DBS C
8.3 Beam characteristics
Rays and beams 167 Separating the real and imaginary parts and demanding a nontrivial solution, we arrive at the following set of coupled equations:
dq(z)
dz = 1, (8.49)
dA(z)
dz =−A(z)
q(z), (8.50)
whose solutions are
q(z) =q0+z, (8.51)
A(z) A(0) = q0
q(z). (8.52)
In writing Eqs. (8.51) and (8.52), we assumed that atz = 0,A(z) =A0 and q(z) = q0. The complex function q can be related to the relevant physical beam parameters as
1
q(z) = 1
R(z)+i λ
πw2(z), (8.53)
whereR(z),w(z) are the real functions ofz. The physical meaning ofRandw will be transparent after we derive the expression of the fundamental Gaussian beam. We measure z from z = 0 where R → ∞. We show below that this amounts to saying that the beam wavefront is planar atz = 0. Similarly we have also assumed thatw(z) atz = 0 is given byw0. We can introduce the so-called Rayleigh rangezR as
zR=iq0=πw20
λ . (8.54)
With this definition we find thatq(z) can be written as 1
q(z)= 1
z−izR = z+izR
z2+z2R. (8.55)
Now comparing the real and imaginary parts of Eq. (8.53) and Eq. (8.55), we find that
R(z) =z
1 +zR
z 2
, (8.56)
w2(z) =w20
"
1 + z
zR
2#
. (8.57)
Similarly, we also find that A(z) A(0) = q0
q0+z = 1−i(z/zR)
1 + (z/zR)2, (8.58)
which can be rewritten as A(z) A(0) =
s 1
1 + (z/zR)2 eiψ= w0
w(z)eiψ, (8.59) where
ψ=−tan−1 z
zR
(8.60) is known as the Guoy phase. Substituting Eqs. (8.56)–(8.59) in Eq. (8.47), we get
U(x, y, z) =U0
w0
w(z)eiψ
exp
ikx2+y2 2R(z)
exp
−x2+y2 w2(z)
, (8.61) whereU0is the normalization factor. This is known as the fundamental mode and labeled by TEM00. Eqs. (8.56) and (8.57) in Eq. (8.61) provide a clear physical meaning to R(z) and w(z) as the radius of curvature of the phase front and the beam spot sizew(z) atz, withw(0) =w0being the beam waist.
In fact, the knowledge of the complex beam parameterq(z) at anyzmakes it possible to recognize all the physical parameters (R, w, ψ) of the fundamental Gaussian beam. As mentioned earlier there are infinite solutions to paraxial equations. When solved in Cartesian coordinates, any general TEMmnis given by [59, 60]
Umn(x, y, z) =
1 π2n+m−1n!m!
1/2
w0
w(z)×
"
Hm
√2x w(z)
! Hn
√2y w(z)
!
exp
ikx2+y2
2R(z) +i(n+m+ 1)ψ(z)
exp
−x2+y2 w2(z)
, (8.62) whereHn,m are the Hermite polynomials. Degeneracy of the modes with the samem+n value is clear from Eq. (8.62). In the cylindrical basis we have the Laguerre-Gaussian modes, which can have vortex character with angular momentum (see Chapter 12).
8.3.2 ABCD matrix formulation for fundamental Gaussian beam
The passage of a beam through optical elements can again be derived using the 2×2 matrix formulation, as with rays. We now propagate the com- plex beam parameter. The input and output beam parameters (q2 and q1, respectively) are related by the following linear fractional transform:
q2= Aq1+B
Cq1+D, (8.63)
Rays and beams 169 and for a sequence of optical elements, we must take the matrix product in reverse order, as in Eq. (8.26). The proof of Eq. (8.63) can be found in Ref. [59];
we do not include it here.
8.3.3 Stability of beam propagation
In the case of rays, stability implied that rays were confined to the axis.
The corresponding eigenvalue problem in Eq. (8.27) had complex eigenvalues assuring oscillations about the axis. In the case of beams, a similar picture holds, albeit with necessary changes. For a stable cavity, the beam (to be precise, the complex beam parameter) has to replicate itself after each period, requiring the following equation to be valid:
q= Aq+B
Cq+D. (8.64)
Eq. (8.64) represents a quadratic equation forq. It is easier to deal with 1/q, whose roots are given by
1
q = D−A 2B ± 1
B s
A+D 2
2
−1. (8.65)
In order to qualify as a Gaussian beam with a finite spot size, 1/q must be complex, requiring
m2=
A+D 2
2
≤1, (8.66)
leading to
1
q =D−A 2B ± i
|B|
p1−m2. (8.67)
We thus arrive at the same stability condition for beams as for rays (compare with Eq. (8.32)). Eq. (8.67) easily leads to the stable beam parametersRand was per Eq. (8.53).
Chapter 9
Optical waves in stratified media
9.1 Characteristics matrix approach . . . 172
9.2 Amplitude reflection, transmission coefficients and dispersion relation . . . 174
9.3 Periodic media with discrete and continuous variation of refractive index . . . 175
9.3.1 Discrete variation of refractive index . . . 176
9.3.2 Continuous variation of refractive index: DFB structures . . . 179
9.4 Quasi-periodic media and self-similarity . . . 182
9.5 Analogy between quantum and optical systems . . . 183
9.5.1 Wigner delay: Fast and slow light . . . 185
9.5.2 Goos-H¨anchen shift . . . 187
9.5.3 Hartman effect . . . 187
9.5.4 Precursor to Hartman effect: Saturation of phase shift in optical barrier tunneling . . . 188
9.5.5 Hartman effect in distributed feedback structures . . . 189
9.6 Optical reflectionless potentials and perfect transmission . . . 190
9.6.1 Construction of the Kay-Moses potential . . . 190
9.6.2 Optical realization of reflectionless potentials . . . 191
9.7 Critical coupling (CC) and coherent perfect absorption (CPA) . 194 9.7.1 Critical coupling . . . 195
9.7.2 Coherent perfect absorption . . . 196
9.8 Nonreciprocity in reflection from stratified media . . . 196
9.8.1 General reciprocity relations for an arbitrary linear stratified medium . . . 197
9.8.2 Nonreciprocity in phases in reflected light . . . 200
9.9 Pulse transmission and reflection from a symmetric and asymmetric Fabry-P´erot cavities . . . 201
9.9.1 Symmetric FP cavity with resonant absorbers . . . 202
9.9.2 Asymmetric FP cavity . . . 203 Often in optics we must deal with layered media when the optical properties change only in one specific direction (say, along thez-axis), while in any plane transverse to this direction, the optical properties are invariant. Such media with dielectric and magnetic responses given by ε(z) and µ(z) as functions of only z are also referred to as stratified media. There are many examples
172 Wave Optics: Basic Concepts and Contemporary Trends
of such media. The most useful one refers to the optical interference coatings that are essential for most optical instruments. Reflection and transmission through these structures can be handled in terms of simple 2×2 matrices when the constituent layers are homogeneous and isotropic. In the case of an anisotropic layered medium, we can develop a 4×4 matrix formulation for uniaxial materials [61, 62]. In this chapter we deal with isotropic homogeneous layers and develop the characteristics matrix formalism to obtain the refection and transmission coefficients (see also Ref. [31]). We discuss how the dispersion in such structures can be engineered to lead to slow and fast light [7]. We apply the technique to investigate the modes of a structure. We probe the effects of finite temporal width of a pulse leading to the Wigner delay [63]. The space equivalent of Wigner delay, also known as the Goos-H¨anchen shift [64], for a spatially finite beam is then discussed. Finally, we show how someone using such structures can realize perfect transmission and coherent perfect absorption. We will define all the necessary notions and concepts as we go along.