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ThF PRORTTON oF pLECTRONAic- 1DIELECTRIC SLABS(U) HARRY DIAMOND LAOS ADEIPHI NDI R STEAD MAY 88 DL-TR-2145
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HDL-TR-2145s May 1988 r.
.4 The Propagation of Electromagnetic Waves In ThinDielectric Slabs
by Michael R. Stead D I
I UN2 1
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The Propagation of Electromagnetic Waves in Thin Dielectric Slabs
* 12 PERSONAL AUTHOR(S) '
Michael R. Stead13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Yea, Month, Day) 1S PAGE COUNT
Fia RMJuly 87 To Nov 87 May 1988 4216 SUPPLEMENTARY NOTATION
HDL project: AE1851, AMS code. 611102.H4400
*11 COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
FIELD IGROUP SUB-GROUP Optical waveguides, near infrared waveguides: energy propagation. 6
20 119 ABSTRACT (Continue on reverse if recessary, and identify by block number)
This report precents the solutions of Maxwell's equations for the TE and TM modes of propagation along a thin dielec-tric slab. From these, the propagation constants are determined and the electric and magnetic field patterns are plotted.These calculations are done with the use of the dielectric coefficients of LiNO'3 and GaAs. The wavelengths used are0.82 ;Am (diode laser), 1.064 IAm (Nd:YAG laser), and 1.55 lAm (diode laser). The thicknesses of the slabs range from 0.1 to
p WI1.0 ;rn.
DTICgELECI E
JUN 22 1988
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Contents '%Page
1. Introd uctio n ............................................................... 5
2. M axw ell's Equations ........................................................ 6
3. Num erical Calculation of Solutions ............................................ 11
4. Calculation of Propagation Parameters ......................................... 17
A cknow ledgem ents ........................................................... 28
L iterature C ited .............................................................. 28 %
D istribution ................................................................. 39 '
Appendix A--FORTRAN Programs Used in Calculating Dielectric WaveguideC haracteri stics .............................................................. 29
Figures
1. Physical representation of problem with transverse electric and magnetic fields .......... 7
2. Plots of both sides of four transcendental equations presented in text ................. 13
3. Electric field versus distance from slab center for two even and two odd TE modes,for a slab of I gim thickness, a dielectric coefficient of 4.80, and free space wavelengthof 1.064 lim .............................................................. 24
4. Magnetic field versus distance from slab center for two even and two odd TM modes .... 25
5. Electric field for TE even mode and magnetic field for TM even mode for a slab of
0.1 gm thickness and a dielectric coefficient of 12.25 ............................. 25 rO ~rl~rnod IDCOPY -1710 --- I _______
K-a"
vi1and/or'it Special
6MEN
Figures
6. Electric field versus distance from slab center for a slab of 0.35 [tm thickness and
dielectric coefficients of 12.25 ............................................... 26
7. Magnetic field versus distance from slab center .................................. 27
8. Highest (fifth) TE even mode for a slab with a thickness of I pAm and a dielectriccoefficient of 12.25, for a free space wavelength of 0.820 pm ....................... 28
Tables
1. Values of it for all possible modes with slab thickness of 0.35 ptm (d = 0.175 pAm),dielectric constant of 12.25, and free space wavelength of 1.064 Am ................. 15
2. Values of u for all possible modes with physical parameters as specified .............. 16
3. Propagation parameters for slab waveguides .................................... 21
* •
1. Introduction
Because dielectric waveguides are frequently used as part of the ex-perinental setups in the laboratory, it is of interest to have general pro-grams that calculate their characteristics. These waveguides generallyconsist of thin sheets (a few wavelengths or less) of a low-loss nonmag-netic dielectric material. The electromagnetic wave is launched at one endof the slab and guided to the sample I 11.
In this report, we idealize the situation to a single slab of infinite len;th in,,two dimensions, and a thickness, 2d, which will be arbitrary. First, we in-troduce the appropriate Maxwell's equations for the problem and derivethe wave equation for the transverse electric (TE) and transverse magnetic(TM) modes. We obtain the general solutions for both modes inside andoutside the dielectric and, by satisfying the boundary conditions at the sur-faces of the slab, obtain four transcendental equations which provide thesolutions to the problem. There is one equation for each mode, the TEeven, TE odd, TM even, and TM odd. Next, we assign values to our Sparameters and numerically solve the transcendental equations. This pro-cedure is repeated for several thicknesses and indices of refraction. Theresults are presented in graphical and tabular form. The appendix containsthe listings of the programs used.
5
2. Maxwell's Equations
Our waveguide extends infinitely in the - and z directions, and from -d tod in the x direction ksee fig. l a). The wave is launched in the y direction.The relevant Maxwell equations for the problem (in centimeter-gram-seconds) are
V x E = ikH (1)
and
-V x H =ikE (2) 0
where F is the dielectric constant. The time dependence is assumed to beexp(-iot), with k = w/c. Throughout the problem, . is considered con-
*stant and equal to one. In all cases considered, we assume that none of thefields vary in the z direction ()/z = 0).
In our first case, the TE mode, E, =0, E = 0, and Ez 0. So, Maxwell'sequations yield
ikH, = ;-E , (3)
ik Hy =- - E , (4)_"4
and
Ty- HX - aHy = ikeE . (5)
Combining these equations gives us the wave equation
Vt E, =-k 2 CE, e=c |ttd( (6)E= I6I>d
V2 a2 +a2where Vp .
6
I- %
Figure 1. Physical repre. (a) (b)sentation of problemwith (a) transverse ___
electric field and /(b) transverse magnetic zfield. / Y
-d dd XA d
E H
S/ /
We will assume a dependence in the y direction of the form exp(ip3y). Wewill also assume odd and even solutions of the form sin kix and cos klxinside the dielectric, and a decaying exponential, exp(-Fx), outside thedielectric. For guided waves in a lossless material, we must have F realand positive.
For even solutions, using equation (6), we have
E, = El cos ki xe i y IxI<d , (7)
E, = E2e-rx e' Y Yxl>d , (8)
where
k 2 +3 2 =k 2 e, (9)
-F 2 + 12 =k2 ; (10)
therefore,
, r=-e (e- 1)-k , (11) 0
We know that E. and Hy must be continuous at the boundaries x = d andx = -d. We need only solve at one of these boundaries, though, because
07
Aof the symmetry of the problem. We can find H by using euto 4eqatowith equations (7) and (8):
HY =k E, k, sin k, x e'O LxI~d ,(12)
HY ikFE2 e- eP [x . (13)
Setting equation (7) equal to (8), and (12) equal to (13) gives us the sys-tern of equations
El cos k, e =E2 e- (14)
El k, sin k, d =E2 Fe" (15)
which is reduced to
k, tan k, d =r (i6)
Throughout the remainder of our discussion, we will be using the follow-ing substitutions:
u=k, d (17)
and
A'-k' (F,-1)d 2 (18)
Using equations (11), (17), and (18), equation (16) becomes
tan u (19)
This transcendental equation will be solved in section 3.
For odd solutions, sine replaces cosine in the material, making our waveequation reduce to
E, = E3 sink kx e'Oy xI<d ,(20)
E, = E4 exei Oy IxI>d .(21)
8
0P
Equations (9), (10), and (11) still hold. Using equation (4) on equations(20) and (2 1) yields
Hy = -1E 3 k, cos kix e'Oy tir <d ,(22)
Hy =IE4 e-rxei Y [lI> d .(23)
At x di, we have a system of equations which produces
-ct , d (24)
Using our substitutions, from equations (11), (17), and (18) we get
-cot U U(25)
This transcendental equation will also be solved in section 3.
The calculations for the TM modes are very similar. We use the sameMaxwell's equations, but now the electromagnetic wave is described byfigure I b:
HX=0, H Y =0, HZ *0
Maxwell's equations yield
Ex = HZ 9(26)
Ey-= H2 (27)
I2= (a ~-~ ~ (28)
* Combining these equation-~ yields the wave equation
V' H, =-k 2 eHz (29)
For even solutions, we assume
H, = H, cos kx ey [xI<d , (30)
H.,=H2 ereiPY R> d , (31.)
as 9
11C
where k, and r are given in equations (9) and (10). E and H2 must becontinuous over x. Using equation (27) with equations (10) and (31) givesus a system of equations at x = d which yields
tan k, d = EF (32)
Or, after making our usual substitutions, with equations (!1), (17), and(8), we get
tan it E A it 3
This will be numerically solved in section 3 with the other transcendentalequations. Iv
For the TM odd mode, a very similar set of calculations arrives at thesolution
-cot fA2 2 (34)It
N,We now have the four transcendental equations providing the even andodd solutions to the TE and TM modes.
00
'-I
.4 oI,
3. Numerical Calculation of Solutions
In section 2 we obtained analytical solutions to our problem in the form ofthese four transcendental equations:
_ A - - 1 2I
TE even tan u - u
U
A2 -2
TE odd --cot t ="it
TM even tan u = eU
-A" 2_ U2V
TM odd -cotu= AU-U2'
We wish to numerically calculate all the solutions to these equations usingN. the Newton-Raphson method. We must first assign values to A and F. To.r"7 assign a value to A we must first break it down to its components. We as-
sume the following:
n = 3.5 (GaAs)
= 12.25 = n2 (GaAs)
d = 0.175 .m
X = 1.064.m
k = 5.93 x 10' .tm
A = 3.466 k2 ( - 1)d2
SB
These values are chosen to illustrate the method of solution to bedescribed, but are not to represent practical values in use in the laboratory121. Although there may be some use of waveguides as thin as this, mostoptical waveguides are in the neighborhood of 1 .m.
Just knowing these parameters is not enough. We must know the numberof solutions, and their approximate values, for the Newton-Raphsonmethod to work well. Figure 2a is a plot of tan iu, -cot u, (A2 - u2)l12/11,and e(A2 - itl versus u. The intersections of the trigonometric func-
0 4
*
tions with the latter two functions are the solutions to our roblem. Theirexact values, so far, are unknown, but they provide a good first guess forthe program, as well as a check on the number and range of our solutions. 0Our initial estimate from figure 2a for the first even solutions was It/2 - d.For each successive even solution, our initial guess was the previous finalapproximation plus n. For the first odd solutions, our initial guess wasit - d. We used the same method for successive solutions (of which therewere none) as in the even case. The markers on figure 2a are the values ofthe solutions generated 'by the program with the Newton-Raphson algo-rithm. Their "u" value is given in table 1.
We now wish to see the result of altering our input parameters--thickness,dielectric coefficient, and wavelength. Table 2 shows values of u for all
possible solutions given these parameters. All combinations were usedfor thicknesses of 0.1, 0.35, and 1.00 .m, and wavelengths of 0.820,1.064, and 1.550 ptm, with a dielectric coefficient of 12.25. We also didone trial with a dielectric coefficient of 4.80 (LiNO 3), a thickness of 1.00.m, and a wavelength of 1.064 .tm.
We can see from this table that increasing the wavelength, decreasing thethickness, or decreasing the dielectric coefficient will decrease the numberof solutions. However, the first even modes (TE and TM) will never dis-appear. If the value of A drops below iT/2, there will be no odd solutions, 5and only one of each even solution. If the value of A is very high, thenumber of solutions will be approximately 4 x A/it (A/it of each type).The separation of the u's for any two consecutive solutions of the sametype approaches it for the lower value modes. These results are illustratedin figures 2a through j. More detailed effects of altering the inputparameters are shown in the next section.
S0
51
0U
(a) (b)
; o
0 _ _
-..
0.0 1.0 2.0 .0 4.0 0.00 1.25 2.50 3.75 5.00 6.25 0.'
u (Arbitrary units) u (Arbitrary units)
(C) (d)
.n I0
o.- , N.
0.0 1.0 2.0 4.0 .0 0.0 2.0 4.0 6. 5.00 10.0
u (Arbitrary units) u (Arbitrary units)
Fig,!,re 2. Plots of both sides of four transeendental equations (19, 25, 33, 34) presented In text. Circles markpoints derived by Ncwton-Raphson approximation. These points define parameters for viable modes of,"-'propagation. Calculations were done for a slab with various thicknesses, dielectric coeffcients, and free 'space wavelengths: (a) 0.35-pim slab thickness, 12.25 dielectric coeffcient, and 1.064-14m free spacewavelength; (b) 1.00-lim slab thickness, 4.80 dielectric coefficient, and 1.064-gm free space wavelength;(c) 1.00-(m slab thickness, 12.25 dielectric coefficient, and 1.550-m free space wavelength; (d) 1.0-gm slab
thickness, 12.25 dielectric coefficient, and 1.064.gm free space wavelength.
0 13
e) ()
i%
C0'
0
oo
0 X
O, 0 , 0 "-'-
0 0 " ,
0.0 2.5 5.0 7.5 10.0 2.5 15.0 0.0 0.2 0.4 0.6 0.8 1.0u (Arbitrary units) u (Arbitrary units)
Wh0 0
0 CS
0 0 0. . . . . 0 . . . .
. .- K ...-.. \ ' \
u (Arbitrary units) u (Arbitrary units)
d Figure 2. Plots of both sides of rour transcendental equations 0!9, 25,33,34) presented in text. Circles markpoints derived by Newton-Raphson approximation. These points define parameters for viable modes ofpropagation. Calculathons were done for a slab with various thicknesses, dielectric coefficients, and freespace wavelengths (cont'd): (e) 1.00-p.m slab thickness, 12.25 dielectric coefficient, and 0.820-im free spacewavelength; (f) 0.10-pm slab thickness, 12.25 dielectric coefficient, and 1.064-1im free space wavelength;(g) 0.10-pm slab thickness, 12.25 dielectric coefficient, and 1.550-p4m free space wavelength; (h) 0.10-pm slabthickness, 12.25 dielectric coefficient, and 0.820-pgm free space wavelength.
% 14j..',I.
0 S
(i) (jo. o
0 L0II
o o o .
o ""'--0\ \. - ---- - .. .. -- , ... ..0 '. - ,
0.0 1.0 2.0 3.0 4.0 5.0 0.0 0.5 ;.0 1.5 2.0 25
u (Arbitrary units) u (Arbitrary units)
Figure 2. Plots (f both sides of four transcendental equations (19, 25,33, 34) presented in text. Circles markpoints derived by Newton-Raphson approximation. These points define parameters for viable modes of
0 propagation. Calculations were done for a slab with various thicknesses, dielectric coefficients, and freespace wavelengths (cont'd): (i) 0.35-pgm slab thickness, 12.25 dielectric coefficient, and 0.820-pgm free spacewavelength; and (j) 0.35-1gm slab thickness, 12.25 dielectric coefficient, and 1.550-pm free space wavelength.
Table 1. Values ofu for all possible modes with slab thickness of0-5 pm (d = 0.175 p4m), dielectric constant of 12.25, and free
pspace wavelength of 1.064 pmTEE TEO TME TMO
1.213211536 2.383396864 1.53064(XX)6 3.0012433533.373444796 - 3.464896202 -S
.15
00
* 0t
Table 2. Values of u for all possible modes with physical parameters as specified
Parameters TEE TEO TME TMO
d = 0.50 Xm 1.336460710 2.660983324 1.514054298 3.014218092e= 4.80 3.954871893 5.168169022 4.462066174 5.585698605X= 1.064 gm . - - -
d = 0.005 gm 0.849031210 - 1.248315096 -
E = 12.25 . -
X = 0.820 4m .u
d = 0.050 gm 0.734798610 - 0.982992113 -
= 12.25 - -
), = 1.064 im .- -
d = 0.050 gm 0.571712613 - 0.678357184 -
= 12.25 - -
) = 1.550in - .- -
d = 0.175 pm 1.281794548 2.541155577 1.541034222 3.065691948E. = 12.25 3.733202219 - 4.378210068 -
X=0.820 gm - -- -
d = 0.175 gm 1.213211536 2.383396864 1.530640006 3.001243353e = 12.25 3.373444796 - 3.464896202 -
X= 1.064 jim - -- -
d = 0.175 pm 1.093350649 2.078845978 1.504289746 2.371007681= 12.25 - - - -
X= 1.550 gm - - -
d=0.500 pm 1.457156897 2.912920237 1.560807824 3.121153116E = 12.25 4.365748882 5.813709259 4.680473804 6.237891197
= 0.820 gm 7.254138947 8.682868004 7.791815758 9.33862972310.092273712 11.464299202 10.866948128 12.30221557612.712786674 - 12.846722603 -
S d= 0.500 gm 1.426275611 2.849714994 1.557798982 3.114547968c = 12.25 4.266947269 5.673201561 4.668779373 6.217437744X = 1.064 gm 7.060328960 8.410255432 7.751673222 9.219581604
9.651021957 - 9.894903183 -
d = 0.500 gm 1.368159413 2.728582621 1.551661611 3.099793911
F- = 12.25 4.070446014 5.372024536 4.636415958 6.116473675X= 1.550 pm 6.552733421 - 6.791202545 -
Note: d = one-haof thicknessE = dielectric coefficient
= free space wavelength
16
4.U,
*XII ?S1
4. Calculation of Propagation Parameters
Once we have found values for u, we can calculate k1, r, and P. given k131 and Ak (the bandwidth of the source), we can also calculate A3 141 (thebandwidth in the material). We shall first find AP for the TE even mode.Differentiating equations (9), (17), (18), and (19) gives us
ki dk + [3c4 = Ekdk (35)
du = dkl d (36)
dA = dkd'E - 1 (37)
2 A (dA du) - u 3 - u8)sec 2 U = u --- u2 u2 (38)__iu A
Substituting equation (37) for dA in equation (38) and rearranging yield
dAdkdui - k1 sec 2 u(u tan it +1) (39)
There are no unknowns on the right side of the equations. We can nowintroduce equations (35) and (36) and solve for dp3. We get
kd ( -I )(40a)dt3 =~ -sec 2u(u tan u + I)4
For the other solution types, we get
dp=d (=- E - 2uE - 1 TE odd (40b)csc u (u cot u + 1)
3~d k -V 2(. 1 TM even (40c)sec 2 u(u tan u +1)
i, 2d k _-__ _ _( __ kA TM odd (40d)csc2 u(u cot u + 1)
17
'I,If we know the propagation parameters k, k1, r, and 3, we can find expres-sions for E2, E4, H2, and H4 in terms of E1, E3, H1, and H3. Solving equa-tion (14) for E2 yields 0
E2 = E, cos kiderd (41)
Similar manipulations give the other desired relationships:
E4 = E3 sin k, erd (42)
H2 = H! cos k, derd (43)
H 4 = H3 sin k derd (44)
These equations are useful for calculating the electric or magnetic field atany point, which will be done later, and for comparing the energy prop-agated inside the slab to that outside the slab.
The time-averaged energy flux at any point is defined [5] by
<S> =Re(- .ExH) . (45)
In the TE case, EZ is defined by equation (7) inside the slab, and equation(8) outside the slab. Ex and Ey are both zero. Taking H* from equations(3) and (4) gives us
H* = El cos ki x e Y (46)
H= k E2 e-rx e- i py [xi > d (47)
* 0'
-ikiHy =-k El sin ki x CO x < d (48)
iFHy =T E2 erxe- PY jxI>d (49)
Inside the waveguide, equation (45) becomes
<S> = - 2Et cos 2/k x xl < d ,(50)8 n
18
and outside we have
cS> - ---E~ jxj>d (1
We define the efficiency of the waveguide as
dS2 f<S> dx
eff = 0, (52)2 J'S> di
0
The denominator can be expressed as
2 j<S>d = 2 f<S>dx +2 f<S>dx (53)0 0) d
We first calculate the integral from zero to d (energy inside the slab):
2 f <S>dx=~ c E I cos 2 kxdx0 =i k f~
c PE21d ( in 2k, d (54Kik (I+ S2 , I xi<d
and outside the slab we have
2 f<S>dx= c A E2 r' e x =IE -8Ie x>d (55d = 4xenr (55)
Now using equation (41), equation (55) becomes
M OE 2 j~cos2 k, d (6
2 f< dx I8Ink (6
and the efficiency of the waveguide becomes
rd + sin it 57
eff= ri+i 57
19
So
for the TE even modes. Similar calculations yield
eff = rd - sin2uI'd (58)
for the TE odd modes. These calculations can also be done for the TMmodes, giving us
eff = erFd + sin 2 u (9erd + I (59)
for TM even modes, and
erd - sin 2ueff "- d (60)
for the TM odd modes.
0We now have a method for determining the propagation parameters of awaveguide, given its physical properties and the wavelength used. Wecan also determine the frequency bandwidth inside the material for eachmode of operation, given the bandwidth of the incident light. The effi-ciency of the waveguide is also calculable for each mode. Theseparameters are presented in table 3(a through j) for various thicknesses,wavelengths, and indices of refraction.
Also presented are plots of E. (or H, for the TM modes) versus x. Theseplots were calculated using, for the TE even case, equations (7), (8), and(41), and assuming that E, = 1. For the other cases, we used correspond-ing equations and assumed either E3 = 1, H = 1, or H3 = 1. These plotsare figures 3 through 8.
An interesting feature of these plots is the bend in each of the TM curves.This means that there must be a discontinuity in the derivative of H,. Thisis correct and is caused by the discontinous polarization current density at
the surface of the slab. In calculations for the TM modes, EY must be con-tinuous. In this case, EY is defined by
*E 1 HEy =-L a- H, .(27)
20
00
IVr
At x = d, e has a jump discontinuity. For Ey to be continuous, JaaxHzmust also have a jump discontinuity.
Interesting conclusions can also be drawn from table 3. We see that AP islarger than Ak and, in many cases, much larger. We see that it is inverselyproportional to P. We also see that 3, as expected, can have a value fromk to (e)i2k. We can also compare the efficiencies of various modes. Sub-sequent solutions for one mode type are always less efficient. The firstTE even solution is always more efficient than the second TE even solu-tion, and so on. This is true for all four mode types. Also, the TM modesare always more efficient than their corresponding TE modes.
The trends evidenced in the sample data cannot necessarily be generalizedfor all situations.
Table 3. Propagation Parameters for Slab Waveguides
Mode ki F 13 A3 efficiency 0
A. Thickness = 1.00000 pin, e = 4.800, wavelength = 1.06400 gm,bandwidth = 0.00100 pm, k = 59052, and A k = 55.501
TEE 26729 111968 126586 123.47 0.99183TEO 53220 102074 117925 138.91 0.95812TME 30281 111060 125784 125.07 0.99988TMO 60284 98067 114474 137.43 0.99931TEE 79097 83636 102382 142.57 0.90889TEO 103363 50669 77811 286.33 0.68176TME 89241 72714 93672 167.92 0.99667TMO 111714 27773 65257 241.68 0.93811
B. Thickness = 0.10000 gm, c = 12.250, wavelength =0.82000 im,bandwidth = 0.00001 pm, k = 76624, and A k = 0.934
TEE 169806 192919 207579 3.36 0.77780TME 249663 60993 97936 8.95 0.97879
C. Thickness = 0.10000 pm, e = 12.250, wavelength = 1.06400 gm,bandwidth = 0.00010 prm, k = 59052, and A k = 5.550
, TEE 146960 132792 145331 19.23 0.66916TME 196598 24084 63775 62.91 0.87576
21
* X
Table 3. Propagation Parameters ror Slab Waveguide (cont'd)
Mode kirAP efficiency
D. Thickness = 0.10000 pmn, e = 12.250, wavelength = 1.55000 gm,bandwidth =0.00400 pim, k = 40537, and A k = 104.611
TEE 1 14343 73567 83996 324.79 0.48295TME, 135671 8917 41506 1248.57 0.60792
E. Thickness = 0.35000 gm, c = 12.250, wavelength = 0.82000 gm,bandwidth = 0.00001 pim, k = 76624, and A k = 0.934
TEE 73245 246.347 257989 3.35 0.98471TEO 145209 212052 225472 4.31 0.91398TME 88059 241449 253315 3.46 0.99998TMO 175182 188050 203062 4.32 0.99986TEE 213326 143331 162527 4.42 0.80362TME 250183 58822 96599 9.08 0.992100
S F. Thickness =035000 lim, E = 12.250, wavelength = 1.06400 gm,ebandwidth = 0.00010 pm, k = 59052, and A k = 5.550
TEE 69326 185539 194710 20.07 0.971 15 40TEO 136194 143813 155465 33.22 0.81213TME 87465 177710 187265 21.44 0.99996TMO 171500) 99090 115352 34.81 0.99908TEE 192768 45512 74556 27.78 0.47274TME 197994 5415 59300 67.53 0.58392
G. Thickness = 0.35000 gm, e = 12.250, wavelength = 1.55000 gm, .
bandwidth = 0.00400 Pm, k = 40537, and A k =104.611
TEE 62477 120759 127382 382.41 0.93218TEO 118791 66143 77577 3650.06 0.34053 40TME 85959 105343 112874 460.23 0.99981TMO 135486 11391 42107 1236.25 0.80131
6 22
Ix06l l lj 1 llf
Table 3. Propagation Parameters for Slab Waveguide (cont'd)
Mode ki r A13 efficiency
H. Thickness = 1.00000 gim, e = 12.250, wavelength = 0.82000 gim,bandwidth = 0.00001 pin, k = 76624, and A k =0.9.M
TEE 29143 255348 266597 3.29 0.99907 0TEO 58258 250315 261780 3.36 0.99589TME 31216 255103 266362 3.29 1.0()(TMO 62423 249309 260819 3.36 1.00000(TEE 87315 241719 253573 3.43 0.991 18TEO 116274 229199 241668 3.69 0.98214TME 93609 239351 251317 3.49 0.99999TMO 124758 224694 2374(X) 3.69 0.99999TEE 145083 212139 225553 3.79 0.97254TEO 173657 189460 204368 4.50 0.95180TME 155836 204369 218262 4.02 0.99997TMO 186773 176544 192455 4.56 0.99993
*TEE 201845 159092 176583 4.65 0.93112TEO 229286 116102 139108 7.26 0.86289TME 217339 137170 157120 5.58 0.99981TMO 246044 74256 106701 8.22 0.99850TEE 254256 37494 85306 7.07 0.65954TME 256934 6041 76862 11.40 0.80353
1. Thickness = 1.00000 gmtn, E = 12.250, wavelength = 1.06400 gm,bandwidth = 0.00010 pim, k = 59052, and A k =53550
TEE 28526 196(X)3 204706 19.58 0.99808TEO 56994 189691 198670 20.39 0.99127 0
pTME 31156 195602 204322 19.65 1.00000)TMO 62291 188018 197074 20.37 0.99999TEE 85339 178741 188243 20.96 0.98132TEO 113464 162348 172754 24.22 0.95957
VTME, 93376 174677 184389 21.77 0.99998TMO 124349 154170 165W93 24.32 0.99995TEE 141207 138894 150927 25.04 0.93603TEO 168205 104585 120105 38.66 0.86209TME 155033 123270 136685 29.37 0.99986TMO 184392 72323 93370 43.0(X) 0.99906TEE 193020 44431 73900) 39.62 0.70521TME 197898 8205 59620 67.29 0.868 10
23
*lilj I0!
Table 3. Propagation Parameters for Slab Waveguide (cont'd)
Mode 13 Af3 efficiency0
J. Thickness = 1.00000 gim, E = 12.250, wavelength =1.55000 4im,bandwidth = 0.00400 pim, k = 40537, and A k 104.611
TEE 27363 133182 139215 371.33 0.99471TEO -54572 124532 1309~63 407.88 0.97413TME 31033 132375 138443 375.23 1.(K)00TMO 61996 12 1007 127617 407.06 0.99998TEE 81409 108898 116198 424.22 0.94437*pTEO 107441 83323 92661 662.16 0.85012TME 92728 99437 107382 483.76 0.99991
* TMO 122329 59344 71868 722.82 0.99924* TEE 131055 36207 54352 665.58 0.66940
TME 135824 6171 41004 1265.66 0.84028
Figure 3. Electric field(arbitrary units) versusdistance from slab centerfor two even and two oddTE modes, for a slab of1 ltm thickneM. adielectric coefficient of -eI
4 ,a fo l w n f gre a n d 0, t- ~C ~~~wavelength of 1.064 4im. .
Vertical line in this,.nallfolowig fgursrep-
resents slabed.
0.0 02 .0 07 .00 12 .5 :5 20
244
1 SJd ' 1 , 1 1I I 'I 1' I ,I I INRIN mu4
Figure 4. Magnetic field(arbitrary units) versusdistance from slab center E n
TM modes. All param-eters are same as for /figure 3.
0,10
0
0
0.00 0.25 0.50 075 I.'00 .'25 .50 1.' 75 2.00Distance from slab center (;AM)
(a) 1( )
le %
0
U ."
4". _ Eo ..
0.0 6.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
i-'-Distance from slab center 6/aM) Distance from sab center (JM)C(C)
,,,Figure S. Electric field for C :
. ,.. ~ ~TE even mode and magnetic i
., r fied for TM even mode o ,
, (only modes possible) for a !-slab of 0.1 gm thickness and
-,€ a dielectric coeffcient ofe12.2S. Free space wave. e
UCC
_ . length is (a) !.550, (b) 1.064, ." ."and (c) 0.820 gm. E o
0 90
0 .0 0.1 0.2 0.3 0.4 0.5,., DDistance from slab center sam)
F r E t ed
TEeenmdead anei
p fil o M vnmd
-. .~(nl moe possble fo a , . . ... ,
A (a) (b)
(UC Cj
~0 ~ 4
---------
0.0 0. 04 .6 6. 0 0.0 .2 050 0:5 .'0 .25 .'0 .7Ditnefo lbcne Mn
0.00 02, 0.a 0.7i100e.2ct.ri.7
Dreiste alnceh fo lbcne m
mod ab thicknessmoe
0a35 (c)~ dielec, wtroiL
% .
freespac wavlengh o26I
(a) (b)
o to
C- 0c-
0a
0.0 0.2 0.4 0.1. .0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
Distance from slab center (yrm) Distance from slab center (Iam)
Figure 7. Magnetic field (C)_____________
versus distance from slabcenter. Slab has thicknessof 0.35 gim, a dielectric
* coefficient of 12.25, andfree space wavelength of B(a) 1.550 W, for one even ..-mode and one odd mode;Z(b) 1.064 gmti, for two even Imodes and one odd mode; C4 o
and (c) 0.820 gim, for twoeven modes and one odd ________
md.0.0 0.2 07 0.6 0.8 1.0Distance from slab center (mam)
0N N
Figure 8. Highest (fifth)TE even mode for a slabwith a thickness of I gm 0
and a dielectric coeffi- 'cient of 12.25, for a free .
space wavelength or Z "0.820 urm. .
0.0 0.2 0., 0.6 0.8 1.0Distance from slab center (jim)
*E Acknowledgements
Thanks to Clyde Morrison for guidance and direction.
Literature Cited-,'.-..
1. Topics in Applied Physics, Volume 7: Integrated Optics, T. Tamir, ed.,Springer-Verlag, Berlin, Heidelberg, NY (1979), pp 13-81.
2. Handbook of Laser Science and Technology, vol V, part 3, sect. 1.1,M. J. Weber, ed., CRC Press, Inc. (1986), pp 20-26.
3. "Laser Focus," 1986 Buyer's Guide, 21 st Edition, M. R. Levitt, ed./pub. (1986).
4. P. R. Bevington, Data Reduction and Error Analysis for the Physical
P Sciences, McGraw-Hill Book Company (1969).
5. J. D. Jackson, Classical Electrodynamics, John Wiley and Sons, Inc.4. (1975), p 3 4 7 .
28
F0 S
N.
A
S
N,
iJ.
* Appendix A.--FORTRAN Programs Used in Calculating Dielectric Waveguide Characteristics S
-N
p~. -~.
0
0
S S
0 0
I0 S
29
S S
UL. -
CSFORT SLABIC$LINI( SLABICSPURGE SLABI.EXECIDEL SLABI.OBJ;*C0C THIS PROGRAM FINDS THE SOLUTIONS FOR ANC INFINITE PLANAR SLAB WAVEGUIDE WITH DIELECTRICC CONSTANT EPS, THICKNESS THK, AND INCIDENTC WAVE LENGTH WL.C
PROGRAM NEWRAPCHARACTER0I ANSDIMENSION TEE(500),TEO(500),TME(500),TMO(50C)TYPE *,' THIS PROGRAM SOLVES FOR THE'TYPE*,' ROOTS TO OIUR SLKS WAVFGUIDE PROBLEM.'TYPE*,:'
C TYPE, ENTER DIELECTRIC CONSTANT,THICKNESS,WAVE LENGTH'CC THIS FILE CONTAINS THE RUN PARAMETERS. IT IS MADE BY THE USER.C YOU MUST STORE THEM IN THE FILE LIKE THIS:C EPSILON THICKNESS(cm) WAVELENGTH(cm) BANDWIDTH(cm)C
OPEN(UNIT=9,TYPE='OLD' ,NAME='SLABRUN.DAT')
THK=THK/2
C SETRUN CONSTANTSC
EFS=1.0P1=3.141592554A=20PI0THK*(EPS-1)0s.5/WL10=IFIX(A/Pl+.5)
CC IF A IS LESS THEN P1/2 THEN WE HAVE ONLY TWO ROOTS, ONE TE EVENC AND ONE TM EVEN. THERE WILL BE NO ODD ROOTS, SO WE GO TO THEC SPECIAL EVEN ONLY ROUTINE. ~C
IF(A.LT.PI/2)GOTO 123100 FORMAT(18x,4F16.9)CC START LOOP, AND SET INITIAL GUESSES FOR THE NEWTON-RAPHSONC ROUTINEC
1=1SE=PI/2 .0-THKSEI=SESO=PI-THK
CC IF OUR GUESS IS GREATER THAN A, THE PROGRAM WILL CRASH. SO WEC INSURE THAT OUR GUESSES ARE LESS THAN A.
IF (SO .GT. A)50=A-THKSO1=SO
C USE SUBROUTINES TO FIND ROOTSC EPS IS EPSILON, EFS IS EPSILON OF FREE SPACECC THERE SHOULD BE ONE SOLUTION OF EACH TYPE - TWO ODD, TWO EVEN,C OF Th AND TE MODES. 6
89 C10 IF (SO GE. A) SO=-A-THK
IF (SOl GE. A) S01=SO* CALL ODD(TEO(I),SO,EFS,A)0
CALL ODD(TMO(I).SO1,EPS,A)CALL EVEN(TEE(I),SE,EFS,A)CALL EVEN(TME(I),SEI,EPS,A)
C
C SET UPPER AND LOWER LIMITS FOR EACH INDIVIDUAL MODEC
'P AI=PIO(201+1)/2.O* AJ=PO(2:1-1 )/2.0
* C AL=PO(2 1- )/2.0
31
CE I' II
C CHECK THAT ROOT IS WITHIN THE SET LIMITSC THAT IS, CHECK THAT NO ROOTS ARE SKIPPED, OR REPEATED.C
IF (TEo(z) .GT. At .OR. TMO) .GT. Al) IFLG=-IFLGIlIF (TEE(I) .GT. AJ .OR. TMdE(I) .GT. AJ) JFLG=-JFLG+1IF (TEO(I) .LT. AJ .OR. TldO(I) ALT. AJ) KFLG=-KFLG+lIF (TEE(I) .LT. AL .OR. TME(I) ALT. AL) LFLG=-LFLG+l
CTYPE 100,TEE( I) ,TEO( I) .TME( I) ,TMO(I)
123 1=1+1CC CHECK TO SEE IF WE'RE DONE.C
IF (I .LE. 10) GOTO 10CC CHECK TO SEE IF WE HAVE ANOTHER PAIR OF EVEN ROOTS.C
IF (A-(I-1)*PI .LE. 0) GOTO 987CC FIND PAIR OF EVEN ROOTS
SE=A-THKSEI1=SE+THK/2
C TYPE0,I00PISE,ACALL EVEN(TEE(I),SE,EFS,A)CALL EVEN(TJJE(I),SEI,EPS,A)IXFL=l
CC IF ANY OF THE FLAGS BELOW ARE NOT ZERO, A MODE WAS SKIPPED
*C OR REPEATED,C987 TYPE *,IFLG,JFLG,KFLG,LFLG
200 FORMAT(Al)
C
OPEN (UN I T=2 ,TYPE-' NEW' ,NAMdE= 'ALROOT. EZG')OPEN (UNIT=1 ,TYPE-'NEW' ,NAMdE='ALROOT .DAT')
*WRITE(,*) 'DO 20 1=1,10,1
WRITE(1,100) TEE(I),TEO(I),TME(I),TMO(I)CC CALCULATE THE VALUES FOR TH1E PLOTTING OF THE INTERSECTIONC OF THE TRANSCENDENTAL EQUATIONS.C
CN=-1 .O/TAN(TEO( I))CNl=-1 .O/TAN(TMO( I))TN=TAN(TEE( I))TNI=TAN(T7k( ))
CC WRITE TO EZG FILE
CRT(,) E()T* WRITE(2,O) TE(I),TN
WRITE(2,') TME(I).TNIWRITE(2,O) TMO(I),CNl
20 CONTINUEIF (IXFL .NE. 1) GOTO 789 (~
C WRITE(2,0) 0.0,99999.WRITE(1,234) TEE(I),TME(I)
*234 FORMAT(16X,F1.9,' '.F189)TN=TAN(TEE( I))WRITE(2.0) TEE(I),TNTN=-TAN(TME(I))WRITE(2,O) 0.0,99999.WRJTE(2.0) TME(I),TN
789 STOPEND
* SUBROUTINE ODD(F,X,E,A)
C THIS SUBROUT INE FINDS THE ROOTS OF ODD SOLUTION MODES
32
C DEPENDING ON E, IT WILL SOLVE FOR TMO, OR TMlEC
DOUBLE PRECISION Q,F2,F3,D1320 Q=1/(AA-X*X)**.5C TYPEv,X,Dl
F2=TAN(X)+X*Q/EF3=COS(X)*' (-2)+Q/E+XOXOQ*03/ED1=-F2/F3IF (ABS(D1) .E. .00001)GOTO 310 .X=X+DlGOTO 320
310 F=XX=X+3. 141592554RETURNENDSUBROUTINE EVEN(F,X,E,A)
CC THIS SUROUTINE FINDS THE ROOTS FOR EVEN SOLUTION MODESC
DOUBLE PRECISION Q,F2,F3,DI420 q=-(A*A-X*X).*.5C TYPE*,XDl
F2=TAN(X)-Q-E/XF3=COS(X)00(-2 )+Q*E/(X*X)+E/QD1=-F2/F3IF (ABS(DI) .E. .00001) GOTO 410X=D1+XGOTO 420
* 410 F=XX=X+3. 141592554RETURN
END
33
CSFORT SLAB2C$LINK SLAB2C$DEL SLAB2.OBJ;*C$PURGE SLAB2.EXEC0C THIS PROGRAM CALCULATES THE PROPAGATION PARAMETERS FOR THEC INDIVIDUAL SOLUTIONS TO THE SLAB WAVEGUIDE PROBLEM.
PROGRAM BETAWDCHARACTER*-i MD(4)REAL K,K1,EFF(4)DATA P1/3.1415926536/
CC THESE CHARACTER STRINGS REPRESENT THE FOUR DIFFERENT MODE TYPES.
MDi= EMD(2)=' TEE
MD(3)=' TMEMD(4)=' TMO
CC OPEN THE RUN PARAMETER FJLE(CREATED BY THE USER).
OPEN(UNIT=QTYPE='OLD' ,NAME='SLABRUN.DAT')C
READ(9,-) EPS,THX.WL,DW4 ' DK=2*Pl*DW/(WL*92-DW*02)
D=-THK/2K=2*PI/WL
C OPEN SOLUTION DATA FILE, CREATED BY SLABI PROGRAM)
* ~OPEN (UNIT=2 ,TYPE=' OLD ,NAME='ALROOT. EZG')0CC OPEN OUTPUT FILE FOR PROPAGATION PARAMETERS, ANDC WRITE FILE HEADER
OPEN (UNIT=1,TYPE='NEW' ,NAME='PARAMS.DAT')WRITE(l,*)' Table . Propogation Parameters for slab waveguide
+ with'WRITE(1,0)WRITE(1,10) THK10000,EPS
10 FORMAT(' thiclcness=',F7.5.' urn',' epsilon=',F7.3)WRITE(1,12) WL*1000.DW*1000
12 FORMAT(' wavelength=',F7.5,' urn',' bandwidth,+F7.5,* urn')
WRITE(l,14) K,dk14 FORMAT(' k'-,F7.0,' delta k=-',F7.3)
write(l,*)'
WRITE( I.)WRITE(l,0) 'mode k gamma beta
+ delta beta efficiency
+write(l,*)'
999 FORMAT(A5,3FI0.0,F11.2,F13.5)* C
C MODE TYPE IS KEPT TRACK OF WITH MTP( 0 BEFORE PROGRAM STARTS)MTP=-O
100 READ(2,*,END=-777) UDUMMYMTP=-MTP+lIF (MP .GT. 4) MTP=-MTP-4IF(U.EQ. 0) GOTO 100
C*C CALCULATE KI,GAMMA
K I=U/DGAN=(K*K*(EPS-I )-K1*Kl )ee.5
CC CALCULATE EFFICIENCY AND BETA "B"
B=(GAM**2+K*K)**.5SU-S I N(U)CU=COS( U)SC=SUCUG~AM*D
GED*GEPSEFF( I)=(GD4SU*02)/(GD+1)
34
EFF(2)=(GD-SU*02)/GDEFF(3)=(GED+SU@*2)/(GED+I)
C EFF(4 )=(GED-SUO*2)/GEDC CALCULATE THE BANDWIDTH OF BETA
CSU2=1/SU*02SCU2=1/CU..2tu~su/cu
S ct~1/tu
IF (MTP .GT. 2) Q=1/EPS'02if (Mtp .eq. 20int(uitp/2)) goto 987Sl~scu2
* f2=tu97 goto 98698 flcsu2
I 2=c t986 db=kdk(eps-qV(eps-l)/(fi6(uf2+1)))/bCC WRITE TO OUTPUT FILE
WRITE(1,999) MD(MTP),K1,GAMd,B,DB,EFF(MTP)
GOTO 100777 ivrite(l,*)'
N CLOSE(UNIT--i)STOP
* END
mlmSm
C$FORT SLAB3C$LIN( SLA133CSDEL SLAB3.OBJ;*CSPURGE SLAB3.EXECC TH IS PROGRAM CALCULATES THE VALUES OF THE CURVES REPRESENTEDC BY THE FOUR TRANSCENDENTAL EQUATIONS FOR OUR SLAB WAVEGUIDE
* MC PROBLEM. THE FOUR CURVES NEED)ED ARE:CC FI(U)=TAN(U) F2(U)=-COT(U) F3(U)=(A*A-U*U)*S F4(U)=EPSOF(3)
a CPROGRAM RTGRADIMENSION F(4)DATA U,WL,PI/O.O, .000108,3.1415928538/
* CC OPEN RUN PARAMETFR FILE (CREATED BY THE USER) AND READ PARAMETERS
OPEN(UNIT=9,TYPE='OLD ,NAME='SLABRUN.DAT)READ(9,-) EPS,THKWL,BW
CC CALCULATE D(HALF THICKNESS) AND A(DESCRIBED IN PAPER)
D=-THK/2A=2*PI0D(EPS-1 ).e.5/WLN=ODU-PI/2000.
CC OPEN FILE FOR OUTPUT OF CURVES. THE OUTPUT IS OF THE FORM:C U F(1) F(2) F(3) F(4)C9 OPEN(UNIT=1,TYPE='NW' ,NAME='SMROOT.DAT')CC BEGIN CURVE CALCULATING LOOP
DO 10 1=0,999CC WE CALCULATE IN BLOCKS OF P1/2 TO AVOID A VERTICLE LINE WHEN'4"C WE PLOT THE TRIGONOMETRIC CURVES WITH EZGRAPH.
U=I DU+NPI/2.IF(U .GT. A) GOTO 11IF (INT(2*U/PI) .EQ. 2@U/PI) GOTO 10F( 1)=TAN(U)F(2)=-1/F( 1)F(3)=(A*A-U*U)00.5/UF(4)=EPSOF(3)DO 20 J=1,4IF(ABS(F(J)) .LT. 100) GOTO 20
C FOR PLOTTING PURPOSES, WE DONT WANT ANY OF THE FUNCT IONS VERY HIGHC IT DOESN'T MATTER MUCH EITHER WAY
F(J)=99.9999*ABS(F(J) )/F(J)920 CONTINUE
g WRITE(1,100) U,F(1),F(2),F(3),F(4)10 CONTINUE100 FORMAT(5F15.1O)
N=N+l11 CLOSE(UNIT=1)
IF(U .LT. A )GOTO 9STOPEND
36
*§
CSFORT SLAB4C$LINK SLAB4CSDEL SLAB4.OBJ;*CPURGE SLAB4.EXEC0C THIS PROGRAM CALCULATES THE VALUES OF THE ELECTRICC OR MAGNETIC FIELDS FOR ANY X, INSIDE OR OUTSIDE THE
a C SLAB. IT ASSUMES A MAXIMUM POSSIBLE FIELD OF ONE, ORC El=I, AND Y=2*PI*n.
PROGRAM MODEPLREAL K,KIDATA PI/3.1415926536/COMMON U.D,K,KI ,B,GAM,EGD,EPSCU,SU,EO
CC OPEN FILE CREATED BY USER WITH RUN PARAMETERS
OPEN(UNIT=9,TYPE= 'OLD' ,NAME=' SLABRUN.DAT')READ(9, ) EPS,THK,WL,BW
CC CALCULATE PARAMETERS USED FOR FINDING FIELDS
D-THK/2K=2*PI/WL
C
C OPEN FILE WITH SOLUTIONS TO TRANSCENDENTALS,CONTAINING VALUES FOR U _C THIS FILE WAS CREATED BY SLABi
OPEN (UNIT=2,TYPE='OLD',NAME='ALROOT.EZG')
C KEEP TRACK OF MODE TYPE WITH MTPMTP=-- !
100 READ(2,*,END=-777) U,DUMMYMTP=MTP+1IF(U.EQ. 0) GOTO 100CU=-COS(U)SU=SIN(U)
Cu os
C CALCULATE PROPAGATION PARAMETERSK1=U/DGAM=(K*K*(EPS-l )-KI*Kl )**.5B.=(GAM*02+K*K)* .5EGD=-EXP(GAM*D)
'a' CC OPEN FILE FOR OUTPUT
OPEN(UNIT=1,TYPE='NEW' ,NAME=SMMODE.DAT')DX=10.OD/1000.
V. CC LOOP TO CALCULATE FIELDS
DO 200 1=0,1000X=I 'DXEO=MTP-2 IFIX(FLOAT(MTP)/2.0)CALL MODCLC(X,V)WRITE(l,*)X*10000.V
200 CONTINUECLOSE(UNIT=i)GOTO 100
777 STOPENDSUBROUTINE MODCLC(X,V)
CC THIS SUBROUTINE CALCULATES THE FIELD DEPENDING ON WHICH TYPE OFC IS BEING PLOTTED
REAL K,KICOMMON U,D,K,KI ,B,GAM,EGD,EPSCU,SU,EO
C CHECK FOR INSIDE OR OUTSIDE OF SLABIF (X .LT. D) GOTO 9
37
Mm WE
CC CHECK FOR EVEN OR ODD MODE .
IF(EO) 2,2,12 AM--CU
GOTO 31 Ald=SU3 V:=AMOEGD*EXP(-~XOGAM)
GOTO 4
9 IF (EO) 6,8.5a V=-COS(K1*X)
GOTO 4 16
5 V=SIN(KIOX)4 RETURN
END
43
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DISTRIBUTION (cont'd)
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ATTN LIBRARY, R-51, TECH RPTS DEPARTMENT OF NUCLEAR ENGINEERING
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* ATTN L. EYRING PRINCETON UNIVERSITYTEMPE, AZ 85281 DEPARTMENT OF CHEMISTRY
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ATTN DR. F. S. RICHARDSON (2 COPIES)KALAMAZOO COLLEGE ATTN DR. M. REID
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HARRY DIAMOND LABORATORIES HARRY DIAMOND LABORATORIES(cont 'd) (cont'd)
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