Download ABCD matrices as similarity transformations of

S. Başkal and Y. S. Kim
Vol. 26, No. 9 / September 2009 / J. Opt. Soc. Am. A
2049
ABCD matrices as similarity transformations of
Wigner matrices and periodic systems in optics
S. Başkal1,* and Y. S. Kim2
1
Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
Department of Physics, University of Maryland, College Park, Maryland 20742, USA
*Corresponding author: [email protected]
2
Received April 14, 2009; accepted July 15, 2009;
posted July 30, 2009 (Doc. ID 110088); published August 24, 2009
It is shown that every ray transfer matrix, often called the ABCD matrix, can be written as a similarity transformation of one of the Wigner matrices that dictate the internal space–time symmetries of relativistic particles, while the transformation matrix is a rotation preceded by a squeeze. The implementation of this mathematical procedure is described, and how it facilitates the calculations for scattering processes in periodic
systems is explained. Multilayer optics and resonators such as laser cavities are discussed in detail. For both
cases, the one-cycle transfer matrix is written as a similarity transformation of one of the Wigner matrices,
rendering the computation of the ABCD matrix for an arbitrary number of cycles tractable. © 2009 Optical
Society of America
OCIS codes: 080.2730, 140.4780, 240.3695.
1. INTRODUCTION
The ABCD matrices have proved to be very useful in depicting quite a variety of optical phenomena such as ray
tracing in geometrical optics, propagation of paraxial
waves with Gaussian apertures in wave optics, in resonators such as lasers [1], or even in dealing with two-port
networks (2PN) in the telephone industry [2].
Perhaps the most prominent advantage of using matrices is to calculate the overall ABCD matrix of the system
composed of different cascaded optical or electrical elements just by matrix multiplications.
If the system has some conserved quantities or stays in
a stable state, for instance when the refractive indices at
the input and output planes are the same, or when the laser cavity is stable [1], then such conditions render the
determinant of the ABCD matrix as one. In the case of
2PN, the transfer function remains unaltered if the points
of excitation and response are interchanged [3].
Apart from those that are purely real as in ray optics,
in general the elements of the ABCD matrices are complex, although there are systems that can particularly be
arranged to yield real matrices, or the constituent complex matrices can be made real by a similarity transformation. The two-by-two complex matrices with unit determinant form the group SL共2 , C兲, which is the covering
group of SO共3 , 1兲, whose four-by-four matrix representations correspond to Lorentz transformations.
It is well established by now that SL共2 , C兲, or its subgroups such as SU共1 , 1兲 and Sp共2兲, provides the underlying mathematics of classical and quantum optics. Specifically, they play a pivotal role in polarization optics [4],
interferometers [5,6], lens optics [7,8], multilayer optics
[9], and laser cavities [10], as well as squeezed states of
light [11,12].
On the basis of the vast amount of literature accumulated on the subject, it is easy to observe now that a com1084-7529/09/092049-6/$15.00
mon mathematical formulation can be established between the physics of concrete setups composed of lenses or
lossless multilayers whose system matrices belong to
SL共2 , C兲 or its subgroups that are isomorphic to SO共2 , 1兲
and special relativity, despite the fact that there does not
seem to be an apparent relation between those distinct
subjects at first glance [13,14].
Successful treatment of stratified media in the context
of periodic systems can be traced back to the work of
Abelès [15], and since then various matrix or group theoretical methods have been exploited by many authors
[16–20]. Mathematical induction can also be a conceivable
approach to a periodic system, where it is possible to assume first that the ABCD matrix is known for n cycles,
and then compute the system for n + 1 cycles.
Recently, one of us has studied multilayer optics based
on exploitation of the properties of the Lorentz group,
where the cycle had to start from the midpoint of one of
the layers [9]. Earlier again, we had to deal with a similar
inconvenience of starting the beam cycle from the midpoint between the two mirrors while calculating the
ABCD matrix for laser cavities using the method of Wigner’s little group [10]. In this paper, all these restrictions
and inconveniences are eliminated by starting the cycles
from arbitrary points. For this purpose, we shall show
that the real ABCD matrix can be cast into one of the
Wigner matrices by a similarity transformation, and that
the similarity transformation is a rotation followed by a
squeeze. This mathematical result proves to be very useful in calculating the overall matrix for periodic systems
where raising the ABCD matrix of one cycle to its nth
power is necessary. We shall study laser cavities and
multilayer optics in this context. In both cases, the multicycle system will be reduced to one cycle.
In Section 2, we construct a similarity transformation
that will bring the ABCD matrix into the form of one of
© 2009 Optical Society of America
2050
J. Opt. Soc. Am. A / Vol. 26, No. 9 / September 2009
S. Başkal and Y. S. Kim
the four Wigner matrices. It is shown that the transformation matrix is a rotation matrix followed by a squeeze
matrix. It is also shown that these four different Wigner
matrices can be combined into one analytical expression.
The present formalism is applicable to one-dimensional
periodic systems. In Section 3, we study laser cavities in
detail. In Section 4, we show how useful Wigner matrices
are in computing the scattering matrix for multilayer optics.
2. SIMILARITY TRANSFORMATIONS
The ray transfer matrix, usually known as the ABCD matrix, is a two-by-two matrix
M=
冉 冊
A B
共1兲
C D
whose determinant is AD − BC = 1 for lossless systems.
Since its elements are real, it has three independent parameters.
This unimodular matrix can be decomposed into three
one-parameter matrices as
M = R共␪1兲B共− 2␭兲R共␪2兲,
共2兲
which is known as the Bargmann decomposition of the
Sp共2兲 matrices [21]. A more manageable form can be obtained when this decomposition is rotated with a rotation
matrix R共␦兲 as
MR = R共− ␦兲MR共␦兲,
共3兲
MR = R共− ␦兲R共␪1兲B共− 2␭兲R共␪2兲R共␦兲,
共4兲
two seemingly unrelated branches of physics; namely,
special relativity and optics. This particular identity [8],
R共␪兲B共− 2␭兲R共␪兲 = S共␩兲W共␻兲S共− ␩兲,
imported from little groups is by no means an exception,
where
S共␩兲 =
冉
e␩/2
0
0
e
−␩/2
冊
共10兲
is the two-by-two matrix representation of boosts along
the z direction, and W共␻兲 is collectively called the Wigner
matrices constituting the set
W共␻兲 = 兵R共2␪*兲,B共2␹兲,N±共␥兲其,
with
N− =
冉 冊
1 −␥
0
1
,
N+ =
冉 冊
1 0
␥ 1
.
MR = S共␩兲W共␻兲S共− ␩兲,
共5兲
where
␪=
␪1 + ␪2
2
,
␦=
␪2 − ␪1
2
M = R共␦兲S共␩兲W共␻兲S共− ␩兲R共− ␦兲.
共6兲
M = Z共␦, ␩兲W共␻兲Z−1共␩, ␦兲,
R共␪兲 =
冉
sin ␪/2
cos ␪/2
冊
共7兲
,
and B共␭兲 is the two-by-two matrix representation of
boosts along the x direction,
B共␭兲 =
冉
cosh ␭/2 sinh ␭/2
sinh ␭/2 cosh ␭/2
冊
.
共8兲
Wigner’s little groups are the subgroups of the Poincaré
group, whose transformations leave the four-momentum
of a relativistic particle invariant. The two-by-two representation of this group dictates the internal space–time
symmetries of relativistic particles [13,22]. Various matrix identities from little groups have been exploited
within the context of optics, specially motivated by the desire to find a common mathematical ground between the
共14兲
共15兲
where the transformation matrix is a product of a rotation
and squeeze in the following order:
共16兲
If we are to deal with periodic systems, ultimately we
shall have the burden of taking the nth power of the
ABCD matrix of one complete cycle to obtain the overall
system matrix Mn. Fortunately, we can circumvent this
difficulty by observing that the Wigner matrices have the
following desirable property:
共17兲
Finally, we have
Mn = Z共␦, ␩兲W共n␻兲Z−1共␦, ␩兲.
Here the rotation matrices R共␪兲 or R共␦兲 are of the form
cos ␪/2 − sin ␪/2
共13兲
Therefore, we have
Wn共␻兲 = 兵R共2n␪*兲,B共2n␹兲,N±共n␥兲其.
.
共12兲
and in view of Eq. (3), the ray matrix becomes
Z共␦, ␩兲 = R共␦兲S共␩兲.
MR = R共␪兲B共− 2␭兲R共␪兲,
共11兲
Now, the rotated ABCD matrix can be expressed as
which becomes
or with the addition of the rotation parameters is expressed in the form
共9兲
共18兲
In order to give the relations between the Bargmann
parameters, which are actually related to the physical parameters of the system under consideration, explicit
forms of Eq. (9) are needed. Let us denote the left-hand
side of this identity as DL共␭ , ␪兲 which after matrix multiplications explicitly reads as
DL共␭, ␪兲 =
冉
共cosh ␭兲cos ␪
− 共cosh ␭兲sin ␪ − sinh ␭
共cosh ␭兲sin ␪ − sinh ␭
共cosh ␭兲cos ␪
冊
.
共19兲
Thus in view of Eq. (5), we have
MR = DL共␭, ␪兲.
共20兲
This matrix has two independent parameters, and the diagonal elements are the same. Now, there are three cases
to be distinguished:
S. Başkal and Y. S. Kim
Vol. 26, No. 9 / September 2009 / J. Opt. Soc. Am. A
(i) If the diagonal elements are smaller than one, then
cosh ␭ sin ␪ ⬎ sinh ␭ and the off-diagonal elements have
opposite signs. So we should use R共2␪*兲 as the Wigner matrix; thus the right-hand side of Eq. (9) becomes
冉
cos ␪*
− e␩ sin ␪*
e−␩ sin ␪*
cos ␪*
冊
3. LASER CAVITIES
A laser cavity consists of two concave mirrors separated
by distance s. The mirror matrix takes the form
冉
共21兲
.
The relation between the Bargmann and the little group
parameters are
1
冉 冊
0 1
sin ␪ + tanh ␭
sin ␪ − tanh ␭
共22兲
.
cosh ␹
e␩ sinh ␹
e−␩ sinh ␹
cosh ␹
冊
共23兲
,
冉
1
0
− 2/R 1
e 2␩ =
− sin ␪ + tanh ␭
冉 冊冉
1 d
1 − ␥e␩
0
1
冊 冉
or
1
␥e
−␩
0
1
冊
共24兲
共25兲
with ␥e␩ = 2 sinh ␭, or for the latter with ␥e−␩ = 2 sinh ␭.
The system matrix M can be expressed in terms of the
little group matrix of Eq. (19) and R共␦兲 as
M = R共␦兲DL共␭, ␪兲R共− ␦兲,
1
0 1
0
1
0
− 2/R 1
冊冉 冊冉
1 s
0 1
共26兲
冉 冊冉
1 d
0 1
冊冉 冊
1 s
0 1
1
0
− 2/R 1
冊冉
A = cos ␪ cosh ␭ + sin ␦ sinh ␭,
L1 =
0
共27兲
To end this section we emphasize that, although the
procedure presented above is established within the
framework of real unimodular ABCD matrices of ray optics, the results are general and can also be applied to different areas of physics whose system matrices can be converted to those of the form of ray optics.
0
− 2/R 1
冊冉
1 s−d
0
1
冊
冉
1 − 2d/R 共1 − 2d/R兲共s − d兲 + d
− 2/R
1 − 2共s − d兲/R
冊
. 共31兲
共32兲
,
冊
共33兲
,
共34兲
L2 = KL1K−1 ,
where
冉 冑 冑冊
1/ s
K=
L2 =
B = − sin ␪ cosh ␭ − cos ␦ sinh ␭,
1
1
keeping in mind that one complete cycle consists of two
repeated applications of the half-cycle (see Fig. 1), which
is L12.
However, the off-diagonal elements of L1 have different
dimensions, while the diagonal elements are dimensionless. In order to deal with this problem, we write this expression as a similarity transformation
and so
D = cos ␪ cosh ␭ − sin ␦ sinh ␭,
共30兲
.
1 s−d
and matrix multiplication yields
and the relations between the elements of this matrix and
the Bargmann parameters are found as
C = sin ␪ cosh ␭ − cos ␦ sinh ␭.
1 s
Thus, half a cycle can be written as
(iii) Either of the off-diagonal elements passes through
zero while going from a positive number to a negative
number, meanwhile cosh ␭ = 1; then either of those elements vanishes, so one of N±共␥*兲 suits as the Wigner matrix. Thus the right-hand side of the identity becomes
冉
冊冉 冊冉
− 2/R 1
L1 =
.
共29兲
.
If we start the cycle at a position d from the mirror, then
one complete cycle becomes
cosh ␹ = cosh ␭ cos ␪ ,
sin ␪ + tanh ␭
共28兲
,
If we start the cycle from one of the two mirrors one complete cycle consists of
0 1
with the relation between the parameters as
冊
where R is the radius of the concave mirror. The separation matrix is
1 s
(ii) If the diagonal elements are greater than one, then
cosh ␭ sin ␪ ⬍ sinh ␭ and the off-diagonal elements have
the same sign. So we should use B共2␹兲 as the Wigner matrix; thus the right-hand side of Eq. (9) is of the form
冉
0
− 2/R 1
cos ␪* = cosh ␭ cos ␪ ,
e 2␩ =
2051
冉
0
0
s
共35兲
,
1 − 2d/R 共1 − 2d/R兲共s − d兲/s + d/s
− 2s/R
1 − 2共s − d兲/R
冊
.
共36兲
The elements of this matrix are now dimensionless. At
this point we choose a unit system where all distances are
measured in terms of the mirror separation with s = 1 and
use notations a = d / s and b = 2 / R. Then the normalized
matrix for a half-cycle can be written as
L=
冉
1 − ab 1 − ab共1 − a兲
−b
1 − b共1 − a兲
We can now rotate L with R共␦兲, where
冊
.
共37兲
2052
J. Opt. Soc. Am. A / Vol. 26, No. 9 / September 2009
S. Başkal and Y. S. Kim
cal parameters a and b of the cavity, and also with
W共w兲 = R共2␪*兲, then half a cycle L starting from an arbitrary plane in the cavity can be written as in Eq. (14) or
as in Eq. (15). Then, for n cycles the overall matrix becomes L2n and explicitly reads as
L2n = Z共␦, ␩兲R共4n␪*兲Z−1共␦, ␩兲.
共43兲
When the cycle starts from the midpoint in the cavity,
a = 1 / 2 and so the half-cycle matrix becomes
L=
冉
1 − b/2 1 − b/4
−b
1 − b/2
冊
共44兲
,
and the angle ␦ becomes zero. We do not need the rotation
matrices that have provided the arbitrariness of the starting plane for the beam in the cavity as in the case we have
illustrated above. The relations between the little group
parameters and the physical parameters of the cavity are
calculated as
b
cos ␪* = 1 − ,
2
Fig. 1. Optical rays in a laser cavity. (a) Multiple cycles in a laser cavity are equivalent to the beam going through multiple
lenses, for which one cavity cycle corresponds to the propagation
of light through a subsystem of two lenses. (b) A laser cavity consisting of two concave mirrors with separation s. In our earlier
paper [10], the cycle in the cavity had to start from 共s / 2兲, the
midpoint between the two mirrors. Now, the cycle can start anywhere, 共s − d兲, including the first mirror.
tan ␦ =
b − 2ab
1 − b − ab共1 − a兲
共38兲
,
LR =
冢
h
1
2
2
共f + g兲
共− f + g兲
h
冣
L=
−b 1−b
g = 兵共b − 2ab兲2 + 关1 − b共1 + a − a2兲兴2其1/2 .
b
1−b
2
,
共1 + b兲 − 关b2 + 共1 − b兲2兴1/2
.
共48兲
4. MULTILAYER OPTICS
共41兲
In multilayer optics, we have to consider two beams moving in opposite directions, one of which is the incident
beam and the other is the reflected beam [23]. We can represent them as a two-component column matrix
冉 冊
E−e−ikz
.
b
共1 + b兲 + 关b2 + 共1 − b兲2兴1/2
E+eikz
f−g
共47兲
.
共40兲
and
e 2␩ =
共46兲
.
As before we can proceed with Eq. (43) to evaluate the
overall matrix of the laser cavity with n cycles.
Now comparing Eq. (39) and Eq. (21) we have
f+g
冊
冉 冊
cos ␪* = 1 −
f = 共1 + b兲 − ab共1 − a兲,
2
1
共39兲
,
b
1
tan ␦ =
e 2␩ =
cos ␪* = 1 −
冉
Now, the relation between the parameters are
,
b
2
共45兲
.
In this case we need the rotation matrix R共␦兲 with
where
h=1−
4b
This is the result we obtained in our previous paper on
laser cavities [10]. The point of this paper is that we can
start the cycle from an arbitrary plane by introducing the
parameter ␦.
When the cycle begins from one of the lenses, a = 0, then
the matrix for the half-cycle becomes
and the rotated matrix LR = R共−␦兲L共a , b兲R共␦兲 becomes
1
4−b
e 2␩ =
共42兲
Thus with the expressions of the little group parameters
␪* , ␩ and the rotation angle ␦ given in terms of the physi-
,
共49兲
where the upper and lower components correspond to the
incoming and reflected beams, respectively. For a given
frequency, the wavenumber depends on the index of refraction. Thus, if the beam travels along the distance d,
S. Başkal and Y. S. Kim
Vol. 26, No. 9 / September 2009 / J. Opt. Soc. Am. A
2053
the column matrix should be multiplied by the two-by-two
matrix [23]
P共␣j兲 =
冉
ei␣j/2
0
0
e
−i␣j/2
冊
共50兲
,
where ␣j / 2 = kjd with j denoting each different medium. If
the beam propagates along the first medium and meets
the boundary at the second medium, it will be partially
reflected and partially transmitted. The boundary matrix
is [23]
B共␮兲 =
冉
cosh共␮/2兲 sinh共␮/2兲
sinh共␮/2兲 cosh共␮/2兲
冊
,
共51兲
with
Fig. 2. Multilayer consisting of two different refractive indices.
In an earlier paper by Georgieva and Kim [9], the cycle started
from the midpoint of the second medium. Now, it can start anywhere including the boundary between the media.
5. CONCLUDING REMARKS
cosh共␮/2兲 = 1/t12,
sinh共␮/2兲 = r12/t12 ,
共52兲
where t12 and r12 are the transmission and reflection co2
2
+ t12
兲 = 1. The
efficients, respectively, and they satisfy 共r12
boundary matrix for the second-to-first medium is the inverse of the above matrix. Therefore, one complete cycle,
starting from the second medium, consists of
We have shown that the ray transfer matrix can be cast
into one of the four one-parameter Wigner matrices
through a similarity transformation, and the similarity
transformation is a rotation matrix followed by a squeeze
matrix. The logarithmic property of these one-parameter
Wigner matrices is transmitted to its similarity transformation through
M = B共␮兲P共␣1兲B共− ␮兲P共␣2兲.
共SWS−1兲n = SWnS−1 ,
共53兲
This complex valued matrix M can be cast into a real matrix by a similarity transformation as
共54兲
M2 = CMC−1 ,
where
1
C=
冑2
冉
ei␲/4
ei␲/4
− e−i␲/4 e−i␲/4
冊
.
共55兲
This transforms the boundary matrix B共␮兲 of Eq. (51) to a
squeeze matrix S共␮兲 of Eq. (10) and the phase shift matrices P共␣j兲 of Eq. (50) to rotation matrices R共␣j兲 of Eq. (7).
Therefore, we have
M2 = S共␮兲R共␣1兲S共− ␮兲R共␣2兲.
共56兲
In view of the Wigner little group identity of Eq. (9) we
rewrite this as
M1 = R共␪1兲B共2␭兲R共␪1兲R共␣2兲,
共57兲
where the relations between the parameters are found to
be
cosh ␭ = cosh ␮关1 − cos2共␣1/2兲tanh2 ␮兴1/2 ,
共58兲
共62兲
and facilitates the calculations for repeated applications
by multiplying the parameter by an integer. We have carried out the procedure for laser cavities and multilayer
systems. It can also find applications in a variety of other
periodic systems, such as one-dimensional scattering
problems in quantum mechanics [24,25], especially in
condensed-matter physics. We can expand our scope to
look into applications in space–time symmetries of elementary particles in view of the fact that the Wigner matrices used in this paper are from Wigner’s 1939 paper on
symmetries in the Lorentz-covariant world [22,26].
Although in this paper the polarization of light waves
was not taken into account, they have transverse electric
and magnetic components, and there are many interesting aspects when the propagation media are not invariant
under rotations around the propagation axis. Here, we
have dealt with the mathematical properties of the rotation and squeeze matrices that are contained in the decomposition of the ray transfer matrix. If these matrices
are applied to the transverse directions, the rotation matrix generates rotations of the two-component polarization vector. The squeeze matrix on the other hand causes
asymmetric dissipations of the amplitude. Of course, the
combination of these two effects leads to interesting results [27].
cos ␪1 = cos共␣1兲/cosh ␮关1 − cos2共␣1/2兲tanh2 ␮兴1/2 . 共59兲
Now the matrix M1 can be simplified to
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共60兲
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共61兲
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