Mathematical Notation Index - Exploring Optics: From Fundamentals to Advanced Applications

This appendix provides a comprehensive reference for the mathematical symbols, operators, and notation used throughout this textbook. The notation follows standard conventions in optics and electromagnetic theory.

1Greek Letters¶

1.1Lowercase Greek Letters¶

Symbol	Name	Common Usage in Optics
$\alpha$	alpha	Absorption coefficient, angle of incidence
$\beta$	beta	Phase constant, propagation constant
$\gamma$	gamma	Complex degree of coherence, damping coefficient
$\delta$	delta	Phase difference, small increment
$\epsilon$	epsilon	Permittivity, small parameter
$\varepsilon_0$	epsilon-naught	Permittivity of free space ( $8.854 \times 10^{-12}$ F/m)
$\zeta$	zeta	Damping ratio
$\eta$	eta	Efficiency, refractive index
$\theta$	theta	Angle, phase
$\kappa$	kappa	Coupling coefficient
$\lambda$	lambda	Wavelength
$\mu$	mu	Permeability, magnification
$\mu_0$	mu-naught	Permeability of free space ( $4\pi \times 10^{-7}$ H/m)
$\nu$	nu	Frequency (Hz)
$\xi$	xi	Normalized frequency parameter
$\rho$	rho	Radial coordinate, density
$\sigma$	sigma	Standard deviation, conductivity
$\tau$	tau	Transmission coefficient, time constant
$\phi$	phi	Phase, angle, electric potential
$\varphi$	varphi	Phase (alternative notation)
$\chi$	chi	Electric susceptibility
$\psi$	psi	Wave function, angle
$\omega$	omega	Angular frequency (rad/s)

1.2Uppercase Greek Letters¶

Symbol	Name	Common Usage in Optics
$\Gamma$	Gamma	Coherence function, reflection coefficient
$\Delta$	Delta	Change in quantity, finite difference
$\Theta$	Theta	Angle
$\Lambda$	Lambda	Period of grating
$\Phi$	Phi	Total flux, phase
$\Psi$	Psi	Wave function
$\Omega$	Omega	Solid angle

2Fundamental Physical Constants¶

Symbol	Constant	Value	Units
$c$	Speed of light in vacuum	$2.998 \times 10^8$	m/s
$h$	Planck’s constant	$6.626 \times 10^{-34}$	J·s
$\hbar$	Reduced Planck’s constant	$h/(2\pi) = 1.055 \times 10^{-34}$	J·s
$e$	Elementary charge	$1.602 \times 10^{-19}$	C
$m_e$	Electron mass	$9.109 \times 10^{-31}$	kg
$k_B$	Boltzmann constant	$1.381 \times 10^{-23}$	J/K

3Electromagnetic Field Quantities¶

Symbol	Quantity	Units
$\vec{E}$ or $\Vector{E}$	Electric field vector	V/m
$\vec{B}$ or $\Vector{B}$	Magnetic field vector	T (Tesla)
$\vec{H}$ or $\Vector{H}$	Magnetic field intensity	A/m
$\vec{D}$ or $\Vector{D}$	Electric displacement field	C/m²
$\vec{S}$ or $\Vector{S}$	Poynting vector (energy flux)	W/m²
$\vec{k}$ or $\Vector{k}$	Wave vector	rad/m
$k$	Wave number magnitude	rad/m
$n$	Refractive index	dimensionless
$I$	Intensity (irradiance)	W/m²
$U$	Complex amplitude	various

4Photon and Quantum Properties¶

Symbol	Quantity	Relation/Value	Units
$E$	Photon energy	$E = h\nu = \hbar\omega$	J
$p$	Photon momentum	$p = E/c = h/\lambda$	kg·m/s
$\lambda$	Wavelength	$\lambda = c/\nu$	m
$\nu$	Frequency	$\nu = c/\lambda$	Hz
$\omega$	Angular frequency	$\omega = 2\pi\nu$	rad/s
$k$	Wave number	$k = 2\pi/\lambda = \omega/c$	rad/m

5Geometric Optics¶

Symbol	Quantity	Description
$f$	Focal length	Distance from lens/mirror to focal point
$s$	Object distance	Distance from object to optical element
$s'$	Image distance	Distance from optical element to image
$m$	Magnification	Ratio of image size to object size
$h$	Object height	Height of object
$h'$	Image height	Height of image
$R$	Radius of curvature	Radius of spherical surface
$n_1, n_2$	Refractive indices	Before and after interface
$\theta_i$	Angle of incidence	Angle relative to surface normal
$\theta_r$	Angle of refraction	Angle relative to surface normal
$\theta_c$	Critical angle	Angle for total internal reflection
$NA$	Numerical aperture	$n\sin\theta_{max}$

6Wave Optics and Interference¶

Symbol	Quantity	Description
$\delta$	Phase difference	Difference in optical path
$\Delta$	Path difference	Physical path difference
$\Delta_{\text{opt}}$	Optical path difference	$n \times \text{physical path}$
$m$	Order of interference	Integer for bright/dark fringes
$d$	Spacing	Distance between slits or grating lines
$\Lambda$	Grating period	Spatial period of periodic structure
$\mathcal{V}$	Visibility (contrast)	$(I_{max} - I_{min})/(I_{max} + I_{min})$
$\gamma_{12}$	Complex degree of coherence	Normalized correlation function
$\ell_c$	Coherence length	Spatial extent of coherence
$\tau_c$	Coherence time	Temporal extent of coherence

7Diffraction¶

Symbol	Quantity	Description
$a$	Aperture width	Width of slit or aperture
$D$	Aperture diameter	Diameter of circular aperture
$\theta$	Diffraction angle	Angle from optical axis
$\text{sinc}(x)$	Sinc function	$\sin(x)/x$
$J_n(x)$	Bessel function	Bessel function of first kind, order $n$
$\mathcal{F}$	Fourier transform	Spatial frequency transform
$k_x, k_y$	Spatial frequencies	Fourier domain coordinates
$u, v$	Normalized spatial coordinates	Dimensionless position variables

8Polarization¶

Symbol	Quantity	Description
$\vec{E}_x, \vec{E}_y$	Electric field components	Perpendicular field components
$A_x, A_y$	Amplitudes	Field amplitudes in x, y directions
$\delta$	Phase difference	Between orthogonal components
$\chi$	Auxiliary angle	Ellipse orientation angle
$\psi$	Azimuth angle	Polarization orientation
$\epsilon$	Ellipticity	Ratio of minor to major axis
$I, Q, U, V$	Stokes parameters	Complete polarization description

9Matrix Optics (Ray Transfer Matrices)¶

Symbol	Quantity	Description
$\mathbf{M}$	Ray transfer matrix	$2 \times 2$ matrix for ray propagation
$A, B, C, D$	Matrix elements	Elements of ray transfer matrix
$y$	Ray height	Distance from optical axis
$\theta$	Ray angle	Angle with optical axis
$n$	Refractive index	Medium refractive index

10Fourier Optics¶

Symbol	Quantity	Description
$F(k_x, k_y)$	Spatial frequency spectrum	2D Fourier transform of field
$k_x, k_y$	Spatial frequency coordinates	Fourier domain variables
$k_r$	Radial spatial frequency	$\sqrt{k_x^2 + k_y^2}$
$H(k_x, k_y)$	Transfer function	System frequency response
$\text{rect}(x)$	Rectangle function	1 if $
$\text{circ}(r)$	Circle function	1 if $r < 1$ , 0 otherwise
$\star$	Convolution operator	Convolution in spatial domain
$\otimes$	Cross-correlation	Correlation operator

11Fiber Optics¶

Symbol	Quantity	Description
$n_1$	Core refractive index	Index of fiber core
$n_2$	Cladding refractive index	Index of fiber cladding
$\Delta$	Relative index difference	$(n_1 - n_2)/n_1$
$V$	Normalized frequency	$V$ -parameter, determines mode count
$a$	Core radius	Physical radius of fiber core
$\alpha$	Attenuation coefficient	Loss per unit length (dB/km)
$\beta$	Propagation constant	Phase change per unit length
$D$	Dispersion parameter	Pulse spreading (ps/(nm·km))

12Laser Physics¶

Symbol	Quantity	Description
$g$	Gain coefficient	Amplification per unit length
$\alpha$	Loss coefficient	Loss per unit length
$R$	Mirror reflectivity	Fraction of light reflected
$L$	Cavity length	Length of laser resonator
$\mathcal{F}$	Finesse	Quality factor of cavity
$Q$	Quality factor	Energy stored / energy lost per cycle
$\Delta\nu$	Linewidth	Spectral width of laser emission
$N_2, N_1$	Population densities	Upper and lower level populations

13Custom LaTeX Macros¶

This textbook uses several custom LaTeX macros for convenience and consistency:

Macro	Expands To	Usage	Example
`\dif`	$\operatorname{d}\!$	Differential operator	`\dif x` → $\operatorname{d}\!x$
`\Vector{E}`	$\boldsymbol{E}$	Bold vector notation	`\Vector{E}` → $\boldsymbol{E}$
`\Real`	$\mathbb{R}$	Real numbers set	`x \in \Real` → $x \in \mathbb{R}$
`\Complex`	$\mathbb{C}$	Complex numbers set	`z \in \Complex` → $z \in \mathbb{C}$

14Mathematical Operators¶

Operator	Symbol	Description
Gradient	$\nabla$ or $\text{grad}$	Spatial derivative operator
Divergence	$\nabla \cdot$ or $\text{div}$	Dot product with gradient
Curl	$\nabla \times$ or $\text{curl}$	Cross product with gradient
Laplacian	$\nabla^2$ or $\Delta$	Second spatial derivative
Partial derivative	$\partial/\partial x$	Derivative with respect to $x$
Total derivative	$\dif/\dif t$	Total time derivative
Complex conjugate	$z^*$ or $\bar{z}$	Complex conjugate of $z$
Magnitude	$	z

15Special Functions¶

Function	Notation	Definition	Usage
Sinc function	$\text{sinc}(x)$	$\sin(x)/x$	Single-slit diffraction
Bessel function	$J_n(x)$	—	Circular aperture diffraction
Error function	$\text{erf}(x)$	$\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$	Gaussian beam analysis
Hermite polynomial	$H_n(x)$	—	Laser mode profiles
Laguerre polynomial	$L_n^m(x)$	—	Optical vortices, orbital angular momentum
Dirac delta	$\delta(x)$	Generalized function	Point sources
Heaviside step	$H(x)$	0 if $x<0$ , 1 if $x \geq 0$	Step functions

16Common Abbreviations¶

Abbreviation	Full Name	Description
EM	Electromagnetic	Relating to electric and magnetic fields
TEM	Transverse Electromagnetic	Mode with no longitudinal field components
TE	Transverse Electric	Mode with no longitudinal electric field
TM	Transverse Magnetic	Mode with no longitudinal magnetic field
LP	Linearly Polarized	Linear polarization mode
NA	Numerical Aperture	Light-gathering capacity
OPD	Optical Path Difference	Path difference including refractive index
OPL	Optical Path Length	Physical path times refractive index
FSR	Free Spectral Range	Spacing between cavity modes
FWHM	Full Width at Half Maximum	Width of peak at half intensity
MTF	Modulation Transfer Function	Spatial frequency response
PSF	Point Spread Function	Image of point source
FT	Fourier Transform	Frequency domain transformation
FFT	Fast Fourier Transform	Efficient FT algorithm
DFT	Discrete Fourier Transform	Digital/sampled FT

17Sign Conventions¶

Throughout this textbook, we adopt the following sign conventions:

Distances: Measured from the optical element
- Object distances ( $s$ ): Positive if object is on the incoming light side
- Image distances ( $s'$ ): Positive if image is on the outgoing light side
Focal lengths:
- Positive for converging lenses/mirrors
- Negative for diverging lenses/mirrors
Heights:
- Positive above the optical axis
- Negative below the optical axis
Angles:
- Measured from the surface normal
- Positive in the counterclockwise direction
Phase:
- Time dependence: $e^{-i\omega t}$ (physics convention)
- Alternative: $e^{+i\omega t}$ (engineering convention) - noted when used
Refractive index: Always positive for passive media

18Equation Numbering Reference¶

This textbook uses automatic equation numbering. Numbered equations can be referenced using:

Inline reference: {eq}equation-label
Formatted reference: {eq}\equation-label``

Example: See equation photon-energy in Chapter 1.

19Units and Notation¶

19.1SI Units¶

Standard SI units are used throughout:

Length: meter (m), with common prefixes (nm, μm, mm, cm)
Time: second (s), with prefixes (fs, ps, ns, μs, ms)
Frequency: Hertz (Hz = 1/s), with prefixes (kHz, MHz, GHz, THz)
Energy: Joule (J), electron volt (eV)
Power: Watt (W = J/s)
Intensity: W/m²

19.2Angle Units¶

Radians (rad) for mathematical calculations
Degrees (°) for practical measurements and visualization
Conversion: $180° = \pi$ rad

20Index of Key Equations¶

20.1Fundamental Relations¶

Photon energy: $E = h\nu = \hbar\omega$
Photon momentum: $p = h/\lambda = E/c$
Wave equation: $\nabla^2 E - \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2} = 0$
Speed of light: $c = \lambda\nu$
Refractive index: $n = c/v$

20.2Geometric Optics¶

Snell’s law: $n_1\sin\theta_1 = n_2\sin\theta_2$
Thin lens equation: $\frac{1}{f} = \frac{1}{s} + \frac{1}{s'}$
Magnification: $m = \frac{h'}{h} = -\frac{s'}{s}$
Lensmaker’s equation: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

20.3Wave Optics¶

Young’s double slit: Bright fringes at $d\sin\theta = m\lambda$
Single slit diffraction: First minimum at $a\sin\theta = \lambda$
Grating equation: $d\sin\theta = m\lambda$
Rayleigh criterion: $\theta_{\text{min}} = 1.22\lambda/D$

20.4Maxwell’s Equations¶

In free space:

$\nabla \cdot \Vector{E} = 0$
$\nabla \cdot \Vector{B} = 0$
$\nabla \times \Vector{E} = -\frac{\partial \Vector{B}}{\partial t}$
$\nabla \times \Vector{B} = \mu_0\varepsilon_0\frac{\partial \Vector{E}}{\partial t}$

21Cross-References¶

For related mathematical topics, see:

Appendix A: Complex Numbers in Optics - Complex Numbers in Optics
Appendix B: Matrix Multiplication in Optics - Matrix Multiplication
Appendix C: Fourier Transform in Optics - Fourier Transform in Optics
Appendix D: Vector Calculus for Electromagnetic Fields - Vector Calculus for Electromagnetic Fields