stopping potential → Higher voltage required for shorter λ

The slopes are independent of light intensity, but current is proportional to light intensity. hν = qV +W Energy of light = (Kinetic energy of the electron) + (Work function) = ν Eph h Energy of quantum of light (Smallest energy unit): Photon

Lih Y. Lin EE 485 2 Photon Energy

E = hν = ω − h = 6.63×10 34 Joule⋅Sec : Plank’s constant  = h/ 2π

1.24 λ (µm) = E (eV)

Lih Y. Lin EE 485 3 Photon Position

2 (∆z)2 (∆p)2 ≥  Heisenberg’s Uncertainty Principle 4

Photon position cannot be determined exactly. It needs to be determined with probability. p(r)dA ∝ I(r)dA

Example: (1) Photon position probability in a Gaussian beam. (2) Transmission of a single photon through a beam-splitter.

(1) (2)

Lih Y. Lin EE 485 4 Photon Momentum

Plane wave E(r,t) = Aexp(− jk ⋅r)exp( j2πνt)eˆ Photon momentum p = k h p = k = λ

Radiation Pressure ∆p h λ h Force: = = N ∆t ∆t λ h p = λ Force Pressure: Area

Example: Photon-momentum recoil versus thermal velocity.

Lih Y. Lin EE 485 5 Optical Tweezer

Forces arising from momentum change of the light ∆P F = a A

∆t m µ 3.2 b B Out In (c-d) (a-b) p Moving a DNA-tethered bead with an q ∆p p P optical tweezer (25 mW) q Q (http://www.bio.brandeis.edu/~gelles/stall/)

pP q c C Q Resultant “gradient” force

D d

E.g., λ = 1064 nm, P = 100 mW, diameter of polystyrene sphere 10 µm sphere = 5 µm → F = 3.18 x 10-12 N. (http://www.phys.umu.se/laser/tweezer1.htm) Ashkin, et al., “Observation of radiation pressure trapping of particles by Lih Y. Lin alternating laser beams,” Phys. Rev. Lett., V. 54, p. 1245-1248, 1985 EE 485 6 Photon Polarization ― Linearly Polarized Photons

Photon polarized along x-direction = + − ω E(r,t) (Axxˆ Ayyˆ)exp( jkz)exp( j t) = + − ω E(r,t) (Ax'xˆ' Ay'yˆ')exp( jkz)exp( j t) 1 1 A = ()A − A , A = ()A + A x' 2 x y y' 2 x y

Example: Transmission of a linearly-polarized photon through a polarizer

Lih Y. Lin EE 485 7 Quantum Communication ― Secured Information Transmission with Single Photons

α β Qubit: |0>+ |1> Alice Bob

Coding With the key given by Alice, Eve obtains the same result as Alice’s. Encryption Without the key given by Alice, (Determine α and β) obtains the wrong result with high probability, and destroys the qubit.

Polarization coding

: |0> Alice: : |1>

Bob: Measure with or basis

Lih Y. Lin EE 485 8 Photon Polarization ― Circularly Polarized Photons and Photon Spin = ()+ − πν E(r,t) AReˆ R ALeˆ L exp( jkz)exp( j2 t) 1 1 eˆ = (xˆ + jyˆ) eˆ = (xˆ − jyˆ) R 2 L 2 Example: (1) A linearly-polarized photon transmitting through a circular polarizer. (2) A right-circularly-polarized photon transmitting through a linear polarizer.

Photon Spin Photon has intrinsic angular momentum. Photon spin: S = ± For right-circularly-polarized photons, S is parallel to k. For left-circularly-polarized photons, S is anti-parallel to k. Linearly-polarized photons have an equal probability of exhibiting parallel and Lih Y. Lin anti-parallel spin. EE 485 9 Photon Interference

Assume the mirrors and beam-splitters are perfectly flat and lossless. Path length difference is d.

Probability of finding the photon at the detector?

If we don’t find the photon at the detector, where is it?

Lih Y. Lin EE 485 10 Photon Time

Heisenberg’s Uncertainty Principle also implies  ∆t ⋅ ∆E ≥ 2

The probability of observing a photon at (r, t) within an incremental area of dA and during the incremental time interval dt following time t:

p(r,t)dAdt ∝ I(r,t)dAdt ∝ U (r,t) 2 dAdt

Lih Y. Lin EE 485 11  photons    Mean Photon Flux Density  sec ⋅ area 

Monochromatic light of frequency ν and intensity I(r) I(r) Photon flux density ϕ(r) = hν Quasi-monochromatic light of central-frequency ν I(r) Photon flux density ϕ(r) = hν

Lih Y. Lin EE 485 12  photons    Mean Photon Flux  sec  and Mean Number of Photons

Mean Photon Flux P Φ = ∫ ϕ(r)dA = A hν P = I(r)dA : Optical power (watts) ∫A

Mean Number of Photons E n = Φ ⋅T = hν E = P ⋅T : Optical energy (joule) T E Time-varying light n = ∫ Φ(t) ⋅ dt = ν 0 h T E = ∫ P(t) ⋅ dt : Optical energy (joule) 0

Lih Y. Lin EE 485 13 Randomness of Photon Flux

Even if the optical power is constant, the time of arrival of a single photon is governed by probabilistic laws.

Lih Y. Lin EE 485 14 Photon Statistics for Coherent Light ― Probability

Mean photon number = n

It’s possible to detect different number of photons at different time intervals.

Probability of detecting n photons is a Poisson distribution: n n exp(−n) p(n) = n!

Lih Y. Lin EE 485 15 Photon Statistics for Coherent Light ― Mean, Variance, and SNR

∞ Mean: n = ∑n ⋅ p(n) n=0

∞ σ2 = − 2 ⋅ Variance: n ∑(n n) p(n) n=0

2 For Poisson distribution, σ = n n

(mean)2 n 2 Signal-to-noise ratio: SNR = = σ2 Variance n

For Poisson distribution, SNR = n

Lih Y. Lin EE 485 16 Photon Statistics for Incoherent Light

  ∝ − En  Probability follows Boltzmann distribution P(En ) exp   kBT  − = × 23 : Boltzmann constant kB 1.38 10 J/k n   hν   hν  p(n) = 1− exp− exp−    kBT   kBT  1 1  n n n = p(n) =    hν  n +1 n +1 exp  −1  kBT  σ2 = + 2 n n n n SNR = <1 No matter how large the optical power is. n +1

Lih Y. Lin EE 485 17