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Scattering matrix analysis of optical resonators with cavity imperfections
1 May 1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
137 (1997) 367-381
Full length article
Scattering matrix analysis of optical resonators with cavity imperfections R. Mavaddat Department
of Electrical and Electronic Etzgitleering, fJniwrsi& Received
16 September
1996: revised 17 December
’ of Western Australia, Nedlands. Australia 6907 1996: accepted
19 December
1996
Abstract The scattering matrix analysis is a powerful tool that is used extensively for the analysis of microwave circuits and occasionally for that of optical devices. Here we apply the scattering matrix analysis to a class of optical resonators with sources external to the cavity. Such an analysis is general in the sense that it allows the inclusion of all higher order cavity modes and the possible situations where the reflection from and transmission through the cavity mirrors, for the waves incident from different directions, are different. The assumption of negligible diffraction losses or large extension of the cavity dimensions relative to the width of the cavity modes has to be made. In this paper single cavities and possible imperfections, such as relative source or mirror misalignments. are considered. The case of multiple coupled cavities is an extension of this analysis and will be dealt with in a separate publication.
1. Introduction The aim of the scattering matrix analysis of an imperfect cavity resonator is to establish a matrix relation between the incident and reflected mode components of the field at the cavity input port, to that of the incident and reflected mode components at the cavity output port. The elements of the matrix are derived from the cavity parameters given for the fundamental and higher order cavity modes of the corresponding perfect cavity resonator. It is assumed that the sources are external to the cavity and the diffraction losses are negligible. For further reference we first briefly review the modes of a perfect cavity.
2. The fundamental
and higher cavity modes of a perfect cavity
Optical resonators or cavities may assume many configurations. Here we consider the basic resonator consisting of a light beam incident on two partially reflecting surfaces, separated by vacuum or air. Each reflecting surface is assumed to be a part of a spherical surface with a specific radius of curvature. 2. I. The fundamental
cavity mode
The modes in an ideal optical resonator mode in the form of [l]
are Transverse
Electromagnetic
’ E-mail: [email protected]. 0030.4018/97/$17,00 Copyright PII SOO30-4018(97)00004-7
0 1997 Elsevier Science B.V. All rights reserved.
Modes (TEM) with the fundamental
TEM,,
Fig. I. Schematic representation of an optical resonator cavity.
where k = 27r/A is the wavenumber system. U,(Z) = k~,dll+~
of the free space propagating
wave. ,6o0 is the field at the origin of the co-ordinate
is the beam spot radius or the radial distance from the cavity axis where the magnitude
of the field reduces to 1/e of its magnitude on the cavity axis. The common axis of the two mirrors defines the :-axis of the = )I’,,= Jp and is co-ordinate system with the origin of the co-ordinate system :: = 0 at a position where TV = CL’(O) a mmimum. z0 is the distance from the waist where the spot size is fi times that of the spot size IV,) at the waist. R(Z) = ~[l + (z,,/:)‘] is the radius of the curvature of the wavefront at a distance Z. At the waist R( :) = R(O) = z. At a distance z = zo, R( z> = R( z,) = 2 :*. The phase factor Q~( Z) is given as qoO(z) = tan- ‘( z/;& When z = 0. qoO(2) = 77JO) = 0. We now consider a resonator cavity with configuration as given in Fig. 1. Let L/, and L< be the position of the /rant and end mirrors along the z-axis respectively, with L = LC - L,- as the length of the cavity. R/ and Rc are the radii of the curvature of the front and end mirrors respectively. It is assumed that the end mirror is always to the right of the front mirror and the mirror curvature is taken as posifirle when the centre of the curvature is to the left of the mirror. For the field to match both mirror curvatures, we require R( L,/) = L/[ 1 + ( ;o/L/.)‘]
= R,/. and R( L,) = L?[ I + ( zo/Lc)‘]
= Rc.
With L= Le - Lr.
, or hence z0 is found in terms of the cavity parameters. In terms of the mirrors radii of curvature R,,, and R?, L, and Lc are given as R/ L/,=,-,/(2 As Rp mirror.
r
Rr
0~) L/-+ 0 and for R/=
and L, = 2 f22,.
- :il
- (2:,,/RC)‘.
Lf = + z. as expected. Similar relations hold for the parameters
Table 1
K,(x) ff,( x)
2X
H?(X) HJ x) J%(X) H,(x)
-2+4x? -12x+8x3 12-48~’ + 16~~ 120x - 160x3 +32x5
I
related to the end
R. Macaddat/ Optics Communications 137 I lY971367-381 2.2. Higher order Hermite-Gaussian
modes
For the higher order modes of cavity known as Hermite-Gaussian
Xexp[-jk++j(l+m+
where H, is a Hermite polynomial 1. 2.3. Mode orthogonality
I)Q~(:)],
are give in Table
and normalisation of Hermite-Gaussian
modes we have
+m
/ --x / -x +I
modes
of order n. For a later reference some lower order Hermite polynomials
Applying the known orthogonality +z
369
E,,(x,y,z)El,,,(.~,y.i)* dxd~=2”“‘-‘I!m!~w~~I?,,
+= E,,(.~,?I,~)E,,,,(*,~,~
)* dxdy=O,
/ -cc / -%
We can now define the normalised E,,,(I,_~,:)
I’=1 and m’=m,
nifm.
I’+1 and/or
fields as
= (2’+‘“-‘l!m!*~~~)-“ZE,,(~,?‘,~),
giving
Xexp[-jkz+j(l+m+ With this normalisation += / -z +x / -x
l)n,,(z)].
we have
+x_ E,,,(x,:,~)El,,,~(x,~,~)*
dxdy=
+=_ E,,(x,y.,-)E,,,,i(x,y,,-)*
dxd_v=O,
/ -?j / -x
1,
m’=m,
l’=/and
/‘#/and/or
m’fm.
3.Scattering matrix formulation The expression for the normalised Z,,,(.r,.v, c)&(;) where Zln,( x,y,z)
= (2’+‘“-‘l!m!n)-“’
modes
--& z
@,,(z)=exp[-jX;-+j(l+m+
of a cavity
H/[&G]
1)7700(z)],
resonator
can be written
HM[fi--&]exp[
as the product
-s-jk&$$-]
E,,(
x,
y,z) =
(3.1)
(3.2)
which is a phase term and a function of z only. Z,,(.r,y,z) defines the field variations in the transverse directions and hence the transverse modes of the cavity. c$Jz) defines the longitudinal cavity modes. The actual field inside the cavity is the weighted sum of all modes that are set up within the cavity as given by (3.1), multiplied by the corresponding phase factor $I,,,( z> of (3.2) for each cavity mode. In general the weight a”” of lm mode in a cavity is a complex number and depends on the cavity configuration, mirror reflection coefficients and losses, as well as the light incident on the cavity mirrors.
R. Maraddat / Optics Communicatioru 137 f 1997) 367-381
370
The total field inside the cavity can be written as -L E(.Y.J.,)
-L
= c c u”nz,m( X..v>Z)&,,( z). /ill II,=0
(3.3)
In a similar way we can consider the field incident on the front mirror from the left as the summation of the weighted modes of the cavity at the plane of the front mirror. This can be written as
The reflected field from the plane of the front mirror can also be written as
For the field incident from the right on the front mirror, we replace _I?,/ (x, y,L/ ), iii?, I!?,/ (x, y,.Q ) and ziy by &(X,Y,L/.), z:‘;, a,( x, .v,$1 and Li$ respectively. The subscript 1, therefore, refers to the left of the mirror and the subscript 2 to the right of the mirror. For the end mirror all subscripts ,L’ change to the subscript e. In general the left and right complex weight parameters of a cavity can be related by a transfer scattering matrix C in the form of 00 C00,ll
00 C00.12
COO IO.12 IO.11 COO
coo 01.1 I
C
c IO.21
c 10.22
C’O 00.11
clo 00.12
C'O 10.1 I
C’O 01.12 IO.12 CAP.,,C’O
c &,
C'O C~oO.22 10.21
00.21
GJ I CO' 00.12 GL, G%?
00
CPd.l
00
C’O IO.22 I
GLl
COO 01.21
.
00 COl.I2
coo
00 00.22
. . .
Cue 01.22
c;p.z,
GY.22
CO’ GE c,o:,, I 01.17 c~fl22G,21 CO’ 01.22
-00
U2e
-
-00
%,
*IO
...
a2,
-10
...
a2,
.
QI
a2,
.
.
to1
a2c
.
Im C00.11 /In
C00.21
I”, C00.12
pn
Cl&2
Cl”’ IO.21
10.1 I
Cl”
p 10.12
C'" IO.22
01.11
C'" 01.21
I,,1 C 01.12
.. .
CA;122
..
.
where ?i$ and &fS are, respectively, the weights of lm mode for the incident and reflected fields from the left-hand of the front cavity. Similarly 2:: and Zie are, respectively, the weights of lrn mode for the incident and reflected fields from the righthand side of the end cavity. In matrix notations
The elements of the cavity transfer scattering matrix C can be calculated from the cavity parameters, as we shall see in the following examples. The above matrix representation is also valid if the cavity elements are not reciprocal.
4. The frequency response of a cavity with an ideal geometry Assuming that the cavity mirrors are partially reflecting, we define the amplitude reflection coefficient ~611’as the ratio of the reflected wave amplitude for the component of the field that is matched to I’m’ mode, to that of the incident wave amplitude for the component of the wave that is matched to the Zm mode of the cavity. Similarly we define the amplitude transmission coefficient f:?’ as the ratio of the transmitted wave amplitude for the component of the field that is matched to I’m mode. to that of the incident wave amplitude, for the component of the wave that is matched to the Im mode.
R. Maoaddat/
Optics Communications
137 (1997) 367-381
371
The transfer scattering matrix for a single mirror with no tilt or mismatch and with the assumption that the modes are not coupled at the mirror surface can be written as Moooql I
M&,
0
0
0
0
...
0
0
M%,
M,&
0
0
0
0
...
0
0
0
0
M::,I M&,
0
0
...
0
0
0
0
...
0
0
.. ...
0
0
0
0
w%,
M,,&
0
0
0
0
M&
0
0
0
0
M,q’.*,M;,&
0
0
I
M;&
M=
.
.
... . 0
0
0
0
0
0
0
0
0
0
0
0
The elements of M in terms of the mirror amplitude reflection
M::,,, = l/t/,“,
. .
4’m”. II 4!L . . 4% I M:mm.22 . ... . ...
and transmission
coefficients
M$',,= -r~~/t~~, M&'2,= r,!,"/t:,", M:l,22 = - [(d$-
are
(t.$/tk
where r:,” and t:,” are respectively, the amplitude reflection coefficient and transmission coefficient of the mirror for the Im mode. Specifically we denote the transfer scattering matrix of the front mirror by the matrix F and we have a Ip = Fa,p and similarly ale = Ea,, where E is the transfer scattering matrix of the end mirror. The weight amplitudes at the left and right planes of the space region between transfer scattering matrix that can be denoted as P and given as
p,g
0
0
0
0
0
. .
0
...
0
pg
0
0
0
0
‘.
0
...
0
0
,I0 PI0
0
0
0
‘.
0
.
0
0
0
PIi
0
0
.‘.
0
...
0
0
0
0
pbq’
0
. ‘.
0
.
0
0
0
0
0
PoOf
. . . . . .
0 .
.
P2
P=
.
0
0
0
0
0
0.
0
0
0
0
0
.
.
0. . . .
or
aI, = Pa,, , where p::=
with
exp( -jS,,)
and p;fz = exp( +jS,,),
0
. . .
.
.
0 p:,m
... ... ... ... ... .. .. ...
the two mirrors are also related by a
377
R. Mat>addat/ Optics Communicatiom 137 (19971367-381
A cavity consisting of a front mirror and an end mirror and a space in between has its total transfer scattering matrix C given by C = FPE. Hence we can relate the mode amplitude weights of the field incident and reflected on the first mirror from left to that of the mode amplitude weights of the field incident and reflected on the end mirror from right in the form of a ,/ = Ca,,
.
where I,,, F/“’ c irt,.il ““‘E:;,, , + F/z,,, p;,::E:;,%,, = I,,, I I P,,,,
p
InI /,ri.I2 = F:,::‘.I I d::‘4?, I2 + F/n,,I2P::: E::l:,x .
p
I,,,.2I = F,% I d::‘4t’. , I + 4%~ P:%‘:,:.~I 3 pi’ _ F/“l ““’ E ;,::,,? + F:;‘z2 P;;;:E;;;.~~. /m,2?- I,,,.?IP/w, Assuming that the wave is incident on the cavity from the front mirror and no wave is incident from the end mirror. we have -/n1 _ CK? I a+,,,* aI (.y” 1 ’ /,,,.I I
(4.1)
(4.2) where
(4.3)
(4.4) We now assume that the light source field intensity and phase distribution at the input mirror (front mirror) is known and be denoted by E,/(.x,_~,$). Then the matrix element 2ii; can be found from (3.4) and the fact that the cavity modes are orthogonal by the expression
Hence we can find the reflected field from the front mirror and the transmitted (4.4) for every mode. 4.1. Matched
Gaussian
field from the end mirror as given by (4.1) to
beam
For the case of the input beam given as E,/x,y,L,) = e,,(.r,.v,Lf)&,,(LF) = exp[-jL$ and matched at the front mirror to the fundamental mode of the cavity, we can write eIp( x.v,$)
+jq,J~~)l
e,/(x,y,$)
= AZ,,( x,Y,L/,),
and hence ~~.=I+rI-rE,,(s,~,L,)*Ae,,,(.~,~,~~)dsd~=A. --5 --* Substituting
into (4.1) and (4.2) we find tjp’r,
1 - r/,py
A
(4.5)
and Z& =
5. Pt< 1 - r/ p2rr
A,
where p = pii = exp( -jS,,).
(4.6) with 6,” = kL + q,J L, ) - qoO( Lc).
373
R. Macaddat/ Optics Communications 137 (1997) 367-381
lr=
0.20
‘:, \
= 0.45
\
r= 0.70 r= 0.95
> 0
-0.2
0.2
Fig. I?. p;,/F,,
0.6
0.4
versus 6,
(X
P)
0.8
1
1.2
with r, = r1 = r as a parameter.
In the above equations tf, t<, r/. and rz are equal, resp:ctively, to ti$, t$G, rO& and ri&. We have plotted the ratio of the transmitted power P2? = (E,J&‘/~&~?/~~ to that of the incident power p,/= (E,//_L.,)‘/‘Z,~Z;~ as a function of 6,, in Fig. 2, with r as a parameter and the assumption of rG = rr= r. If the spacing is changed over a few wavelengths, the variation of Q,&L,) - q&L,) with the cavity spacing is negligible. For this reason the given curves can also be considered as the normalised powers, plotted as a function of the fraction of a half-wavelength changes in the cavity length. A 7~ radian change in 6,, corresponding to a A/2 change in the cavity spacing. With n is an integer and provided that nh/2 <( L, the transverse field configuration is almost identical if the length of the cavity is changed by nh/2. These different longitudinal solutions with identical transverse fields, constitute what is known as the longitudinal cavity modes. 4.2. Tilted Gaussian beam Here we assume that a Gaussian beam. initially matched to the fundamental mode of the cavity, is incident on the front mirror of the resonator. The axis of the beam is then tilted by O.r?,*radians in the x--z plane about the position of the front mirror on the :-axis. We wish to determine the fields transmitted through and reflected from the cavity. Here with the assumption of a small tilt, we include in the analysis only TEM,, and TEM,, modes and neglect all higher order modes. In this case the incident field can be written as E,/(x,y,L/)=exp[-jkL~-+j?7,,(4)]e,/,(x,v,L~). while for small H,, we can write e,/(x,y,+)
=AZ,(x._v,L,,)exp(
-jkO,,x)
=A&&,y,Lp)[l
-jkO,,
x-
ik20.r: x2].
The total power incident on the front mirror of the cavity can be found by integration over the area of the mirror in the form of
indicating that within our approximation, of the mirror tilt.
the incident power has remained unchanged from its value prior to the introduction
R. Mavaddar/
37-k
Opiics Communications
137 (1997) 367-381
The incident field e,, (x, y,L/) can then be expanded in terms of the fundamental
modes of the cavity as
+r L% -00 _ ~,~“(~,.v,L~)*e,/,(l,y,l,)dxdv (II,/ - / -7 / --%
+Ic
+%
x
/ --x
’
=
/
2exp[-$$-]Aexp[--$$][i-jkO_Y4~~-’k’O~$~’]d~dv
7rw(L,)
=A(1 -+[fkw(L/)e,,$]
=A(1
-f@/)
and -10 = U,/‘.
+r.
+x ~,o(“,y,L~)*e,/,(x,~,l/-)dxdy
f --x / -z
= ~+X~+X~,,(~,y,~~)* -Cc --LL =
+=
-jkO,,x-~k%$x’]
AZ,,(x,y,L/.)[l
f-r
4x ?rW(L/)
/ --P / -=
=A{ -jikw(+)Oxg}
3 Aexp[ - 2~z~~~~‘]
[l -jkOrBx-$k20;kx2]
d.rdy
dxd?;
= -jA$,,-,
now verify that the sum of the input power of the individual modes is the same as the total where B, = $kw($.)O,,.We input power we have just calculated by forming
=A2(~0/p0)“28,‘,.
(ea//_~~)“%;J:~;* With modes orthogonal, p;/=
the total power is the sum of above terms given as
( E&,,)“~(
q;,q;:
) ‘I A’( E&,-,)“~
+ z$i;;*
which, within our approximation, is the same as found by direct integration. The weights of the transmitted and reflected modes are hence
for the fundamental
TEM,,
for the TEM,, mode. In the above equations &=kL.+n00($-~00(4)
mode, and
poo = pii = exp(-j6,) and
and plo =pig = exp(-j6,,)
where
610=kL+7710(L~)-77~~(Lc),
with noo(z) = tan-‘(z/z,) and ~~~(2) = 2tan-‘(z/z,). I;addition, y= rti,,/ = rt&/= r$2f= r{&, is the amplitude reflection coefficient of the front mirror and 5 = rg,‘< t&f= t&/= = rlo.lC = roozr = r10.7r is the amplitude reflection coefficient of the end mirror. Similarly tr= t$lr= are the front mirror and the end mirror amplitude transmission coeffic’ents, and ry”L $$,~~ tl&, = too.?< O” = tlo.z< lo G,2/ respectively.
R. Maraddat/ Optics Communications 137 (19971367-381
Fig. 3. F2,/<,. The incident power was given by ?,/= expression
and the transmitted
versus 6,
A’(E,,/~~)‘/‘.
(X
T)
with 8,,, as a parameter.
The reflected power from the first mirror can be found from the
power from the end mirror from the expression
p’,, =A2@;*g;
+~~*~:“,)(E&,)“~,
where we have used the fact that the modes are orthogonal. In Fig. 3 we have compared the variations of p’,,/?,, as a function of 6,, for different values of e,,, as calculated by the above approximate analysis, and by the numerical method described in the following paragraph. We have assumed the cavity to be lossless with r/.= 0.95, r< = 0.97 and tanP’(R,Jz,,) - tan-‘(R
=A~,,(L/)~,(r,L*+AL).
instead of E,p(r,L/)=A~,,(L/)e,(r,Lp),
where r’=x’+.v’
and
R. Maraddut/
376
In the above expression Hence with
a
--
a;
for Z,,,(r,c),
I 1 I
137 (1997) 367-381
the most sensitive term to the variations in ,-, is the term containing
the variable k.
aR(z)
+(;)=-I_=
R(L)
Optics Communicntior~s
R(z)’
a,-
we have
or
and hence ,E,/(‘,L/)
=
&,(LF)e,/.(r,L~.)
=A~,,,,(L/)e,,(r.L~.)[l
-jkft(L,)r’AL-ik’5(L~‘)‘r4AL’]
The total power incident on the front mirror of the cavity can be found by the following mirror as
=
AZ,(r,LF~)* wPo)“2~~-~~o
integration
over the area of the
AV,,(r,L~)rdBdr=A’(q,/p0)“2,
indicating that within our approximation, the incident power has remained unchanged. Now we need to find the elements of the weight matrix. From the fact that the modes are orthogonal
I
2r’
2X2i7
= / r=O 7rw(L,)z
Aexp
[
-
WZ(L/)
1
[l -jkf[(Q)r’AL-
we can write
~k2~(Lf)‘r4AL’]rdr
where
~~L=+~~(L/)w(L/)~AL=
-
1 - ( zo/L,/)’ 1 + (i,,/LJ
As we have seen, the sum of the two modes TEM,,
AL co
and TEM ,,? produces an axially symmetric
field. With the incoming
R. Macaddat/Optics
light axially symmetric, we assume an equal contribution cylindrical coordinate system we can write
=
lx
~zv[Z20(r,L~)
371
Communications 137 (1997) 367-381
from these modes to the input field of the cavity. Hence in a
+Z,,(r,L/)]*
AZ,,(r,L/,)[l
$1
Aexp[ ----$$I
-j+kg(L,)r’AL]rd0dr
r=O H=O
=
z
2a
/ I= 0 2TW( L//)
? [ -4+
kY(L/ )’ p+ 4
[l -jik<(L,,)r’AL]rdr
2;;~;)S]A+jk2~(~;)S[4+-8+]AAL
It is of interest again to verify that the input power has remained unchanged.
We can verify this by forming
and
With modes orthogonal,
the total power is the sum of above terms given as
Having found the incident field on the front mirror of the cavity, the transmitted following expressions
In the above S,, = a,2 = In Fig. 4 lossless and
poo = pii = exp( - jS,) and p20 Tjg,,iLl) with qoO(z) = as a function of we have plotted P,?/P,/ also r/ = 0.95, r, = 0.97 and tan-‘(R//q,) equations
-jkL
+ jqzo(L,)
and the reflected fields can be found by the
= pii = exp( - jS,,>, where So0 = - jkL + jrloo(LC) -jqoO( 4) and tan-‘(z/z,) and Q~(z) = T~_o_(:)= 3tan-‘(c/z,). S,, for different values of AL. We have assumed the cavity to be - tan-‘(R?/:,) = 3~/8 and hence a20 - 6,, = 3n/4.
F28 147 0.
5
VL 0.04
2.L = 0.30
j-
\
G 0:
.o:!-
L-
0.c:
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
6ooW>
Fig. 4. p’,,/p,,
versus S,
( X a)
with 2 L as a parameter.
37X
R. Maraddat/
Optics Communications
137 (1997) 367-381
5. Optical cavity frequency response - non-ideal cavities In this section we consider the situation when the cavity itself deviates from a perfect configuration. As an example we examine the case of an optical cavity with a small tilt in either of its mirrors. The incident field can be considered as being a Gaussian beam matched to a perfect cavity of the same configuration. Other imperfections can be dealt with in a similar manner. If the plane of a mirror m is tilted about the >I-axis by an angle 19,,~ towards the z-axis as in Fig. 5, any field incident from the left and reaching the new plane of the mirror, is modified and is multiplied by a phase factor exp( -jk0,, x-1. If H,,,, is small, this phase term can be written as
exp( -jkH,,,, X) = 1 -jk0 .,,,,.Y - $k$;,
2.
The modification of the field converts a portion of the fundamental mode of the cavity to higher order cavity modes. Assuming that the effect of mirror tilt on the weight amplitudes is limited to the TEM,, and TEM,, modes, the conversion of the weight amplitudes due to a tilt can be represented by a scattering matrix in the form of 0
0
en2=
00 00.21
p
0 .,a
0 @
10 00.21
010
OO.l?.n,
.m
0
00.12.n/
0
000
lO.lZ.YR
0
000
10.21.~, 0
@ ‘0
IO.2l.m
($10
IO.l2.,J8
’
0
where
zzz1-~[fk~(L_)H,/]2=l--~.~~. where 6,,,) = fkwx(L,,,)B .,,,1. An examination of power under the above matrix transformations TEM,,, is incident on a tilted mirror with a unity reflection coefficient,
Fig. 5. Schematic representation
will show that if a field of unit power matched to the normalised reflected power in TEM,, mode will
of a mirror tilt.
R. Mauaddat/ Optics Communications 137 (19971367-381
379
be approximately equal to (1 - 2 i& )(I - 2a&) * f 1 - 43,2, and in TEM,, to (-j28,_)(-j28,_)* = 4812_, adding to unit reflected power. However, if the incident wave is matched to TEM,,, the reflected power in TEM,, is (-j28J-j28,J* =4@& but the reflected power in TEM,, is (1 -6e,?_)(l -68,2,,)* = 1 - 12e:,,z and the total power will not add to unity. In the latter case, for the right power relation to be hold, it is necessary to include also the conversion of the TEM,, field to TEM,,, which can be written as 020 ,o.,z ,,,, = O$‘2,.,,, = @$‘, = -jfi[$kw(L,~)O,,]
= /_~~~_~i;,o(.~~v.~,,,)*~,o(x,~,L_)[l
-jkO,_x-
fk20Z,,,~‘] dxd?
= -jfi&_.
This will contribute ( - j2 J2 e,_>( - j2J2e,,,,l* = 8 a:,,< to the total power which then will add to unity. For a more rigorous analysis, therefore, we need to include the TEM 2. also. Here to avoid this additional complication, we take @ ‘0
10.12./R
=
o&,,,
=
o;:.,
=
1 -
+q,“,
which will give the right power relations and implicitly includes the effect of partial conversion of TEM,, to TEM2, and back to TEM,, mode. The above tilt scattering matrix thus found has to be converted to transfer scattering matrix in the form of @,‘,o~_/A@
- @;$‘_/A@
where A0 = O&Oii&
- O,$‘, /A@
0
0
0’0 oo.,?
000 oo.
0 fl_ = 0
0
0
@:,_/A@
000
IO,“/
0
’
0’0
0
IO.,/,
- 0”0O.mOE,W1.
5.1. Tilted front mirror In this case the cavity matrix C can be written as C = F$PE, where F, P and E are as defined before and fi/ is the tilt transfer matrix for the front mirror. By evaluating the above matrix, the reflected and transmitted field through the cavity can be found. In Fig. 6 we compare the plotted values of PxF/P,/ as a function of 6, for different values of e,,, as calculated by the above approximate
Fig. 6. F2,/<,,
versus 6,
(X 7~) with 8,,
as a parameter.
R. Macbaddat/Optics Communications 137 (1997) 367-381
II
1
1;
2
,I, 3
n
4
‘-
5
C.6
:.’
3.e
to.9
;
sooG4 Fig. 7. p’,,/<,
analysis and by the numerical iterative rji’= 0.95. r, = 0.97 and tan-‘(R,/z,) .q&V,.
versus 6,
(X
a)
method described before. - tan- ‘(R
with H,< as a parameter
We have assumed the cavity to be lossless and also and hence S,, - S,, = 37r/8. In this figure “; =
5.2. Tilted end mirror In this case the cavity matrix C can be written as C = FPLn
6. Conclusion We have applied the scattering matrix analysis to the analysis of optical cavity resonators. Specifically it is applied to the situations where the input beam is misaligned or its curvature at the resonator input mirror is not matched to the cavity. We have also considered the case when the cavity itself deviates from a perfect configuration. The analysis can be easily extended to the a number of coupled optical cavities. With the proposed optical interferometers for gravitational wave detection, the analysis of complex coupled cavities has assumed a great importance. The formal method of analysis developed here can be a very useful tool for the performance evaluation of such systems.
Acknowledgements In this paper the results obtained by the scattering method were compared with an iterative numerical code described in Section 4. The latter code was initially developed at the Department of Physics and Theoretical Physics of the Australian National University by A. Tridgell, D. McClelland and C. Savage [2,3]. Later the code was modified and adopted by the author for evaluation of the coupled cavities.
R. Mar~addat/Optics
Conzmunications
137
f IYY7J
367-381
381
The author is indebted to Y. Hafetz of Massachusetts Institute of Technology for the discussion he had several years back on the analysis of the coupled cavities. Hafetz had produced a matrix formulation for analysis of coupled cavities with intention of explaining a specific experimental situation. Although somewhat similar, the formulation and the analysis here using scattering matrices is fundamentally different. The scattering matrix analysis was suggested by the author prior to this discussions
[4,5].
The numerical calculations
in this paper
was performed
by the Mathematics
code.
References [I] [2] [3] [4] [5]
A. Yariv, Introduction to Optical Electronics (Halt, Rinehart and Winston, 1971). A. Tridgell. Honours Thesis. Department of Physics and Theoretical Physics, Australian National University, Canberra. Australia, D.E. McClelland. CM. Savage. A. Tridgell and R. Mavaddat, Phys. Rev. D 48 (1993) 5475. R. Mavaddat. Classical Quantum Gravity 10 (1993) 811. R. Mavaddat, Network scattering parameters. in: Advanced Series in Circuit and Systems (World Scientific. Singapore, 1996).
1990.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
137 (1997) 367-381
Full length article
Scattering matrix analysis of optical resonators with cavity imperfections R. Mavaddat Department
of Electrical and Electronic Etzgitleering, fJniwrsi& Received
16 September
1996: revised 17 December
’ of Western Australia, Nedlands. Australia 6907 1996: accepted
19 December
1996
Abstract The scattering matrix analysis is a powerful tool that is used extensively for the analysis of microwave circuits and occasionally for that of optical devices. Here we apply the scattering matrix analysis to a class of optical resonators with sources external to the cavity. Such an analysis is general in the sense that it allows the inclusion of all higher order cavity modes and the possible situations where the reflection from and transmission through the cavity mirrors, for the waves incident from different directions, are different. The assumption of negligible diffraction losses or large extension of the cavity dimensions relative to the width of the cavity modes has to be made. In this paper single cavities and possible imperfections, such as relative source or mirror misalignments. are considered. The case of multiple coupled cavities is an extension of this analysis and will be dealt with in a separate publication.
1. Introduction The aim of the scattering matrix analysis of an imperfect cavity resonator is to establish a matrix relation between the incident and reflected mode components of the field at the cavity input port, to that of the incident and reflected mode components at the cavity output port. The elements of the matrix are derived from the cavity parameters given for the fundamental and higher order cavity modes of the corresponding perfect cavity resonator. It is assumed that the sources are external to the cavity and the diffraction losses are negligible. For further reference we first briefly review the modes of a perfect cavity.
2. The fundamental
and higher cavity modes of a perfect cavity
Optical resonators or cavities may assume many configurations. Here we consider the basic resonator consisting of a light beam incident on two partially reflecting surfaces, separated by vacuum or air. Each reflecting surface is assumed to be a part of a spherical surface with a specific radius of curvature. 2. I. The fundamental
cavity mode
The modes in an ideal optical resonator mode in the form of [l]
are Transverse
Electromagnetic
’ E-mail: [email protected]. 0030.4018/97/$17,00 Copyright PII SOO30-4018(97)00004-7
0 1997 Elsevier Science B.V. All rights reserved.
Modes (TEM) with the fundamental
TEM,,
Fig. I. Schematic representation of an optical resonator cavity.
where k = 27r/A is the wavenumber system. U,(Z) = k~,dll+~
of the free space propagating
wave. ,6o0 is the field at the origin of the co-ordinate
is the beam spot radius or the radial distance from the cavity axis where the magnitude
of the field reduces to 1/e of its magnitude on the cavity axis. The common axis of the two mirrors defines the :-axis of the = )I’,,= Jp and is co-ordinate system with the origin of the co-ordinate system :: = 0 at a position where TV = CL’(O) a mmimum. z0 is the distance from the waist where the spot size is fi times that of the spot size IV,) at the waist. R(Z) = ~[l + (z,,/:)‘] is the radius of the curvature of the wavefront at a distance Z. At the waist R( :) = R(O) = z. At a distance z = zo, R( z> = R( z,) = 2 :*. The phase factor Q~( Z) is given as qoO(z) = tan- ‘( z/;& When z = 0. qoO(2) = 77JO) = 0. We now consider a resonator cavity with configuration as given in Fig. 1. Let L/, and L< be the position of the /rant and end mirrors along the z-axis respectively, with L = LC - L,- as the length of the cavity. R/ and Rc are the radii of the curvature of the front and end mirrors respectively. It is assumed that the end mirror is always to the right of the front mirror and the mirror curvature is taken as posifirle when the centre of the curvature is to the left of the mirror. For the field to match both mirror curvatures, we require R( L,/) = L/[ 1 + ( ;o/L/.)‘]
= R,/. and R( L,) = L?[ I + ( zo/Lc)‘]
= Rc.
With L= Le - Lr.
, or hence z0 is found in terms of the cavity parameters. In terms of the mirrors radii of curvature R,,, and R?, L, and Lc are given as R/ L/,=,-,/(2 As Rp mirror.
r
Rr
0~) L/-+ 0 and for R/=
and L, = 2 f22,.
- :il
- (2:,,/RC)‘.
Lf = + z. as expected. Similar relations hold for the parameters
Table 1
K,(x) ff,( x)
2X
H?(X) HJ x) J%(X) H,(x)
-2+4x? -12x+8x3 12-48~’ + 16~~ 120x - 160x3 +32x5
I
related to the end
R. Macaddat/ Optics Communications 137 I lY971367-381 2.2. Higher order Hermite-Gaussian
modes
For the higher order modes of cavity known as Hermite-Gaussian
Xexp[-jk++j(l+m+
where H, is a Hermite polynomial 1. 2.3. Mode orthogonality
I)Q~(:)],
are give in Table
and normalisation of Hermite-Gaussian
modes we have
+m
/ --x / -x +I
modes
of order n. For a later reference some lower order Hermite polynomials
Applying the known orthogonality +z
369
E,,(x,y,z)El,,,(.~,y.i)* dxd~=2”“‘-‘I!m!~w~~I?,,
+= E,,(.~,?I,~)E,,,,(*,~,~
)* dxdy=O,
/ -cc / -%
We can now define the normalised E,,,(I,_~,:)
I’=1 and m’=m,
nifm.
I’+1 and/or
fields as
= (2’+‘“-‘l!m!*~~~)-“ZE,,(~,?‘,~),
giving
Xexp[-jkz+j(l+m+ With this normalisation += / -z +x / -x
l)n,,(z)].
we have
+x_ E,,,(x,:,~)El,,,~(x,~,~)*
dxdy=
+=_ E,,(x,y.,-)E,,,,i(x,y,,-)*
dxd_v=O,
/ -?j / -x
1,
m’=m,
l’=/and
/‘#/and/or
m’fm.
3.Scattering matrix formulation The expression for the normalised Z,,,(.r,.v, c)&(;) where Zln,( x,y,z)
= (2’+‘“-‘l!m!n)-“’
modes
--& z
@,,(z)=exp[-jX;-+j(l+m+
of a cavity
H/[&G]
1)7700(z)],
resonator
can be written
HM[fi--&]exp[
as the product
-s-jk&$$-]
E,,(
x,
y,z) =
(3.1)
(3.2)
which is a phase term and a function of z only. Z,,(.r,y,z) defines the field variations in the transverse directions and hence the transverse modes of the cavity. c$Jz) defines the longitudinal cavity modes. The actual field inside the cavity is the weighted sum of all modes that are set up within the cavity as given by (3.1), multiplied by the corresponding phase factor $I,,,( z> of (3.2) for each cavity mode. In general the weight a”” of lm mode in a cavity is a complex number and depends on the cavity configuration, mirror reflection coefficients and losses, as well as the light incident on the cavity mirrors.
R. Maraddat / Optics Communicatioru 137 f 1997) 367-381
370
The total field inside the cavity can be written as -L E(.Y.J.,)
-L
= c c u”nz,m( X..v>Z)&,,( z). /ill II,=0
(3.3)
In a similar way we can consider the field incident on the front mirror from the left as the summation of the weighted modes of the cavity at the plane of the front mirror. This can be written as
The reflected field from the plane of the front mirror can also be written as
For the field incident from the right on the front mirror, we replace _I?,/ (x, y,L/ ), iii?, I!?,/ (x, y,.Q ) and ziy by &(X,Y,L/.), z:‘;, a,( x, .v,$1 and Li$ respectively. The subscript 1, therefore, refers to the left of the mirror and the subscript 2 to the right of the mirror. For the end mirror all subscripts ,L’ change to the subscript e. In general the left and right complex weight parameters of a cavity can be related by a transfer scattering matrix C in the form of 00 C00,ll
00 C00.12
COO IO.12 IO.11 COO
coo 01.1 I
C
c IO.21
c 10.22
C’O 00.11
clo 00.12
C'O 10.1 I
C’O 01.12 IO.12 CAP.,,C’O
c &,
C'O C~oO.22 10.21
00.21
GJ I CO' 00.12 GL, G%?
00
CPd.l
00
C’O IO.22 I
GLl
COO 01.21
.
00 COl.I2
coo
00 00.22
. . .
Cue 01.22
c;p.z,
GY.22
CO’ GE c,o:,, I 01.17 c~fl22G,21 CO’ 01.22
-00
U2e
-
-00
%,
*IO
...
a2,
-10
...
a2,
.
QI
a2,
.
.
to1
a2c
.
Im C00.11 /In
C00.21
I”, C00.12
pn
Cl&2
Cl”’ IO.21
10.1 I
Cl”
p 10.12
C'" IO.22
01.11
C'" 01.21
I,,1 C 01.12
.. .
CA;122
..
.
where ?i$ and &fS are, respectively, the weights of lm mode for the incident and reflected fields from the left-hand of the front cavity. Similarly 2:: and Zie are, respectively, the weights of lrn mode for the incident and reflected fields from the righthand side of the end cavity. In matrix notations
The elements of the cavity transfer scattering matrix C can be calculated from the cavity parameters, as we shall see in the following examples. The above matrix representation is also valid if the cavity elements are not reciprocal.
4. The frequency response of a cavity with an ideal geometry Assuming that the cavity mirrors are partially reflecting, we define the amplitude reflection coefficient ~611’as the ratio of the reflected wave amplitude for the component of the field that is matched to I’m’ mode, to that of the incident wave amplitude for the component of the wave that is matched to the Zm mode of the cavity. Similarly we define the amplitude transmission coefficient f:?’ as the ratio of the transmitted wave amplitude for the component of the field that is matched to I’m mode. to that of the incident wave amplitude, for the component of the wave that is matched to the Im mode.
R. Maoaddat/
Optics Communications
137 (1997) 367-381
371
The transfer scattering matrix for a single mirror with no tilt or mismatch and with the assumption that the modes are not coupled at the mirror surface can be written as Moooql I
M&,
0
0
0
0
...
0
0
M%,
M,&
0
0
0
0
...
0
0
0
0
M::,I M&,
0
0
...
0
0
0
0
...
0
0
.. ...
0
0
0
0
w%,
M,,&
0
0
0
0
M&
0
0
0
0
M,q’.*,M;,&
0
0
I
M;&
M=
.
.
... . 0
0
0
0
0
0
0
0
0
0
0
0
The elements of M in terms of the mirror amplitude reflection
M::,,, = l/t/,“,
. .
4’m”. II 4!L . . 4% I M:mm.22 . ... . ...
and transmission
coefficients
M$',,= -r~~/t~~, M&'2,= r,!,"/t:,", M:l,22 = - [(d$-
are
(t.$/tk
where r:,” and t:,” are respectively, the amplitude reflection coefficient and transmission coefficient of the mirror for the Im mode. Specifically we denote the transfer scattering matrix of the front mirror by the matrix F and we have a Ip = Fa,p and similarly ale = Ea,, where E is the transfer scattering matrix of the end mirror. The weight amplitudes at the left and right planes of the space region between transfer scattering matrix that can be denoted as P and given as
p,g
0
0
0
0
0
. .
0
...
0
pg
0
0
0
0
‘.
0
...
0
0
,I0 PI0
0
0
0
‘.
0
.
0
0
0
PIi
0
0
.‘.
0
...
0
0
0
0
pbq’
0
. ‘.
0
.
0
0
0
0
0
PoOf
. . . . . .
0 .
.
P2
P=
.
0
0
0
0
0
0.
0
0
0
0
0
.
.
0. . . .
or
aI, = Pa,, , where p::=
with
exp( -jS,,)
and p;fz = exp( +jS,,),
0
. . .
.
.
0 p:,m
... ... ... ... ... .. .. ...
the two mirrors are also related by a
377
R. Mat>addat/ Optics Communicatiom 137 (19971367-381
A cavity consisting of a front mirror and an end mirror and a space in between has its total transfer scattering matrix C given by C = FPE. Hence we can relate the mode amplitude weights of the field incident and reflected on the first mirror from left to that of the mode amplitude weights of the field incident and reflected on the end mirror from right in the form of a ,/ = Ca,,
.
where I,,, F/“’ c irt,.il ““‘E:;,, , + F/z,,, p;,::E:;,%,, = I,,, I I P,,,,
p
InI /,ri.I2 = F:,::‘.I I d::‘4?, I2 + F/n,,I2P::: E::l:,x .
p
I,,,.2I = F,% I d::‘4t’. , I + 4%~ P:%‘:,:.~I 3 pi’ _ F/“l ““’ E ;,::,,? + F:;‘z2 P;;;:E;;;.~~. /m,2?- I,,,.?IP/w, Assuming that the wave is incident on the cavity from the front mirror and no wave is incident from the end mirror. we have -/n1 _ CK? I a+,,,* aI (.y” 1 ’ /,,,.I I
(4.1)
(4.2) where
(4.3)
(4.4) We now assume that the light source field intensity and phase distribution at the input mirror (front mirror) is known and be denoted by E,/(.x,_~,$). Then the matrix element 2ii; can be found from (3.4) and the fact that the cavity modes are orthogonal by the expression
Hence we can find the reflected field from the front mirror and the transmitted (4.4) for every mode. 4.1. Matched
Gaussian
field from the end mirror as given by (4.1) to
beam
For the case of the input beam given as E,/x,y,L,) = e,,(.r,.v,Lf)&,,(LF) = exp[-jL$ and matched at the front mirror to the fundamental mode of the cavity, we can write eIp( x.v,$)
+jq,J~~)l
e,/(x,y,$)
= AZ,,( x,Y,L/,),
and hence ~~.=I+rI-rE,,(s,~,L,)*Ae,,,(.~,~,~~)dsd~=A. --5 --* Substituting
into (4.1) and (4.2) we find tjp’r,
1 - r/,py
A
(4.5)
and Z& =
5. Pt< 1 - r/ p2rr
A,
where p = pii = exp( -jS,,).
(4.6) with 6,” = kL + q,J L, ) - qoO( Lc).
373
R. Macaddat/ Optics Communications 137 (1997) 367-381
lr=
0.20
‘:, \
= 0.45
\
r= 0.70 r= 0.95
> 0
-0.2
0.2
Fig. I?. p;,/F,,
0.6
0.4
versus 6,
(X
P)
0.8
1
1.2
with r, = r1 = r as a parameter.
In the above equations tf, t<, r/. and rz are equal, resp:ctively, to ti$, t$G, rO& and ri&. We have plotted the ratio of the transmitted power P2? = (E,J&‘/~&~?/~~ to that of the incident power p,/= (E,//_L.,)‘/‘Z,~Z;~ as a function of 6,, in Fig. 2, with r as a parameter and the assumption of rG = rr= r. If the spacing is changed over a few wavelengths, the variation of Q,&L,) - q&L,) with the cavity spacing is negligible. For this reason the given curves can also be considered as the normalised powers, plotted as a function of the fraction of a half-wavelength changes in the cavity length. A 7~ radian change in 6,, corresponding to a A/2 change in the cavity spacing. With n is an integer and provided that nh/2 <( L, the transverse field configuration is almost identical if the length of the cavity is changed by nh/2. These different longitudinal solutions with identical transverse fields, constitute what is known as the longitudinal cavity modes. 4.2. Tilted Gaussian beam Here we assume that a Gaussian beam. initially matched to the fundamental mode of the cavity, is incident on the front mirror of the resonator. The axis of the beam is then tilted by O.r?,*radians in the x--z plane about the position of the front mirror on the :-axis. We wish to determine the fields transmitted through and reflected from the cavity. Here with the assumption of a small tilt, we include in the analysis only TEM,, and TEM,, modes and neglect all higher order modes. In this case the incident field can be written as E,/(x,y,L/)=exp[-jkL~-+j?7,,(4)]e,/,(x,v,L~). while for small H,, we can write e,/(x,y,+)
=AZ,(x._v,L,,)exp(
-jkO,,x)
=A&&,y,Lp)[l
-jkO,,
x-
ik20.r: x2].
The total power incident on the front mirror of the cavity can be found by integration over the area of the mirror in the form of
indicating that within our approximation, of the mirror tilt.
the incident power has remained unchanged from its value prior to the introduction
R. Mavaddar/
37-k
Opiics Communications
137 (1997) 367-381
The incident field e,, (x, y,L/) can then be expanded in terms of the fundamental
modes of the cavity as
+r L% -00 _ ~,~“(~,.v,L~)*e,/,(l,y,l,)dxdv (II,/ - / -7 / --%
+Ic
+%
x
/ --x
’
=
/
2exp[-$$-]Aexp[--$$][i-jkO_Y4~~-’k’O~$~’]d~dv
7rw(L,)
=A(1 -+[fkw(L/)e,,$]
=A(1
-f@/)
and -10 = U,/‘.
+r.
+x ~,o(“,y,L~)*e,/,(x,~,l/-)dxdy
f --x / -z
= ~+X~+X~,,(~,y,~~)* -Cc --LL =
+=
-jkO,,x-~k%$x’]
AZ,,(x,y,L/.)[l
f-r
4x ?rW(L/)
/ --P / -=
=A{ -jikw(+)Oxg}
3 Aexp[ - 2~z~~~~‘]
[l -jkOrBx-$k20;kx2]
d.rdy
dxd?;
= -jA$,,-,
now verify that the sum of the input power of the individual modes is the same as the total where B, = $kw($.)O,,.We input power we have just calculated by forming
=A2(~0/p0)“28,‘,.
(ea//_~~)“%;J:~;* With modes orthogonal, p;/=
the total power is the sum of above terms given as
( E&,,)“~(
q;,q;:
) ‘I A’( E&,-,)“~
+ z$i;;*
which, within our approximation, is the same as found by direct integration. The weights of the transmitted and reflected modes are hence
for the fundamental
TEM,,
for the TEM,, mode. In the above equations &=kL.+n00($-~00(4)
mode, and
poo = pii = exp(-j6,) and
and plo =pig = exp(-j6,,)
where
610=kL+7710(L~)-77~~(Lc),
with noo(z) = tan-‘(z/z,) and ~~~(2) = 2tan-‘(z/z,). I;addition, y= rti,,/ = rt&/= r$2f= r{&, is the amplitude reflection coefficient of the front mirror and 5 = rg,‘< t&f= t&/= = rlo.lC = roozr = r10.7r is the amplitude reflection coefficient of the end mirror. Similarly tr= t$lr= are the front mirror and the end mirror amplitude transmission coeffic’ents, and ry”L $$,~~ tl&, = too.?< O” = tlo.z< lo G,2/ respectively.
R. Maraddat/ Optics Communications 137 (19971367-381
Fig. 3. F2,/<,. The incident power was given by ?,/= expression
and the transmitted
versus 6,
A’(E,,/~~)‘/‘.
(X
T)
with 8,,, as a parameter.
The reflected power from the first mirror can be found from the
power from the end mirror from the expression
p’,, =A2@;*g;
+~~*~:“,)(E&,)“~,
where we have used the fact that the modes are orthogonal. In Fig. 3 we have compared the variations of p’,,/?,, as a function of 6,, for different values of e,,, as calculated by the above approximate analysis, and by the numerical method described in the following paragraph. We have assumed the cavity to be lossless with r/.= 0.95, r< = 0.97 and tanP’(R,Jz,,) - tan-‘(R
=A~,,(L/)~,(r,L*+AL).
instead of E,p(r,L/)=A~,,(L/)e,(r,Lp),
where r’=x’+.v’
and
R. Maraddut/
376
In the above expression Hence with
a
--
a;
for Z,,,(r,c),
I 1 I
137 (1997) 367-381
the most sensitive term to the variations in ,-, is the term containing
the variable k.
aR(z)
+(;)=-I_=
R(L)
Optics Communicntior~s
R(z)’
a,-
we have
or
and hence ,E,/(‘,L/)
=
&,(LF)e,/.(r,L~.)
=A~,,,,(L/)e,,(r.L~.)[l
-jkft(L,)r’AL-ik’5(L~‘)‘r4AL’]
The total power incident on the front mirror of the cavity can be found by the following mirror as
=
AZ,(r,LF~)* wPo)“2~~-~~o
integration
over the area of the
AV,,(r,L~)rdBdr=A’(q,/p0)“2,
indicating that within our approximation, the incident power has remained unchanged. Now we need to find the elements of the weight matrix. From the fact that the modes are orthogonal
I
2r’
2X2i7
= / r=O 7rw(L,)z
Aexp
[
-
WZ(L/)
1
[l -jkf[(Q)r’AL-
we can write
~k2~(Lf)‘r4AL’]rdr
where
~~L=+~~(L/)w(L/)~AL=
-
1 - ( zo/L,/)’ 1 + (i,,/LJ
As we have seen, the sum of the two modes TEM,,
AL co
and TEM ,,? produces an axially symmetric
field. With the incoming
R. Macaddat/Optics
light axially symmetric, we assume an equal contribution cylindrical coordinate system we can write
=
lx
~zv[Z20(r,L~)
371
Communications 137 (1997) 367-381
from these modes to the input field of the cavity. Hence in a
+Z,,(r,L/)]*
AZ,,(r,L/,)[l
$1
Aexp[ ----$$I
-j+kg(L,)r’AL]rd0dr
r=O H=O
=
z
2a
/ I= 0 2TW( L//)
? [ -4+
kY(L/ )’ p+ 4
[l -jik<(L,,)r’AL]rdr
2;;~;)S]A+jk2~(~;)S[4+-8+]AAL
It is of interest again to verify that the input power has remained unchanged.
We can verify this by forming
and
With modes orthogonal,
the total power is the sum of above terms given as
Having found the incident field on the front mirror of the cavity, the transmitted following expressions
In the above S,, = a,2 = In Fig. 4 lossless and
poo = pii = exp( - jS,) and p20 Tjg,,iLl) with qoO(z) = as a function of we have plotted P,?/P,/ also r/ = 0.95, r, = 0.97 and tan-‘(R//q,) equations
-jkL
+ jqzo(L,)
and the reflected fields can be found by the
= pii = exp( - jS,,>, where So0 = - jkL + jrloo(LC) -jqoO( 4) and tan-‘(z/z,) and Q~(z) = T~_o_(:)= 3tan-‘(c/z,). S,, for different values of AL. We have assumed the cavity to be - tan-‘(R?/:,) = 3~/8 and hence a20 - 6,, = 3n/4.
F28 147 0.
5
VL 0.04
2.L = 0.30
j-
\
G 0:
.o:!-
L-
0.c:
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
6ooW>
Fig. 4. p’,,/p,,
versus S,
( X a)
with 2 L as a parameter.
37X
R. Maraddat/
Optics Communications
137 (1997) 367-381
5. Optical cavity frequency response - non-ideal cavities In this section we consider the situation when the cavity itself deviates from a perfect configuration. As an example we examine the case of an optical cavity with a small tilt in either of its mirrors. The incident field can be considered as being a Gaussian beam matched to a perfect cavity of the same configuration. Other imperfections can be dealt with in a similar manner. If the plane of a mirror m is tilted about the >I-axis by an angle 19,,~ towards the z-axis as in Fig. 5, any field incident from the left and reaching the new plane of the mirror, is modified and is multiplied by a phase factor exp( -jk0,, x-1. If H,,,, is small, this phase term can be written as
exp( -jkH,,,, X) = 1 -jk0 .,,,,.Y - $k$;,
2.
The modification of the field converts a portion of the fundamental mode of the cavity to higher order cavity modes. Assuming that the effect of mirror tilt on the weight amplitudes is limited to the TEM,, and TEM,, modes, the conversion of the weight amplitudes due to a tilt can be represented by a scattering matrix in the form of 0
0
en2=
00 00.21
p
0 .,a
0 @
10 00.21
010
OO.l?.n,
.m
0
00.12.n/
0
000
lO.lZ.YR
0
000
10.21.~, 0
@ ‘0
IO.2l.m
($10
IO.l2.,J8
’
0
where
zzz1-~[fk~(L_)H,/]2=l--~.~~. where 6,,,) = fkwx(L,,,)B .,,,1. An examination of power under the above matrix transformations TEM,,, is incident on a tilted mirror with a unity reflection coefficient,
Fig. 5. Schematic representation
will show that if a field of unit power matched to the normalised reflected power in TEM,, mode will
of a mirror tilt.
R. Mauaddat/ Optics Communications 137 (19971367-381
379
be approximately equal to (1 - 2 i& )(I - 2a&) * f 1 - 43,2, and in TEM,, to (-j28,_)(-j28,_)* = 4812_, adding to unit reflected power. However, if the incident wave is matched to TEM,,, the reflected power in TEM,, is (-j28J-j28,J* =4@& but the reflected power in TEM,, is (1 -6e,?_)(l -68,2,,)* = 1 - 12e:,,z and the total power will not add to unity. In the latter case, for the right power relation to be hold, it is necessary to include also the conversion of the TEM,, field to TEM,,, which can be written as 020 ,o.,z ,,,, = O$‘2,.,,, = @$‘, = -jfi[$kw(L,~)O,,]
= /_~~~_~i;,o(.~~v.~,,,)*~,o(x,~,L_)[l
-jkO,_x-
fk20Z,,,~‘] dxd?
= -jfi&_.
This will contribute ( - j2 J2 e,_>( - j2J2e,,,,l* = 8 a:,,< to the total power which then will add to unity. For a more rigorous analysis, therefore, we need to include the TEM 2. also. Here to avoid this additional complication, we take @ ‘0
10.12./R
=
o&,,,
=
o;:.,
=
1 -
+q,“,
which will give the right power relations and implicitly includes the effect of partial conversion of TEM,, to TEM2, and back to TEM,, mode. The above tilt scattering matrix thus found has to be converted to transfer scattering matrix in the form of @,‘,o~_/A@
- @;$‘_/A@
where A0 = O&Oii&
- O,$‘, /A@
0
0
0’0 oo.,?
000 oo.
0 fl_ = 0
0
0
@:,_/A@
000
IO,“/
0
’
0’0
0
IO.,/,
- 0”0O.mOE,W1.
5.1. Tilted front mirror In this case the cavity matrix C can be written as C = F$PE, where F, P and E are as defined before and fi/ is the tilt transfer matrix for the front mirror. By evaluating the above matrix, the reflected and transmitted field through the cavity can be found. In Fig. 6 we compare the plotted values of PxF/P,/ as a function of 6, for different values of e,,, as calculated by the above approximate
Fig. 6. F2,/<,,
versus 6,
(X 7~) with 8,,
as a parameter.
R. Macbaddat/Optics Communications 137 (1997) 367-381
II
1
1;
2
,I, 3
n
4
‘-
5
C.6
:.’
3.e
to.9
;
sooG4 Fig. 7. p’,,/<,
analysis and by the numerical iterative rji’= 0.95. r, = 0.97 and tan-‘(R,/z,) .q&V,.
versus 6,
(X
a)
method described before. - tan- ‘(R
with H,< as a parameter
We have assumed the cavity to be lossless and also and hence S,, - S,, = 37r/8. In this figure “; =
5.2. Tilted end mirror In this case the cavity matrix C can be written as C = FPLn
6. Conclusion We have applied the scattering matrix analysis to the analysis of optical cavity resonators. Specifically it is applied to the situations where the input beam is misaligned or its curvature at the resonator input mirror is not matched to the cavity. We have also considered the case when the cavity itself deviates from a perfect configuration. The analysis can be easily extended to the a number of coupled optical cavities. With the proposed optical interferometers for gravitational wave detection, the analysis of complex coupled cavities has assumed a great importance. The formal method of analysis developed here can be a very useful tool for the performance evaluation of such systems.
Acknowledgements In this paper the results obtained by the scattering method were compared with an iterative numerical code described in Section 4. The latter code was initially developed at the Department of Physics and Theoretical Physics of the Australian National University by A. Tridgell, D. McClelland and C. Savage [2,3]. Later the code was modified and adopted by the author for evaluation of the coupled cavities.
R. Mar~addat/Optics
Conzmunications
137
f IYY7J
367-381
381
The author is indebted to Y. Hafetz of Massachusetts Institute of Technology for the discussion he had several years back on the analysis of the coupled cavities. Hafetz had produced a matrix formulation for analysis of coupled cavities with intention of explaining a specific experimental situation. Although somewhat similar, the formulation and the analysis here using scattering matrices is fundamentally different. The scattering matrix analysis was suggested by the author prior to this discussions
[4,5].
The numerical calculations
in this paper
was performed
by the Mathematics
code.
References [I] [2] [3] [4] [5]
A. Yariv, Introduction to Optical Electronics (Halt, Rinehart and Winston, 1971). A. Tridgell. Honours Thesis. Department of Physics and Theoretical Physics, Australian National University, Canberra. Australia, D.E. McClelland. CM. Savage. A. Tridgell and R. Mavaddat, Phys. Rev. D 48 (1993) 5475. R. Mavaddat. Classical Quantum Gravity 10 (1993) 811. R. Mavaddat, Network scattering parameters. in: Advanced Series in Circuit and Systems (World Scientific. Singapore, 1996).
1990.