1

EE 485Introduction to Photonics― Photon Optics and Photon Statistics

2Lih Y. LinEE 485

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

WqVh +=ν

ILight

Clean metal Vstop

ν = c/λ

Different metals, same slope

Current flows for λ < λ0, for any intensity of light

Slope = h/q

Apply stopping potential→ Higher voltage required for shorter λ

The slopes are independent of light intensity, but current is proportional to light intensity.

Energy of light = (Kinetic energy of the electron) + (Work function)

ν= hEph Energy of quantum of light (Smallest energy unit): Photon

3Lih Y. LinEE 485

Photon Energy

π=⋅×=

ω=ν=−

2/SecJoule 1063.6 34

hh

hE

�

�

: Plank’s constant

(eV) 24.1m)(

E=µλ

4Lih Y. LinEE 485

Photon Position

4)()(222 �≥∆∆ pz

dAIdAp )()( rr ∝

Heisenberg’s Uncertainty Principle

Photon position cannot be determined exactly. It needs to be determined with probability.

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

(1) (2)

5Lih Y. LinEE 485

Photon Momentum

erkrE ˆ)2exp()exp(),( tjjAt πν⋅−=

kp �=

λ== hkp �

Plane wave

Photon momentum

Radiation Pressure

λ= hp

Force:λ

=∆

λ=∆∆ hN

th

tp

Pressure:AreaForce

Example: Photon-momentum recoil versus thermal velocity.

6Lih Y. LinEE 485

Optical Tweezer

Moving a DNA-tethered bead with an optical tweezer (25 mW) (http://www.bio.brandeis.edu/~gelles/stall/)

(http://www.phys.umu.se/laser/tweezer1.htm)

3.2

µm10 µm sphere

Forces arising from momentum change of the light

tPF

∆∆= a

b

c

d

A

B

C

D

pq

PQ

In(a-b)

Out(c-d)

pq ∆p

pPq

Q

Resultant “gradient” force

E.g., λ = 1064 nm, P = 100 mW, diameter of polystyrene sphere = 5 µm → F = 3.18 x 10-12 N.

Ashkin, et al., “Observation of radiation pressure trapping of particles by alternating laser beams,” Phys. Rev. Lett., V. 54, p. 1245-1248, 1985

7Lih Y. LinEE 485

Photon Polarization― Linearly Polarized Photons

( ) ( )yxyyxx

yx

yx

AAAAAA

tjjkzAAttjjkzAAt

+=−=

ω−+=

ω−+=

21 ,

21

)exp()exp()'ˆ'ˆ(),(

)exp()exp()ˆˆ(),(

''

'' yxrEyxrE

Photon polarized along x-direction

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

8Lih Y. LinEE 485

Quantum Communication― Secured Information Transmission with Single Photons

BobAlice

EveWith the key given by Alice, obtains the same result as Alice’s.

Without the key given by Alice, obtains the wrong result with high probability, and destroys the qubit.

Qubit: αααα|0>+ββββ|1>

Coding

(Determine αααα and β)β)β)β)

Encryption

: |0>

: |1>

Polarization coding

Alice:

Bob: Measure with or basis

9Lih Y. LinEE 485

Photon Polarization― Circularly Polarized Photons and Photon Spin

( )

)ˆˆ(2

1ˆ )ˆˆ(2

1ˆ

)2exp()exp(ˆˆ),(

yxeyxe

eerE

jj

tjjkzAAt

LR

LLRR

−=+=

πν−+=

Example: (1) A linearly-polarized photon transmitting through a circular polarizer.(2) A right-circularly-polarized photon transmitting through a linear polarizer.

Photon SpinPhoton has intrinsic angular momentum. �±=SPhoton spin:

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 anti-parallel spin.

10Lih Y. LinEE 485

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?

11Lih Y. LinEE 485

Photon Time

2�≥∆⋅∆ Et

Heisenberg’s Uncertainty Principle also implies

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:

dAdttUdAdttIdAdttp 2),(),(),( rrr ∝∝

12Lih Y. LinEE 485

Mean Photon Flux Density

ν=ϕhI )()( rr

Monochromatic light of frequency ν and intensity I(r)

Photon flux density

Quasi-monochromatic light of central-frequency

Photon flux densityν

=ϕhI )()( rr

ν

⋅ areasecphotons

13Lih Y. LinEE 485

Mean Photon Flux andMean Number of Photons

secphotons

(watts)power Optical : )(

)(

dAIPhPdA

A

A

∫

∫

=ν

=ϕ=Φ

r

rMean Photon Flux

Mean Number of Photons

(joule)energy Optical : TPEhETn

⋅=ν

=⋅Φ=

Time-varying light

(joule)energy Optical : )(

)(

0

0

∫

∫

⋅=

ν=⋅Φ=

T

T

dttPE

hEdttn

14Lih Y. LinEE 485

Randomness of Photon Flux

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

15Lih Y. LinEE 485

Photon Statistics for Coherent Light― Probability

Mean photon number = nIt’s possible to detect different number of photons at different time intervals.

Probability of detecting n photons is a Poisson distribution:

!)exp()(

nnnnp

n −=

16Lih Y. LinEE 485

Photon Statistics for Coherent Light― Mean, Variance, and SNR

Mean: ∑∞

=⋅=

0)(

nnpnn

Variance: ∑∞

=⋅−=σ

0

22 )()(n

n npnn

For Poisson distribution, nn =σ2

Signal-to-noise ratio: 2

22

Variancemean)(SNR

n

nσ

==

For Poisson distribution, n=SNR

17Lih Y. LinEE 485

Photon Statistics for Incoherent Light

Probability follows Boltzmann distribution

−∝

TkEEPB

nn exp)(

J/k 1038.1 23−×=Bk : Boltzmann constant

n

BB Tkh

Tkhnp

ν−

ν−−= expexp1)(

1exp

1

−

ν=

Tkh

n

B

n

nn

nnp

++=

111)(

22 nnn +=σ

11

SNR <+

=nn

No matter how large the optical power is.