- •Contents
- •Preface
- •How to use this book
- •Chapter 1 Units, constants, and conversions
- •1.1 Introduction
- •1.2 SI units
- •1.3 Physical constants
- •1.4 Converting between units
- •1.5 Dimensions
- •1.6 Miscellaneous
- •Chapter 2 Mathematics
- •2.1 Notation
- •2.2 Vectors and matrices
- •2.3 Series, summations, and progressions
- •2.5 Trigonometric and hyperbolic formulas
- •2.6 Mensuration
- •2.8 Integration
- •2.9 Special functions and polynomials
- •2.12 Laplace transforms
- •2.13 Probability and statistics
- •2.14 Numerical methods
- •Chapter 3 Dynamics and mechanics
- •3.1 Introduction
- •3.3 Gravitation
- •3.5 Rigid body dynamics
- •3.7 Generalised dynamics
- •3.8 Elasticity
- •Chapter 4 Quantum physics
- •4.1 Introduction
- •4.3 Wave mechanics
- •4.4 Hydrogenic atoms
- •4.5 Angular momentum
- •4.6 Perturbation theory
- •4.7 High energy and nuclear physics
- •Chapter 5 Thermodynamics
- •5.1 Introduction
- •5.2 Classical thermodynamics
- •5.3 Gas laws
- •5.5 Statistical thermodynamics
- •5.7 Radiation processes
- •Chapter 6 Solid state physics
- •6.1 Introduction
- •6.2 Periodic table
- •6.4 Lattice dynamics
- •6.5 Electrons in solids
- •Chapter 7 Electromagnetism
- •7.1 Introduction
- •7.4 Fields associated with media
- •7.5 Force, torque, and energy
- •7.6 LCR circuits
- •7.7 Transmission lines and waveguides
- •7.8 Waves in and out of media
- •7.9 Plasma physics
- •Chapter 8 Optics
- •8.1 Introduction
- •8.5 Geometrical optics
- •8.6 Polarisation
- •8.7 Coherence (scalar theory)
- •8.8 Line radiation
- •Chapter 9 Astrophysics
- •9.1 Introduction
- •9.3 Coordinate transformations (astronomical)
- •9.4 Observational astrophysics
- •9.5 Stellar evolution
- •9.6 Cosmology
- •Index
172 |
Optics |
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8.7Coherence (scalar theory)
Mutual |
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(t)ψ (t+ τ) |
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coherence |
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Γ12(τ) = |
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ψ |
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(t)ψ (t+ τ) |
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γ12(τ) = |
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Complex degree |
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of coherence |
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Γ12(τ) |
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(8.99) |
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[Γ11(0)Γ22(0)]1/2 |
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Itot = I1 + I2 + 2(I1I2) |
1/2 |
[γ12 |
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intensitya |
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(8.100) |
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Fringe visibility |
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2(I1I2)1/2 |
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(8.101) |
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V (τ) = |
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I1 + I2 |
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if |γ12(τ)| is a |
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Imax − Imin |
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constant: |
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Imax + Imin |
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if I1 = I2: |
V (τ) = |γ12(τ)| |
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(8.103) |
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ψ |
1 |
(t)ψ (t+ τ) |
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Complex degree |
γ(τ) = |
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1 |
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(8.104) |
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|ψ1(t)2| |
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of temporal |
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b |
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I(ω)e−iωτ dω |
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coherence |
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I(ω) dω |
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Coherence time |
∆τc = |
∆lc |
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(8.106) |
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and length |
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∆ν |
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ψ |
1 |
ψ |
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Complex degree |
γ(D) = |
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2 |
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(8.107) |
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of spatial |
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[|ψ1|2 |ψ2|2 ]1/2 |
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= % |
I(sˆ)eikD sˆ dΩ |
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(8.108) |
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coherence |
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I(sˆ) dΩ· |
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Intensity |
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I1I2 |
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= 1 + γ2(D) |
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correlation |
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2 |
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1/2 |
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I2 |
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Speckle |
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intensity |
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pr(I) = |
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e−I/ I |
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distributione |
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Speckle size |
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λ |
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(coherence |
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∆wc α |
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(8.111) |
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width) |
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Γij mutual coherence function
τtemporal interval
ψi (complex) wave disturbance at spatial point i
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mean over time |
γij |
complex degree of coherence |
complex conjugate
Itot |
combined intensity |
Ii |
intensity of disturbance at |
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point i |
real part of
Imax max. combined intensity
Imin min. combined intensity
γ(τ) degree of temporal coherence
I(ω) specific intensity
ωradiation angular frequency
cspeed of light
∆τc coherence time ∆lc coherence length
∆ν spectral bandwidth
γ(D) degree of spatial coherence
Dspatial separation of points 1 and 2
I(sˆ) specific intensity of distant extended source in direction sˆ
dΩ di erential solid angle
sˆ unit vector in the direction of dΩ
kwavenumber
pr probability density
∆wc characteristic speckle size
λwavelength
αsource angular size as seen from the screen
aFrom interfering the disturbances at points 1 and 2 with a relative delay τ. bOr “autocorrelation function.”
cBetween two points on a wavefront, separated by D. The integral is over the entire extended source.
dFor wave disturbances that have a Gaussian probability distribution in amplitude. This is “Gaussian light” such as from a thermal source.
eAlso for Gaussian light.
8.8 Line radiation |
173 |
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8.8 Line radiation
Spectral line broadening
Natural |
I(ω) = |
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broadeninga |
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Natural |
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half-width |
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broadening |
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pressure |
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Doppler |
I(ω) = |
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half-width |
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I(ω) normalised intensityb
τlifetime of excited state
ωangular frequency (= 2πν)
∆ω half-width at half-power
ω0 centre frequency
τc mean time between collisions
ppressure
de ective atomic diameter
mgas particle mass
kBoltzmann constant
Ttemperature
cspeed of light
I(ω) |
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aThe transition probability per unit time for the state is = 1/τ. In the classical limit of a damped oscillator, the
e-folding time of the electric field is 2τ. Both the natural and collision profiles described here are Lorentzian.
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bThe intensity spectra are normalised so that I(ω) dω = 1, assuming ∆ω/ω0 1.
cThe pressure-broadening relation combines Equations (5.78), (5.86) and (5.89) and assumes an otherwise perfect gas of finite-sized atoms. More accurate expressions are considerably more complicated.
Einstein coe cientsa
Absorption |
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R12 = B12Iν n1 |
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transition rate, level i → j (m−3 s−1) |
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Iν |
specific intensity of radiation field |
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R21 |
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A21 |
Einstein A coe cient |
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number density of atoms in quantum |
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A21 |
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Planck constant |
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Coe cient |
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174 Optics
Lasersa
R1 |
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light out
L r2
Cavity stability |
0 ≤ |
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cavity modesb |
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2L |
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Cavity Q |
λ[1 − (R1R2)1/2] |
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4πL |
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λ(1 |
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Cavity line |
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width |
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Q |
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Schawlow– |
∆ν |
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glNu |
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Townes line |
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νn |
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P |
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glNu |
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guNl |
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Threshold |
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lasing condition |
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r1,2 radii of curvature of end-mirrors
Ldistance between mirror centres
νn mode frequency
ninteger
cspeed of light
Qquality factor
R1,2 |
mirror (power) reflectances |
λ |
wavelength |
∆νc |
cavity line width (FWHP) |
τc |
cavity photon lifetime |
∆ν |
line width (FWHP) |
Plaser power
gu,l |
degeneracy of upper/lower levels |
Nu,l |
number density of upper/lower |
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levels |
αgain per unit length of medium
βloss per unit length of medium
aAlso see the Fabry-Perot etalon on page 163. Note that “cavity” refers to the empty cavity, with no lasing medium present.
bThe mode spacing equals the cavity free spectral range.