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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(linalg)
:with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected n
ames norm and trace have been redefined and unprotected\n" }}{PARA 7 "
" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined
\n" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 27 "Fresnel Kummer RAY Surface
s" }}{PARA 19 "" 0 "" {TEXT -1 32 "fresnel.mws   R.M.Kiehn 11/05/97" }
}{PARA 256 "" 0 "" {TEXT -1 16 "updated 12/23/99" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 339 "This program will compute and plot the Fresnel
 RAY surfaces for a generalized Hermitean constitutive matrix.  The su
rfaces are specializations of a Kummer quartic surface, and are relate
d to the Clifford algebra Cl(3,3). Specialized forms have been selecte
d for easy visualization.  The formulas can be modified to handle the \+
general case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "RAYS ARE IN THE DIRECTION OF THE ENERGY FLUX" }}{PARA 0 "
" 0 "" {TEXT -1 48 "WAVES ARE IN THE DIRECTION OF THE MOMENTUM FLUX." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 91 "The constitutive tensor is a 6 x 6 comple
x hermitian  matrix partitioned into 3x3 matrices." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "constitutiv
e_tensor:=matrix([[epsilon,gamma],[conj(gamma),1/mu]]);" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4constitutive_t
ensorG-%'matrixG6#7$7$%(epsilonG%&gammaG7$-%%conjG6#F+*&\"\"\"F1%#muG!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 242 "The on-diagonal upper 3x3 matrix is the epsilon \+
matrix. The epsilon matrix real part describes birefringence. The lowe
r on-diagonal 3x3 matrix is the reciprocal mu matrix. The complex part
 of the mu matrix represents magnetic Faraday effects." }}{PARA 0 "" 
0 "" {TEXT -1 596 "  The off-diagonal  3x3 matrix has a real part whic
h represents Fresnel-Fizeau effects, and a complex part which represen
ts optical activity.  Typical entries are shown below.  The elements h
ave been scaled such that the \"position vector\" to a fresnel RAY sur
face point in this space has a value equal to the reciprocal of the \"
effective\" index of refraction in that direction.  The ray speed (or \+
group velocity)  in that direction is c/n, where c is defined as 1/sqr
t(epsilon*mu).  The ray speed should be viewed as at the speed of the \+
energy  flux (ExH) in the direction of the position vector." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "For the Fresne
l WAVE surface which defines the propagation speed of the momentum flu
x (DxB), usually described as the \"phase\" velocity,  see" }}{PARA 0 
"" 0 "" {TEXT -1 45 "http://www22.pair.com/csdc/maple/fresnel.zip " }}
{PARA 0 "" 0 "" {TEXT -1 195 "Note that when the media is dispersive, \+
the energy flux and the momentum flux do not propagate at the same spe
ed, but the product of the group speed times the phase speed is always
 1/(epsilon.mu)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" 
{TEXT -1 33 "The epsilon (permittivity) matrix" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 72 "eps:=(matrix([[A*epsilon,I*fd,0],[-I*fd,B*epsi
lon,0],[0,0,C*epsilon]]));" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$epsG-%'matrixG6#7%7%*&%\"AG\"\"\"%(epsilo
nGF,*&^#F,F,%#fdGF,\"\"!7%*&^#!\"\"F,F0F,*&%\"BGF,F-F,F17%F1F1*&%\"CGF
,F-F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "The real (symetric) par
t of the epsilon matrix (above) is the permittivity matrix and leads t
o bifrefringence;" }}{PARA 0 "" 0 "" {TEXT -1 70 " the complex antisym
metric part represents dielectric Faraday effects." }}{PARA 18 "" 0 "
" {TEXT -1 43 "The Gamma matrix (Real and Imaginary Parts)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "GR:=matrix([[0,r,0],[-r,0,0],[0,0,0
]]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#GRG-%'matrixG6#7%7%\"\"!%\"rGF*7%,$F+!\"\"F*F*7%F*F*F*" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The real part of the off-diagonal \+
gamma matrix is given above." }}{PARA 0 "" 0 "" {TEXT -1 97 "The Fresn
el-Fizeau and Sagnac effect is contained the real part of the off diag
onal gamma matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "GI:=matrix([[I*g,0,I*p],[0,I*g,0],[I*p,0,I*
g]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GIG-%'matrixG6#7%7%*&^#\"
\"\"F,%\"gGF,\"\"!*&F+F,%\"pGF,7%F.F*F.7%F/F.F*" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 84 "The optical activity is due to the complex part of t
he off diagonal submatrix above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "G:=evalm(GR+GI);GA:=evalm(tr
anspose(+GR+GI));" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"GG-%'matrixG6#7%7%*&^#\"\"\"F,%\"gGF,%\"rG*&F+F,%
\"pGF,7%,$F.!\"\"F*\"\"!7%F/F4F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#GAG-%'matrixG6#7%7%*&^#\"\"\"F,%\"gGF,,$%\"rG!\"\"*&F+F,%\"pGF,7%F/F*
\"\"!7%F1F4F*" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 51 "The complex con
jugate transpose of the Gamma Matrix" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "GD:=evalm(transpose(GR)-transpose(GI));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#GDG-%'matrixG6#7%7%*&^#!\"\"\"\"\"%\"gGF-,$%
\"rGF,*&F+F-%\"pGF-7%F0F*\"\"!7%F1F4F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 74 "The Gamma matrix need not be Hermitian.  It can be anti-h
ermitian as well." }}{PARA 18 "" 0 "" {TEXT -1 54 "The inverse mu matr
ix (reciprocal permeability matrix)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "mmu:=matrix([[a/(mu),I*fm,0],[-I*fm,b/(mu),0],[0,0,c/
(mu)]]);" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$mmuG-%'matrixG6#7%7%*&%\"aG\"\"\"%#muG!\"\"*&^#F,F,%
#fmGF,\"\"!7%*&^#F.F,F1F,*&%\"bGF,F-F.F27%F2F2*&%\"cGF,F-F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The real (symetric) part of the mu
 matrix is the magnetic birefringence part;" }}{PARA 0 "" 0 "" {TEXT 
-1 67 "the complex antisymmetric part represents magnetic Faraday effe
cts." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "muinv:=inverse(mmu)
;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&muinvG-%'matrixG6#7%7%,$*(%\"bG\"
\"\"%#muGF-,&*&%\"aGF-F,F-!\"\"*&)%#fmG\"\"#F-)F.F6F-F-F2F2**^#F-F-F5F
-F.F6F/F2\"\"!7%**^#F2F-F5F-F.F6F/F2,$*(F1F-F.F-F/F2F2F:7%F:F:*&F.F-%
\"cGF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 188 "The matrices are now s
caled for ease of computation. Values of n = [x.y.z]  > 1 imply a phas
e velocity less than c =1/sqrt(epsilon*mu).   Values of n < 1 imply ph
ase speeds greater than c." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
18 "" 0 "" {TEXT -1 53 "The N matrix (Inverse \"Index\" of Refraction \+
Operator)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "cc:=simplify(
(epsilon)^(1/2)*(mu)^(1/2),sqrt);N:=matrix([[0,z/cc,-y/cc],[-z/cc,0,x/
cc],[y/cc,-x/cc,0]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ccG*&-%%s
qrtG6#%(epsilonG\"\"\"-F'6#%#muGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"NG-%'matrixG6#7%7%\"\"!*(%\"zG\"\"\"%(epsilonG#!\"\"\"\"#%#muGF/,$
*(%\"yGF-F.F/F2F/F07%,$F+F0F**(%\"xGF-F.F/F2F/7%F4,$F8F0F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 319 "The N matrix above is the scaled \"index
 of refraction matrix\" which acts like a cross product operator.  The
 N matrix has three components which form the position vector to the K
ummer surface.  The magnitude of N is the \"index\" of refraction, n, \+
in the direction of the vector N. The phase speed is then related to 1
/n." }}{PARA 0 "" 0 "" {TEXT -1 99 " The elements of the constitutive \+
matrix have been scaled (below) for algebraic reduction purposes." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 145 "fd:=fda*epsilon;p:=pa*epsilon^(1/2)/(mu)^(1/2);g:=ga*epsilon^(1
/2)/(mu)^(1/2);r:=ra*epsilon^(1/2)/(mu)^(1/2);fm:=fma/mu;fd:=0;ra:=0;;
fm:=0;pa:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#fdG*&%$fdaG\"\"\"%(epsilonGF'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG*(%#paG\"\"\"%(epsilonG#F'\"\"#%#muG#!\"\"F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*(%#gaG\"\"\"%(epsilonG#F'\"\"#%
#muG#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG*(%#raG\"\"\"%(e
psilonG#F'\"\"#%#muG#!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fmG
*&%$fmaG\"\"\"%#muG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fdG\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#raG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#fmG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#paG\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Now compute the various ma
trices whose determinant create the Kummer equation." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 72 "eps:=(matrix([[A*epsilon,I*fd,0],[-I*fd,B
*epsilon,0],[0,0,C*epsilon]])):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "
M:=innerprod(N,eps):MM:=innerprod(M,N):" }}{PARA 18 "" 0 "" {TEXT -1 
30 "The Fresnel LAGRANGIAN  Matrix" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "HHH:=-evalm(mmu)+innerprod(G,N)+innerprod(N,-GD)-eval
m(MM);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$HHHG,(-%'matrixG6#7%7%*&%
\"aG\"\"\"%#muG!\"\"*&^#F-F-%#fmGF-\"\"!7%*&^#F/F-F2F-*&%\"bGF-F.F/F37
%F3F3*&%\"cGF-F.F/F/*&\"\"#F--F'6#7%7%F3**F1F-%#gaGF-F.F/%\"zGF-**F6F-
FCF-F.F/%\"yGF-7%**F6F-FCF-F.F/FDF-F3**F1F-FCF-F.F/%\"xGF-7%**F1F-FCF-
F.F/FFF-**F6F-FCF-F.F/FJF-F3F-F--F'6#7%7%,$*&,&*&)FDF=F-%\"BGF-F-*&)FF
F=F-%\"CGF-F-F-F.F/F/**FFF-F.F/FZF-FJF-**FDF-F.F/FWF-FJF-7%Fen,$*&,&*&
FVF-%\"AGF-F-*&)FJF=F-FZF-F-F-F.F/F/**FDF-F.F/F\\oF-FFF-7%FfnF_o,$*&,&
*&FYF-F\\oF-F-*&F^oF-FWF-F-F-F.F/F/F/" }}}{EXCHG {PARA 18 "" 0 "" 
{TEXT -1 37 "The Kummer Fresnel Quartic Polynomial" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 34 "LAGRANGIAN:=factor(det(HHH)*mu^3);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+LAGRANGIANG,H*(%\"aG\"\"\"%\"bGF(%
\"cGF(!\"\"**)%\"zG\"\"#F(%\"BGF(F)F(F*F(F(*,F-F(F0F(F)F()%\"yGF/F(%\"
AGF(F+**)F.\"\"%F(F0F(F4F(F*F(F+**F2F(%\"CGF(F)F(F*F(F(**)F3F7F(F9F(F)
F(F4F(F+**F'F(F)F(F2F(F4F(F(**F'F(F)F()%\"xGF/F(F0F(F(**F'F(F-F(F4F(F*
F(F(**F'F(F>F(F9F(F*F(F(**F'F()F?F7F(F9F(F0F(F+**F7F()%#gaGF/F(F-F(F*F
(F(**F7F(FEF(F2F(F)F(F(*,F-F(F0F(F>F(F9F(F*F(F+*,F2F(F9F(F)F(F>F(F0F(F
+*,F2F(F9F(F-F(F4F(F*F(F+*,F'F(F-F(F4F(F>F(F0F(F+*,F'F(F>F(F9F(F2F(F4F
(F+**F7F(F'F(FEF(F>F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 491 "Now c
hoose values for the matrix elements. A,B,C are numeric factors times \+
epsilon.  a,b,c are numeric factors times mu.  The Optical Activity pa
rt is scaled by the impedance of \"free space\" or sqrt(epsilon/mu).  \+
The algebraic method presented permits symbolic factorization.  The di
electric Faraday fd is scaled by epsilon.  The magnetic faraday is sca
led by mu. The example below is for a chiral Optical Activity coeffici
ent gamma of  sqrt(2)/2.  Note that the phase velocity of one of the" 
}}{PARA 0 "" 0 "" {TEXT -1 82 " polarization states is faster than the
 speed of light and the other is slower.!!!" }}{PARA 0 "" 0 "" {TEXT 
-1 102 "Select the effects to be studied algebraically by elimating al
l effects but one or two of the scalars," }}{PARA 0 "" 0 "" {TEXT -1 
16 "fma,fda,ra,pa,ga" }}{PARA 0 "" 0 "" {TEXT -1 94 "The example studi
es optical activity algebraically, by setting all factors to zero, exc
ept ga:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "LAGR:=subs(fma=0,fda=0,pa=0,ra=0,LAGRANGIAN);eval(%);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%LAGRG,H*(%\"aG\"\"\"%\"bGF(%\"c
GF(!\"\"**)%\"zG\"\"#F(%\"BGF(F)F(F*F(F(*,F-F(F0F(F)F()%\"yGF/F(%\"AGF
(F+**)F.\"\"%F(F0F(F4F(F*F(F+**F2F(%\"CGF(F)F(F*F(F(**)F3F7F(F9F(F)F(F
4F(F+**F'F(F)F(F2F(F4F(F(**F'F(F)F()%\"xGF/F(F0F(F(**F'F(F-F(F4F(F*F(F
(**F'F(F>F(F9F(F*F(F(**F'F()F?F7F(F9F(F0F(F+**F7F()%#gaGF/F(F-F(F*F(F(
**F7F(FEF(F2F(F)F(F(*,F-F(F0F(F>F(F9F(F*F(F+*,F2F(F9F(F)F(F>F(F0F(F+*,
F2F(F9F(F-F(F4F(F*F(F+*,F'F(F-F(F4F(F>F(F0F(F+*,F'F(F>F(F9F(F2F(F4F(F+
**F7F(F'F(FEF(F>F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,H*(%\"aG\"\"
\"%\"bGF&%\"cGF&!\"\"**)%\"zG\"\"#F&%\"BGF&F'F&F(F&F&*,F+F&F.F&F'F&)%
\"yGF-F&%\"AGF&F)**)F,\"\"%F&F.F&F2F&F(F&F)**F0F&%\"CGF&F'F&F(F&F&**)F
1F5F&F7F&F'F&F2F&F)**F%F&F'F&F0F&F2F&F&**F%F&F'F&)%\"xGF-F&F.F&F&**F%F
&F+F&F2F&F(F&F&**F%F&F<F&F7F&F(F&F&**F%F&)F=F5F&F7F&F.F&F)**F5F&)%#gaG
F-F&F+F&F(F&F&**F5F&FCF&F0F&F'F&F&*,F+F&F.F&F<F&F7F&F(F&F)*,F0F&F7F&F'
F&F<F&F.F&F)*,F0F&F7F&F+F&F2F&F(F&F)*,F%F&F+F&F2F&F<F&F.F&F)*,F%F&F<F&
F7F&F0F&F2F&F)**F5F&F%F&FCF&F<F&F&" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 
-1 41 "Reduced Fresnel Kummer quartic polynomial" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 13 "HAMRED:=LAGR;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%'HAMREDG,H*(%\"aG\"\"\"%\"bGF(%\"cGF(!\"\"**)%\"zG\"\"#F(%\"BG
F(F)F(F*F(F(*,F-F(F0F(F)F()%\"yGF/F(%\"AGF(F+**)F.\"\"%F(F0F(F4F(F*F(F
+**F2F(%\"CGF(F)F(F*F(F(**)F3F7F(F9F(F)F(F4F(F+**F'F(F)F(F2F(F4F(F(**F
'F(F)F()%\"xGF/F(F0F(F(**F'F(F-F(F4F(F*F(F(**F'F(F>F(F9F(F*F(F(**F'F()
F?F7F(F9F(F0F(F+**F7F()%#gaGF/F(F-F(F*F(F(**F7F(FEF(F2F(F)F(F(*,F-F(F0
F(F>F(F9F(F*F(F+*,F2F(F9F(F)F(F>F(F0F(F+*,F2F(F9F(F-F(F4F(F*F(F+*,F'F(
F-F(F4F(F>F(F0F(F+*,F'F(F>F(F9F(F2F(F4F(F+**F7F(F'F(FEF(F>F(F(" }}}
{EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "Set anisotropic coefficients:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "HAMA:=subs(A=2,B=3,C=1,a=1,
b=1,c=1,HAMRED):eval(%):HAMA:=factor(HAMA);ga:=2^(1/2)/2;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%%HAMAG,<!\"\"\"\"\"*&\"\"&F')%\"zG\"\"#F'F'*(
\"\")F'F*F')%\"yGF,F'F&*&\"\"'F')F+\"\"%F'F&*&\"\"$F'F/F'F'*&F,F')F0F4
F'F&*&F4F')%\"xGF,F'F'*&F6F')F;F4F'F&*(F4F')%#gaGF,F'F*F'F'*(F4F'F?F'F
/F'F'*(\"\"*F'F*F'F:F'F&*(F)F'F:F'F/F'F&*(F4F'F?F'F:F'F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#gaG,$*$-%%sqrtG6#\"\"#\"\"\"#F+F*" }}}{EXCHG 
{PARA 18 "" 0 "" {TEXT -1 43 "Set the numeric values for the coefficie
nts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "HAM:=factor(subs(fma
=0,fda=0,ra=0,pa=0,HAMA));eval(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$HAMG,6!\"\"\"\"\"*&\"\"(F')%\"zG\"\"#F'F'*(\"\")F'F*F')%\"yGF,F'F&*
&\"\"'F')F+\"\"%F'F&*&\"\"&F'F/F'F'*&F,F')F0F4F'F&*&F2F')%\"xGF,F'F'*&
\"\"$F')F;F4F'F&*(\"\"*F'F*F'F:F'F&*(F6F'F:F'F/F'F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,6!\"\"\"\"\"*&\"\"(F%)%\"zG\"\"#F%F%*(\"\")F%F(F%)%
\"yGF*F%F$*&\"\"'F%)F)\"\"%F%F$*&\"\"&F%F-F%F%*&F*F%)F.F2F%F$*&F0F%)%
\"xGF*F%F%*&\"\"$F%)F9F2F%F$*(\"\"*F%F(F%F8F%F$*(F4F%F8F%F-F%F$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Coefficients were chosen to displ
ay chiral effect due to optical activity on top of biaxial birefringen
ce" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "KUMM:=factor(HAM);LOR
ENTZ:=(x^2+y^2+z^2)^2-1;factor(KUMM-LORENTZ);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%KUMMG,6!\"\"\"\"\"*&\"\"(F')%\"zG\"\"#F'F'*(\"\")F'F
*F')%\"yGF,F'F&*&\"\"'F')F+\"\"%F'F&*&\"\"&F'F/F'F'*&F,F')F0F4F'F&*&F2
F')%\"xGF,F'F'*&\"\"$F')F;F4F'F&*(\"\"*F'F*F'F:F'F&*(F6F'F:F'F/F'F&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(LORENTZG,&*$),(*$)%\"zG\"\"#\"\"\"
F-*$)%\"yGF,F-F-*$)%\"xGF,F-F-F,F-F-F-!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,4*$)%\"zG\"\"#\"\"\"\"\"(*(\"#5F(F%F()%\"yGF'F(!\"\"*&
F)F()F&\"\"%F(F.*&\"\"&F(F,F(F(*&\"\"$F()F-F1F(F.*&\"\"'F()%\"xGF'F(F(
*&F1F()F:F1F(F.*(\"#6F(F%F(F9F(F.*(F)F(F9F(F,F(F." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 44 "Now for an x=0 section to get more detail.  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "KUMMX:=subs(x=0,KUMM);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&KUMMXG,.!\"\"\"\"\"*&\"\"(F')%\"zG
\"\"#F'F'*(\"\")F'F*F')%\"yGF,F'F&*&\"\"'F')F+\"\"%F'F&*&\"\"&F'F/F'F'
*&F,F')F0F4F'F&" }}}{EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "X=0 section o
f Fresnel Kummer RAY Vector Surface" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 117 "implicitplot(KUMMX=0,y=-2.0..2.0,z=-2..2,numpoints =
 5000,scaling=CONSTRAINED,title=`OPT ACT. + Biref ga=sqrt(2)/2`);" }}
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" 0 "" {TEXT -1 85 "It would appear that the propagation velocity (gro
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-1 38 "3-D Plot of Fresnel RAY Vector Surface" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 210 "implicitplot3d(KUMM=0,x=-0..2,y=-2..2,z=-2..2,s
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