DBS C
8.2 Ray propagation through linear optical elements
Rays and beams 161 independent of the same along another ray. In order to draw a parallel with the hydrodynamics of incompressible fluid, we multiply Eq. (8.7) byAyielding A2∇ ·(nˆs) + 2A∇A·(nˆs) = 0. (8.19) In writing Eq. (8.19) we have made use of the relation∇2S =∇·∇S=∇·(nˆs).
In compact form Eq. (8.19) reads as
∇ ·(nA2ˆs) =∇ ·J= 0, (8.20) where we introduced the current densityJ=nA2ˆs. Eq. (8.20) represents the well-known continuity equation for a stationary incompressible fluid. In our case J gives the light flux density. It is proportional to the (time)-averaged Poynting vectorhSigiven by [31]
hSi=vhwiˆs, (8.21)
wherev, hwiare velocity and the time-averaged energy density, respectively.
Thus light flows along the narrow light tubes formed by rays.
the axis is small enough so that tanθ∼sinθ∼θ. We now describe a method to evaluate the changes inR(z) for passage through linear optical elements.
Let the initial ray state vector (before it enters the optical element) be given byR1and the ray state after it byR2. Using a 2×2 matrix to characterize the optical element/elements,R1 andR2can be related as
R2=MR1, M =
A B
C D
. (8.24)
Matrix M is known as the ABCD matrix for the particular optical ele- ment/elements. For non-lossy elements, we further have det(M) = AD − BC= 1.ABCD matrix elements for several linear optical elements are given in Table 8.1. They are also listed in many other standard textbooks [59, 60].
TABLE 8.1:ABCD matrices for typical optical elements.
Homogeneous medium of lengthd and refractive indexn
1 dn 0 1
Dielectric interface with n1(n2) as the entry (exit) medium refractive index
1 0 0 nn12
Spherical mirror of radiusR
1 0
−R2 1
Spherical dielectric interface of ra- diusR(R > 0) withn1(n2) as the entry (exit) medium refractive in- dex
1 0
−n2n−1Rn1 n1
n2
Thin lens of focal lengthf(f >0)
1 0
−f1 1
Rays and beams 163
1 2 3 n-1 n
FIGURE 8.3: Ray propagation through a sequence of optical elements, each represented by circles.
8.2.1 Sequence of optical elements
Consider the sequence of optical elements arranged one after the other as shown in Fig. 8.3. TheABCD matrix for this system can be evaluated as follows. The relation between the ray states before and afternoptical elements is given by
Rn+1 =MnRn =MnMn−1Rn−1=MtotalR0, (8.25) whereMtotal is theABCD matrix for the entire system
Mtotal=MnMn−1· · ·M2M1. (8.26) Proper attention is to be paid to the order in which the product of matrices is taken in Eq. (8.26), as the matrix product is not commutative in general.
Note also that det(Mtotal) = 1 since the product of unimodular matrices is again unimodular.
8.2.2 Propagation in a periodic system: An eigenvalue problem
Consider a periodic system of linear optical elements, each represented by its own ABCD matrix. We pose a general eigenvalue problem to find the stability of the ray propagation. Ray propagation is understood to be stable ifris finite for anyz along the axis. On the contrary, divergingrimplies an unstable system where the ray is no longer confined near the axis. Let the ABCD matrix for one period be denoted by M. We show below that the eigenvalues ofM determine the character of ray propagation and whether the ray is confined near thez-axis. The eigenvalue problem can be stated as [59]
MR=λR, (8.27)
where the eigenvalueλsatisfies the algebraic equation
λ2−2mλ+AD−BC= 0, (8.28)
with m= A+D2 . As our system is loss-free, we have AD−BC = 1. Solving forλyields
λ± =m±p
m2−1. (8.29)
LetR±represent the eigenvectors corresponding toλ±, which are orthogonal (for distinct roots). Any rayR0 at the input andRnafter passage throughn
periods can be written as the linear combination ofR±:
R0=c+R++c−R−, (8.30)
Rn =c+λn+R++c−λn−R−. (8.31) Eq. (8.31) is obtained by then-fold application ofM on Eq. (8.30). It is now clear why the evolution of the ray will depend on the nature of λ± (as λn± signifies converging or diverging solutions asn→ ∞). Based on the value of m, two cases are possible, namely,m2≥1, andm2 <1 (see Eq. (8.29)). We now consider these two cases separately and draw necessary conclusions.
• Case (a)
m2≤1, or −1≤
A+D 2
≤1. (8.32)
We replacemby cosθsince|m| ≤1. Eigenvaluesλ± now take the form λ± = cosθ±ip
1−cos2θ=e±iθ. (8.33) The eigenvectorRn in Eq. (8.31) can be written as
Rn=c+einθR++c−e−inθR−, (8.34a)
=c+(cosnθ+isinnθ)R++c−(cosnθ−isinnθ)R−, (8.34b)
=a0cosnθ+b0sinnθ, (8.34c)
witha0=c+R++c−R−andb0=i(c+R+−c−R−). It can be seen from Eq. (8.34c) that the ray state oscillates about the axis and, as mentioned earlier, this kind of system is called a geometrically stable system.
• Case (b)
|m|>1,
A+D 2
>1. (8.35)
As before we parametrizemby coshθ(θ6= 0). Eq. (8.29) can be rewrit- ten as
λ±= coshθ±p
cosh2θ−1 =e±θ. (8.36) Let the eigenvectors corresponding toλ± be ˜R±. The ray state aftern periods ˜Rn, in terms of the new basis vectors, can be written as
R˜n=c+enθR˜++c−e−nθR˜−, (8.37a)
=c+(coshnθ+ sinhnθ) ˜R++c−(coshnθ−sinhnθ) ˜R−, (8.37b)
= ˜a0coshnθ+ ˜b0sinhnθ, (8.37c)
with ˜a0=c+R˜++c−R˜−and ˜b0= (c+R˜+−c−R˜−). The ray state after traversingn periods grows exponentially as given by Eq. (8.37c). As a consequence such systems are referred to as unstable systems.
Rays and beams 165 (a)
(b)
d
d d z
R2 R1
f2 f1
FIGURE 8.4: Schematics of (a) a spherical mirror cavity and (b) its equiv- alent lens waveguide.
We now present an example to understand the stability of rays in a res- onator. Consider the system shown in Fig. 8.4(a) consisting of two mirrors (having radii R1 and R2) separated by a distanced. This system is periodic since the rays in the cavity can retrace their path, being bounced by the mir- rors repeatedly. We evaluate the conditions under which the ray propagation is stable. Noting the equivalence of a thin lens and a spherical mirror, the system can be modeled by a periodic arrangement of lenses (also known as a lens waveguide) as shown in Fig. 8.4(b) with the focal lengthsf1=R1/2 and f2=R2/2. The ABCDmatrix for one period (see Fig. 8.4(b)) is given by
M =
1 d 0 1
1 0
−R22 1
1 d 0 1
1 0
−R21 1
, (8.38)
=
1−R2d1 1−R2d2
−R2d1 d 2−R2d2
−R22
1−R2d1
−R21 1−R2d2
. (8.39) In writing Eq. (8.39) we have made use of Eq. (8.26) and Table (8.1). For the half-trace of theABCD matrix we have
A+D
2 = 2
1− d
R1 1− d
R2
−1, (8.40)
and using Eq. (8.32) we find that the resonator will be stable if 0≤
1− d
R1 1− d
R2
≤1 or (8.41)
0≤g1g2≤1, (8.42)
whereg1,2= (1−d/R1,2).