ct

C'("'u~tc\ Clc.~lfic.c.-tf(.l(\S=::::~:E-::==:==========-

-r~tro.( 0. 00 1-~::::~~~~ 'X:::: 0 a. 0 'n.= fi-;;'

c'vb\t. () 0 Q

~~~pl~~: ~lum WIOfJ~.> di~YI\onct

\J'()\axi~\ [0.. 0 01-tf\~O~\ x:: 0 0. 0 'no= ~ ~E'7 1)0 1~(~v-e.JI

.-te~o~()(\c\ 0 0 b 'nE= Fb ~E"<nD \lnoftJa.tf;~'1

A\"\I~'c \" \I ,eX ~O'1'\2. ~"1Tve.. ~afI vc.r-- -, ,-- -.'Y1o 'T1f'"

--e~a.~pl~.. ,\"c~) c:rl.la~) cqICife:.) tovrrY\~l;t'\(.. 'l'ct. I.'j~ I.~LO

(Aid!!. 1.(q5'8 1.~8(q

"~ '6 I",. .,fit: \a.'X,IQ. \Q 0 01 r;-:-: '}1 , ~ ~ I +Q

-tric,\inic. X:: 0 b ~ 1) -1T+b.I 0 0 C, 1-

mOY)Qc.I/~Ic. 'Y1o:: F

~~~'*ti~.--C)'('fhorhovnb,'c,

~1-o~\~ ~ G~psv~) MI'Ca..) fOf)o2..

dn a'("b/-hc~ dl'r~ 1~n ' c2..

~~~~ 8<j a~=~~~~~~~==~~ =

.~~ V. '0 .= ff...:.. ~ ~ ~ ~

V. B = 0 D.= Eo'E + P~ ~ -:.. -4 /-"' ~Vx E = -~ H= '1i:8~ 1'-1

at ;-~ -:.. aD ~

VxH= * +J+.

lets CtJY"IS~er a. med,'uW1 writ'l no m~()e:t15;rft(Yl (H::=.())

aY1d 'm ~ cho~ or currents

~ ~

V.D =-0

~~V. :B == 0. ~-:..VxE ---~

at~ ~ -oE oP

Vx'8 -)Aoto ~ +10 at

.-:.. ~-:..:for a. 1:nee:.v- bu+- a1')lS!Jt1!:,)~c medfuW'\.. P = Eo X E

tokrn~ %e. curl ~ ~ 3rd Qqn..

...:.. -:.. ~ ~ ( ~ - ) d2£ ~ a 2.fV y. v x E =- -"at rJ>< 'B = ~oEo ~2. -JAo Eo 'X. ~

A

51VeY'\ :t-he.t X tS Sj'fl1Mem'e ay..£i -t;h~f&e d{~Ot}a.{ in rh.e-

~'('intl~cl a-x;e.S and. ~uYi\i'~ ~ ~ve a harvY\O¥1iC pla\'1e.

wave.-:.. -So {~.; -wt-)

.E .::::. Eo e

--".._""."'.-

-

r;'j~

...:) ...;:.Vxf:=- -

and

M for n Q j n ~y rr'I.S aF n-e..gr<! r

we.. use.. Cl ~~- hend ncrtrot'ltO/\ -it> '~/aie. -£he. CtAr-1 of e

--

a. b:= ~b~ +a'jb~-+Qz.~ =' ~ Qib,' \41h~ ~kJ "11~~il'lp/!e~ a su n,

I_' C .'.L~ b I Q.b, t-Q2.bt. tO3bjr ~V1. -\'/\1.0 ~~~ 0

8 a)l. b = (Ei~ ~ ad b,,) .x.i .'. .~ vnr+ V&+t,r In 1-11t cl..m.~" "xl~ X

~=g6.i1b.~ +1 i,j,b. ':ltJ'c. CL1~=E312.-t.2SI= +/

-1 '~j,k Qrrtic~rJic. e:1~=f:.13a.= 6.2,-a=-)

O!he;noi~ 0 't'111R ~tQ~~ E!I"= e~:. t:,3)It,= 0

~ -

-1(~.r-l4)t)foe(~ ~"'\ .a r' , ( 8)"\Ix E Ji = eJ~'A 'ai' ER ;:::. Eij~ ,,~k.j nk- =- -~ '

~ j 1 , ) 7.

,'. V~E = ~JCE ~--~ i~xn 'Eoe.i(~'1"-\CJt:)

i(kx.h) Eo -iwBo r1B1\ ~ 'R (A "I"t:>o °B = 'Bo = ~ kxn) Eo

-~ -

8 Q( ~ ) -A (REo ) -i(k.~-w't) .n == kxnU '('It -n& -e ) &

CA). "- ( Eo) -i("~-klt) ~r 'BCY'(lc)=- 'ne, -r;: e

"'

c3

'the. time.. de(\'~~ ~Impl~ brlnj dt>L<Jr\. fa~ of 1'1.1)

.g: riE~ 'L'(R.r-~-I:-)] -( .,2. ~ i(~'r-(,I)t)dt~ L oe -~lW) Co e

aM -!:he 'S~cl d~'('I1vot1\.tS bri~ ~~n a. fac.k c>f- k

"'1 x. V x (Eoe'l'(~r-I.AJt:)) = tR x;~ x EoeiC"~-IJf:)

So -chot we.. ~'(\ worrte.. '1h~ wa.ve.. ~n, Cl5

~ ~ ~ w~ .-:.. <.cJ1- ~ ~

-kx.\«xE'= -;E +~?CECO ~

~~ '

w here X t: I r\ -\he pt\'nciplt l Q.~e.~ II~. ('XII () () )(E )~ ~ = 0 x.o.'1.. 0 E;

0 0 'X~~ '£;2-

'tl':in~ -f;~ ved'Dr Idexrl1~ k x ~)( ~ == ~ (t. .E) -E (k. k)

= ~(~Ex.+t~t~+~~E'2..) -~(bo.c1.+R~'L+R:)

the- Wa.~ ~ua1t~ \I~~~~~~~~.:::~~:Itt:)~~~~t~tJtt8 ItS

~ ( \ ...:0..( '1... 2. 1 1..) (J'J..( e.:..) ~ - R ~xEx. +~tlj + kz.Ez) -E R)( t k~ +[.(.z. + q jl+ x. E -0

wnldt ~ hi- W'Citler\ (J.:5 Q mwrx tirY\~ fu ~~

COYY'\ ~ -h of E: 0.~~.

c.4

j)'~ ---

~.-.;' '()2.R~-k2. ~2.I 0 ~ -z ~x ~~ kxRz. E~

R'j ~~ h~ 't<o 12..- R; -ki ~~ R-z. E-.!!J == 0

Rz.kx Rz.~.'.::\ n~ Ro'L -R: -k'j2.. £2

.whe.(t.. the. ~r,ncipc\ incli C.eS oF ~frqc..rtf(Yl ere. n .= ~ -'1+ "Y , ' .\ 2. f)

\- J ~ 1 -t f-jj' -;1= I I,)

2-.., an~ R =. ~

0 Co2..

Tr1 ()rd.er ..fa<- 0- ,(\DY1-fr\VIC\ so)f1. in .ex\~-t (i.t. j:)r)I;~~ ~ ~:~ +l.cn )

tV1e. de-to m~t van\"Sn. 'j

\ det A=-O P:2...

L &-tine.~ a. -thret'-~'M~rdO(1o\ su.r.f~e iY1 k -spoce- 1-;;-,:;r!j-. Iz 'J:.

wha.t do~ -tht~ ~u..ff&e. tDok Iik ~ --~ k=:t

Ct>n~idef ~e.- ~\c\"\e.. in ~-"3?&.e wneY'e.. Kz-==-o

1he. e-~'{). de t A = 0 redlAC-~ -to the... plvciLlct of 2. -f'attVf3

[ Y1~Ro'- -R~-p.~l.[( y)~Ro2.-~;)~~R~-R;) -R~~~ ==0

' y--~\f -th~ 1~t Rac,wr \'$ zemj we- Cjet tf1e.. eqn. of- Cl c;rc.-/-e..

1. 'l. 2..h'LR ~ +- ~'j = '(); Ko

\'f {-he- 2"'d ~c...{'f)(' LS ze('o) we- C?Jet the. e-~Y\. of d1) .e.111'rJse.2. ",",2.. r

~-t-~=-1(1: R; n.2. p:

c5"~!~::i,~;',,;. ".. c ~:,

'::~;:,:;,-

..,,'~.;'!

f", ...c..,o-;c

For Rz =::. 0 ";:

R ~D ~ """" ~fS~ pDlar~tf~ ~mI- -it> curve

"",2. I"'V~

Rx R\j-+-=-1n 2.. n 2- 'w""

.2. J ~j1

I'\..) k J;::?:c§ t.- "" f..".J kx/6 Kx =~ :'~'~:C:

.Ro'!

~~~: fc>lariza1f{)'l ty) ~ di~tI{)'L ii~;"

/"'-' 2. ,-.-J 2- 2. ~:

OptiC ailS n Rx -'t- kj -= ns :;i-(--- \ .-

;:=:i::::.;;,:.

.:;':-"~:',,'~

i,-;.fl

'"~.,'" ,

b cia\oj

.for ~ CIr-cule.Y" loc.us of S(){n~ ~k)C.-

.the fX'la'fi1dl1{)1 \S alo~ Z (~er1t; ~ ~e 5~d.i'yY) .jU~tt.)

.-l-he m~Y")\'-tude of ~e.. k vec.{Dr IS independtnt-

of direc,'UCY\.. ,\'n the Rx: -kj plan-e.

lk 1= ko n~ (pn~ipql "I\~ .,t:refn:t~)

for * .ell 'p1Jcc-1 1t!.US ot soln$..::..:: ! ::--

.the. po\. 1S ta~t -tP ~e Jlps.e

0 the ~n,'tl.tde of- l".b.l de-rends on the dir-e.c:1t~

.RIj lRI =- Ro ne

~A f'- <"'-k = Ro r1e CO~-t7 Kx + kt) n.o ~in{) ~~

~-~ ' ~T~

Rf; II<I-J-

R?'e;lli'p~n I

(~~~~~ 2.. + ~~~~-~~§) ~ 1

'Yl:J..~ ko ~ 'n,3..ko 2-

cos2.e s{n2..e 1--+ =- = --;;-to 2.. .-

11,1. '11, 1')6

\n ~e. Rx-~ pw.°. for a ajVe-n dit'ecltC)')J.. ~ere a-re. ~ di,fftxe¥1f re%advve ltr1d,'C~

(j '_V" VI.'-"~.J V"--

.1)'13 for ,col. alOl\j Z .l-fo k)(~b~ f~

rle frx- pol. ,'n the k)(-~ plorre, I b" Wifttbut Q. rr..!.\<l1w-

~Cj ..!!:!LC1ptr~~ _5 -~~~ e- ~nd ~e. w~ -b--c-~ .phdS-e. de~

c'f-

For a wave traveling in a principal plane and in a.given direction 8, there are two choices for the k-vector.

One for which the 0 and E vectorsare parallel and the index of refrac-tion is independent of 8.

k

. One for which 0 and E aren't parallel...E is tangent to the ellipsewhile 0 vector is normal to k and

D E the index of refraction...does depend on 8

k

.

0 t " " 0 ." c8~P IC axIs ptlC axIs

kZ Z k

no.ne E,

,

j

iI

iI! 0 wave e wave

fa) (b)

Figure 6.3-9 (a) Variation of the refractive index n(8) of the extraordinary wave with 8 (theangle between the direction of propagation and the optic axis). (b) The E and D vectors for theordinary wave (0 wave) and the extraordinary wave (e wave). The circle with a dot at the centersignifies that the direction of the vector is out of the plane of the paper, toward the reader.

Special Case: Uniaxial CrystalsIn uniaxial crystals (nl = nz = no and n3 = ne) the index ellipsoid is an ellipsoid of

.revolution. For a wave traveling at an angle () with the optic axis the index ellipse hashalf-lengths no and n(O), where

1 cosz() sinzOz = z + Z' (6.3-15)

n «()) n n .0 e Refractive Index of the

Extraordinary Wavecc~~~ RaY k surface ~Ray Wavefronts k surface

/

Wavefronts 7/S S /

/V

.0 0

(a) Ordinary (bfExtraordinary

Figure 6.3-10 Rays and wavefronts for (a) spherical k surface, and (b) nonspherical k surface.

~ ~

c8bkJ.

eko k2

Figure 6.3-11 Intersection of the k surface with the y-z plane for a uniaxial crystal.

S

.@'//;~~ kE,D

k3 kJ

noko

k2 neko k2

(a) Ordinary (b) Extraordinary ~

Figure 6.3-12 The normal modes for a plane wave traveling in a direction k at an angle 8 withthe optic axis z of a uniaxial crystal are: (a) An ordinary wave of refractive index no polarized ina direction normal to the k-z plane. (b) An extraordinary wave of refractive index n(8) [given by

.(6.3-15)] polarized in the k-z plane along a direction tangential to the ellipse (the k surface) atthe point of its intersection with k. This wave is "extraordinary" in the following ways: D is notparallel to E but both lie in the k-z plane; S is not parallel to k so that power does not flow alongthe direction of k; rays are not normal to wavefronts and the wave travels "sideways."

(.Cj

-_.,~:~ j3~:~~~:::1._~:~~:~:1 ~.~} .D i~ -L -to t since 7, D = 0 (~de:fjY)~ +h.e.. W8Ve:~rh )

~ Vet~~

E i'5 tahCtexT't ~ A~uyke.. /. h F.Jn I >J ~.s ow .-:J 1Q,~-I.2

the t!ow at eY)-e~~ ('POljrl-tiYlj vet.'fL)r)~ ~~. I ~S =- .E x. ~ 1S ~ -to suY"'.tttce. (llralj dr're.c..1't())lj

p~t~~__~l~ ~ ' ~

"t=.Y2- ~ ~ -~.1~1 Ron -'r1e

§~~-~ ~ ~

~ == ~ -.V~w(~)d. R t--y--/

th;~ [~a.. 'lec..to<" norrr1C1 -to -th-e.

~ -s~ce ~u.ffttce /I-tv""t

\l ( dA'ret1YrI1of)-!-he.. direc.~O'l of e.n~ -+ra.n.5~rt- theA UShf r'a-'<1S ~~~

M -the l\.e.x.iYoo\dina~' ~~~ of irc3.vd;hj Q+ an.ob\;que.. ~~\e wi1h ~e. wa.veR11nis '/

~.J='~ t,3 -10-)

ctD

/he. Wa.V.e.. -Vec.flX"' surfa('.t..-

.-so.faY' we have ~~d our qtt.et11YC4f"\ -to W~ve v~ "n fu~~£~~-

rvV'- ,Sj'(11lla.r e1m. &Y"e. obtained .fof' fu ~'Xk2- and k'j kz. planeS

.the 'l'n-tercepl- oF ~e k surfaCt w,.~ each C(t>rciJ~+e ?/dn{

coY)$is.{s of one circle 8\1\d Dne.. ellIpse

0 -the coYr'iplcle k ;surfate 1S QQ~~l~

conSj~-h oF 8'(\ \Y)ner sheet and an ou/-er sheet-

.for 8'1\!j <J"VeX1 d,rec.--nCt\. of #1e. w~v.e.. vedr;r ~ th~

a.~ two ~ible. Volues .fcI- the. Wc'l/enuMbtt k (1'~ I~/IJ)

8nd ihere.k -two d,'S'-k'nct phase vebcr'-&'es

.CDrf'eS~d;nj m or-th~dY1c I fiJl.

.the. i'(\'(Ie'(' and cute-r :shet"ts -tDuch at Q. cerotcin. poiht .

ihi~ ~()\\rrt de.-tin.eS -the diret'tim of thIL Optic. a-x:i.s-two )q\oY13 which -the wavenu'rI1be-rs (i~'~ of'- re-fraSfM

CD\~~'~ 10 ~ -two pol. ate.. ~uQ.Q.

.

c.11

.lOW'

R'j

rJ. k& ;J(:

lOW

°p-b'c. Q'ttS

.

ct2.

CoY1slcler

..~~::~~~~::~:~~~~:::::~~~:~j~;t~~1~::==-

rr) I:::: 112:::: mo ~ 3'::: 'YJe..

de-tA=O

th.e. t Vet-WI :Sl)..~CL" ~iYY\pli.ffe<5 -to

~R2.- n'J..~'l)(~2'L-I- ~ -R ~= 000 n'L hi 0e 1'0

I-' l r "

.s~here. ell ip2lO(d of 'fev(!)l~

L =- h ~ R,.2+Rz. '"L+- R3'2.. -L.. 2-

F<. () 0 ---~Dn1. n2..e, ().

Ir}E > 1)0 pO~I'-bVt.- uniaxial

~F < no ~~ unJaxlO I

I( . { 'IUY1la~iB be.catL~ there. 'lS on~ a1e.

otJlic.. Q~is -alOt'lj k3

.-

c.1~

174 OPTICS OF SOLIDS

.Optic axis Optic axis

(a)

tic axis

~/ \I \

\

\ 0 ).\ !

(b) ,/

7'/

!

/;!

!

\

\ ]\

(c) \ 1

~

Figure 6.9. Intercepts of the wave-vector surfaces in the Xl plane for (a) Ibiaxial crystals; (b) uniaxial positive crystals; (c) uniaxial nega- i

.tive crystals.

,,*" .,*""' .,

c.14

Uo v b I e. fefr cX..1f (Y'\.=-

wQ.\Je%Mts at ihe.. 1Y'1c:derrl; a~ refracied Wa~ ~~T

be- ~~d at ~und~~ ( Snell$ /8W)

11- R.

R\,sin-G, = k~;n.f}

Ron, sinf), =- koYl ~in(;)"111

b~ ~e a.();~IC merJJUtt1 ~up~ ~()

.rn~ of- dl..fte~, (fe..frocitve md;~) fh~ ve1o~e:8or-e.-

one.. e:(pecb 1t<JC> ~tted r~~J -frx- eac.-h fl.

'fr r\,=:.1 (air) -~~~~~~~--- 10 solve th/sJ We. clrdlO ~e- \nk~~S\nel.:: nfJ sinG of +hi- p~e of jnCJ'denCR. and *

~ -V~t.W 'Sl.tf~u. Qt1cl Jrok.

~~~~~ -toy- o.Y\~les tJ whtfch s~m-Rj~~7'j~V"-V V' 'Id' -"0 tht~ ~n.

-:3iIi~~~t~ ~

~.We;~~~~~~~.t;$¥ri"~~.f:SE J

c--&t~~ 19J~ ~~.

ciS""

Case.. 1"0 u'()oax~c( crtjsfu I) ~la.he ~ Inc(\cley>C,e II -to ~tc. ClXJ.j~ ::-. --~-

..~~ ~ -su'('face. wil\ ihfe-~eC+ ~ plcne of

1't'\oldentt. in 0- circle + e.llc'p~

<~l~ Co,3-{3)~'T61-T'e.I'dt

the G~jnc~ '~j (IE") E fY:>1~ .l. fu pldV1e af Inc.)

OVollY'lCrlj \IlaNe..

SinG,.::: nos)n.eo (.sr}e.1I~ law holds)

-t\i.e- ex+roDrdmc~ (Qj -i) -tht'j t5ni real~ a s"ellS ta~ "LIt1~

(a) ~ ~ CoIIst, and..s-, Y'lt7l.

.SI'()~I=- n(.oe,)SJnee.. ~)C{Z(MLlthc/ r-ofa<trmswhkh doht- ch3h~ eJ

WI'li ch~~ h(~~) and

~eJ'e.~ ee

~ a unl\8.~'l:,\ crlj~k I c'tJ1- 1/ io optic. Q7CI:.:s

and vll'~ p(,,"~.e... of- Incidence.. ..l- -to opt1c.. QXI~-thl. ~ -surfttCt. w~l\ Ir\ ~ct ik P I~ oF- 1V\c)del'lC{ t'h WO crrcl-e..3

.~ we 5et two S net [s I BLUS

'noSln f3o ::: ~I'(I&, /

><"...h~ &,14 Fowles'r1£.3I' n GtE' = 6": 11 e I CQ,Se. (Co)

'\"{\ #I's Q2~

..io~ I in fernt3f refl.ec1ftlVl for one pol. aVlol 1rdi"l5rn/~/tV\

~ ~.e. o~.et- Is -fI1e. fj)n"nCiPlt ~ o~~ for

-th-e. 61ch rol~rI2l~ pri~r'rl <show h3 Co.I5" ~~j

c.1l()

.-.

~ I

k surface (air)

I I

~ ""i~ ~Iko sin 81 ko sin 81

Figure 6.3-13 Deteffilination of the angles of refraction by matching projections of thevectors in air and in a uniaxial crystal.

,".I

Air

r I.

.Incident rayt

Figure 6.3-14 Double refraction at normal incidence.

problem of determining <1>, given the value of (J, is thus not a simple nE is great,erone. One way is to solve for <P graphically as suggested by Figure negative cry~6.13. <PE ~ <Po for

In the case of uniaxial crystals, as we have seen, one of the parts are illustrate..of the k surface is a sphere. The corresponding wavenumber k is con- waves are II

stant for all directions of the wave in the crystal, and Snell's law is are indicate(obeyed. This wave is known as the ordinary wave, and we have given wave \

sin (J face at the Isi~ = no (6.112) ence [5].

c..!?Polarizing PI

Uniaxial positive Uniaxial negative the inside of, ...

"'-.,. optIC aXIs IS,-- Figure 6.14(1,

~-=~~: :::~:== = circles, as sl===:~ -,~== :,: nary wave al,--",--,- --,- .

~ -::::::::.::::: (a) -medIUm be B"--'- ----,--

==--- ---= ===~::~====~=~:::='+:::::::::~{~' ,:=::::::::::::

where (J is t~ ~ angles of reft

.1 -'T T 1 ;~:P;i~~~~Z~

I I (b) I wave is para:

Ii" i:: i, , ! I .Suppose II ,' I ! li:I'i il I 'i'

l illi:i , i! ,'j cite, and that, ! I,!" I, , , , J ! j I ! J

-~ ",- ,--- In this case \

.not for the

.pletely polar:(c) ..I.clple for prc

.One of tJ.,- ", ..,'. pnsmshown

h calcite cut soFigure 6.14. Wave vectors for double refraction in uniaxial crystals. (a) T e n..

o t d.. I of 001 Un e so

optiC axIs parallel to the boundary and parallel to the pane b

incidence; (b) the optic axis perpendicular to the boundary t e~~een theand parallel to the plane of incidence; (c) the optic axis parallel to dena. If an

.the boundary and perpendicular to the plane of incidence. egrees..

~g -

~",1 cl%

(b)

(c)

Figure 6.15. (a) Separation of the extraordinary and ordinary rays at theboundary of a crystal in the case of internal refraction; (b) con-struction of the Glan polarizing prism; (c) the Nicol prism.

,

""';" ;

.:;,:. .'..., .

~(a) (b) (c)

Figure 6.16. Three types of prisms for separating unpolarized light into two 1divergent orthogonally polarized beams. (a) The Wollaston 1prism; (b) the Rochon prism; (c) the Senarmont prism. All )prisms shown are made with uniaxial positive material (quartz).

c \<3 .,

ISlf~e, (r.~ht- is InclJeiIt oYl a bo,..,J&~ 0+ a. i

.-~~(:d.~5+a1 arJ air : ~r~rn~IM -ft:t-- ordlV\a~ 1'2j +-fI>fel 'l'rVemal.~ re-Pler:1r~

,fC:I"- exfr2b~'ne~ ~

~__~_~I;~~~:t-:- .(~p~~ ~ rYI )

y--rL~ JT.'\

R oldit\t.~ = Ro n ordillt."j l

~f:.x.ffQofcJ. = Ro ~~I~ j l n c.1\jsk l

h

circle.of radI~o k '"" Ve.t.fDf' 5I.A.~

'r)e.

eo-thi~ ~.c... \ fI::;, ~" c.ircJeol,-radi~ce~roct be. / n 'SQHstie.ol ~, a'r

I ~ -ved'I>r- .

.~~~"j ~~~ ~ frr>jecltO'\S q,1bV\~ bDf)Yl~ ~u.5t be ~uaA.for :nCJ'dfftt +- Y"f.~iJed Willie:.-

n Sin e := naif' Sin eo0

'()e,.SIY1e- =- nQ,'(' SinG!..

.