Notations-and-Definitions_1976_Beam-and-Fiber-Optics

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Notations and Definitions 
The system of units used is the international system of units (meter, 
kilogram-mass, second, ampere). The field of traveling waves is denoted 
exp[/(A:z — cot)]. As the wave propagates in the + z-direction, its phase 
thus advances, k = 2π/λ denotes the wavenumber and λ the wavelength 
in the medium. The free space wavelength is denoted λ0. When we discuss 
transformations through optical systems, primed quantities are used in the 
object space, and unprimed quantities in the image space, where most of 
the transformations are carried out. The radius of a gaussian beam is 
defined at the l/e point of the beam irradiance rather than at the \/e 
point of the field modulus. If the former radius is denoted £ and the latter 
w, we have w = V2~|. The ξ-notation is selected because it provides more 
symmetrical expressions between far and near fields than the w-notation. 
Furthermore, | , rather than w, corresponds to the classical turning point. 
The ^-notation is always used in quantum mechanics. The field of scalar 
modes in circularly symmetric fibers is denoted ψ^, where μ = 0, ±1 , ±2, 
. . . is the azimuthal mode number and a = 0, 1, 2, . . . the radial mode 
number. For step-index fibers, the notation LP^^a + l (where LP stands for 
"linearly polarized") has been used in place of ψ^ in some recent works. 
This alternative notation, however, is not consistent with that commonly 
used for graded-index fibers. The normalized frequency of a fiber, which 
we denote F, is often denoted V. Note also that the numerical aperture 
(NA) of a fiber is defined from the radiation pattern in air. Vectors and 
matrices are distinguished by boldface, usually with lowercase and capital 
letters, respectively. Scalar products are denoted a · b, ab, or simply, when 
no confusion with a tensor product is possible, ab. 
XV 
XVI Notations and Definitions 
Latin Letters 
D-
a: 
am\ 
a: 
A: 
A: 
Ai(x): 
b: 
b: 
B: 
B: 
c: 
c: 
^ : 
C: 
e 
d 
Z) 
D 
ds 
dC 
dS: 
■(u/kXdk/dd) 
D„ D2 
e 
e 
E 
E 
/: 
F 
g 
G(x; x'): 
G 
G 
h 
h 
h(z) 
H(k, x) = 0: 
H 
radius of the core of a fiber; radius of an aperture; normal-mode 
amplitude; asymmetry parameter 
element of a ray matrix 
coefficient of spectral expansion 
potential vector 
element of a ray matrix; area 
4-potential vector = (a, iV) 
Airy function 
normalized phase velocity 
element of a ray matrix 
element of a ray matrix; radiance 
magnetic field; binormal vector 
velocity of light in free space; transverse coupling 
element of a ray matrix 
coupling 
element of a ray matrix; capacitance per unit length; fiber axis curva­
ture, = 1/radius of curvature = 1/p 
capacitance 
distance between two mirrors; half slab thickness 
element of a ray matrix 
element of a ray matrix; diffraction walk-off parameter; spacing 
electric induction; spectral matrix of an operator 
elementary ray length 
vector perpendicular to a planar contour, with magnitude equal to the 
elementary arc length 
vector perpendicular to a surface, with magnitude equal to the ele­
mentary area 
dispersion parameter 
differentiation matrices 
inhomogeneous material dispersion parameters 
= 2.718 . . . ; electron charge 
electron charge divided by h 
magnitude of the electric field; pulse energy 
electric field 
focal length; frequency; force; ray density in phase space 
Fresnel number; finesse; normalized frequency = (k2 — k2)1/2 X (a or 
d) (denoted V in other works) 
electromagnetic field 6-vector; hamiltonian parameter 
resonator parameter, = 1 — d/R 
hamiltonian parameter 
Green's function 
geometrical walk-off parameter 
Hamiltonian parameter 
Planck's constant = 2π X h 
= 1.054 X 10 - 3 4 joule X second 
curvature of the surface of wave vectors 
hamiltonian function 
magnetic induction 
Hermite polynomial of order m 
Notations and Definitions xvu 
Hem ,2(*1> *2>: 
/: 
Γ, i: 
I: 
Im(): 
/: 
JF: 
*«: 
Ä:: 
*.: 
k0: 
k: 
K: 
K 
/ 
L 
L 
e 
e 
L 
L 
m 
m: 
m* 
m0, 
M 
n: 
N: 
N2 
n 
N: 
P 
/> 
Λ 
P, P+ 
?ω 
e 
modified Hermite polynomial in two variables. 
(—l)1/2; time dependence denoted: exp(— ίωί); in subscript: imaginary 
part 
angles of incidence and of refraction defined with respect to the normal 
to the surface 
electric current; Lagrange ray invariant 
imaginary part 
intermediate frequency current density 
adiabatic invariant 
Bessel function of the first kind of order μ 
Maxwell current 
Fock current with components j (transverse) and p (axial) 
free wavenumber at some point r 
Fiber core free wavenumber, — 2ττ/λ 
cladding or substrate free-wavenumber 
wavenumber on-axis or in the upper medium 
wave vector, with components kx, k , kz or P\,p2> kz 
4-wave vector, with components kx, k , k2, ίω/c; wave function opera­
tor; third material matrix (6 X 6) 
Boltzmann constant, = 1.38 X 10~23 joule/°C; normalized coupling 
modified Bessel function of the second kind of order μ 
axial mode number; relative loss 
loss in dB or loss per unit length in dB/km; inductance per unit length 
lagrangian of the beam axis 
lagrangian density; inductance 
average lagrangian density 
second material matrix with submatrices c , ξ , £, μ 
surface of ray vectors 
Lagrange function 
electron mass; transverse mode number 
= m/h 
effective mass 
mass per unit length 
transverse mode numbers 
ray matrix; material matrix (6 X 6) with submatrices M n , M12, M21, 
M22 
refractive index, = λ 0 /λ = k/k0; mode number 
power coupling in two dimensions, mode number density; number of 
transverse dimensions 
power coupling in three dimensions 
hamiltonian parameter 
hamiltonian parameter 
numerical aperture, = sine of half-radiation angle in air 
transverse ray momentum with components/?,,^ or kx, ky 
power; power in the core 
power in the cladding 
total power 
complex matrical ray momenta (2 X 2) 
complex ray 
positive and negative frequency components of a complex ray 
mismatch parameter; complex ray 
Notations and Definitions 
Q,Q f 
r 
R: 
Re(): 
s: 
s: 
S(x): 
S(x; x'): 
S: 
t: 
T: 
T: 
u: 
u0: 
uz\ 
u: 
U: 
Ur 
u r 
v: 
W 
W 
W* 
W g 
X 
X 
Xl> X2> X 3 
X 
X 
y 
Y 
z 
z 
Z 
complex ray 
complex matrical rays (2 X 2) 
radius in space or in a transverse plane; field reflectivity; as a subscript, 
denotes real part 
wavefront mismatch parameter; mirror radius of curvature; power re­
flectivity, - rr* = \ - T. Also, R = r2, where r denotes radius; field 
strength parameter 
real part 
maximum ray radius squared 
normalized susceptance with dimension 1/length; integer 
ray vector; 4-vector in spatial phase space 
eikonal (phase) 
point-eikonal, = phase shift along a ray 
energy flux 
time 
power transmission, — \ — R 
rotation matrix; Lorentz transformation matrix (6 X 6) 
magnitude of group velocity; ray parameter; circular rod parameter 
average axial group velocity 
axial group velocity 
point-eikonal parameter; group velocity 
electric potential; point-eikonal parameter 
generating point-eikonal parameter (two dimensions) 
generating point-eikonal parameter (three dimensions) 
phase velocity of free waves in a medium; ray parameter; circuit rod 
parameter 
axial phase velocity, = v/kz 
axial group velocity 
point-eikonal parameter 
scalar (electric) potential; point-eikonal parameter 
generating point-eikonal parameter (two dimensions) 
generating point-eikonal parameter (three dimensions) 
vT X gaussian beam half-width £ 
point-eikonal parameter (two dimensions); energy density 
point-eikonal parameter (three dimensions); modal matrix of operators; 
dispersion curvature matrix (3 X 3) 
generating point-eikonal parameter (two dimensions) 
generating point-eikonal parameter (three dimensions) 
transverse coordinate 
defines a point in space with components Χ , / , Ζ Ο Γ Χ , , X2, X3 
period vectors 
= x2 
defines a point in space-time (x, ict) 
transverse coordinate 
admittance; also Y = y2 
axial coordinate 
normalized axial coordinate 
Fabry-Perot response parameter; ray period 
Greek Letters 
Notations and Definitions xix 
Δη;characteristic parameter of resonators; 
ky, ω, *,y) 
a: angle of a ray to the axis; loss coefficient; integer for summation; radial 
wavenumber 
a: generating vector with components <χλ, a2 
β: axial propagation constant (alternative notation, kz)\ integer for summa­
tion 
integer for summation 
Dirac symbolic function 
prism angle; also = (1 — 
1 if i = j ; 0 otherwise 
small variation, e.g., kkx 
aberration; Laplacian operator; = δ2/2 « Δ Λ / Λ 
small quantity; medium permittivity 
free-space permittivity = (4ττ X 9 X 109) - 1 MKSA 
= /i2, n refractive index 
permittivity tensor (3 X 3) 
modified axial coordinate; displacement of a surface 
material submatrix (3 X 3); 5-vector in the phase space (kx 
waveguide efficiency 
beam axial phase shift; normalized axial ray angular momentum 
normalized phase shift 
phase shift of axial rays 
beam phase shifts in three dimensions 
power-law parameter; also stands for c or μ 
wavelength; eigenvalue 
eigenvalue 
wavelength in free-space 
wavelength in oblique directions 
medium permeability; azimuthal scalar wave number; complex 
wavefront curvature 
free-space permeability, = 4π X 10 ~7 MKSA 
permeability tensor (3 X 3) 
azimuthal e.m. wavenumber; characteristic angle of helical fibers; rota­
tion angle 
gaussian beam half-width, — \/e point of the beam irradiance 
normalized gaussian beam half-width 
gaussian beam waist half-width 
half-width of matched gaussian beams 
material submatrix (3 X 3); 6-vector in space-time phase space 
= 3.14159 . . . 
wavefront curvature radius; helix curvature radius 
normalized wavefront curvature radius 
ray parameter; rms pulse width 
summation sign 
proper time; relative time of flight of an optical pulse; spatial rate of 
rotation 
normalized relative time of flight 
azimuthal coordinate 
phase shift (e.g., under total reflection) 
Notations and Definitions 
Φν φ2> Φ3'· Pn ase shifts per period 
φ: electromagnetic field 6-vector 
χ: normalized transverse vector with components χ , , χ 2 
ψ: wavefunction « (A:0)1/2 X electric field transverse component 
\l·', Ψ~ ~ : wavefunctions of modes of order m and / W | , AW2, respectively 
Ψ: wavefunction for off-set beams 
ω: angular frequency, = 2π X frequency 
wc: cutoff angular frequency 
Ω: focusing strength; solid angle; path twist angle 
Ω0: spatial period of perturbation 
Ω: focusing strength matrix 
3: partial differentiation 
V: gradient operator with rectangular components: d/dx, d/dy> d/dz 
V · : divergence operator 
V x : rotational operator 
Other Symbols 
*: complex conjugation 
~: transposition (omitted on first vectors of products) 
| : adjointness 
·: one upper dot; first derivative with respect to argument 
two upper dots; second derivative with respect to argument 
!: factorial, a ! = 1 X 2 X · · · X a 
• : scalar product, a · b = ab = ab 
X : vector product 
(*): -a\/[(a-b)\b\] 
« : approximately equal 
~ : order of magnitude; asymptotic expansion 
oc: proportional 
Definitions 
Angular frequency: 2m X frequency 
Bianisotropy: dependence of the electric and magnetic inductions (D, H) on both the electric 
and magnetic fields (E, B) 
Canonical momentum: product of the wavevector (k) and an adiabatic invariant (J) 
Dispersion hypersurface (or hypersurface of 4-wave vectors): hypersurface in the kx, ky, kz, ω 
space. Restriction to the kx, k■ , kz space is called the dispersion surface. Restriction to the 
ω, kz space is called the dispersion curve 
Distribution ( / ) : density of rays in phase space (k, x) 
Inertial coordinate system: frame of reference in which a particle not submitted to a 
recognized force (e.g., electrical force) has constant velocity 
Inhomogeneous dispersion: spatial variation of D = v/u = (ω/'kyßk/άω) 
Irradiance: power radiated by an extended source per unit area, in watts/meter2 
Material dispersion: M = (ω 2 /k\d 2 k /dco 2 ) 
Radiance: power radiated by an extended source per unit projected area and unit solid angle 
in some given direction, in watt/(steradian X meter2). For a lambertian source, the 
radiance is independent of direction, denoted B