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SOLUTION SET
Chapter 2
WAVE NATURE OF LIGHT-THE INTERACTION OF LIGHT WITH MATERIALS
"LASER FUNDAMENTALS"
Second Edition
By William T. Silfvast
CH 2-
1. Show how to go from the integral form to the differential form of Maxwell's four equations.
(b)
lC)
~ {ver~el1ce .lla.o reVV\
f E· iS t J \7· E Jv = r~ ( ~)Jv 5 v v
~ t r\7· 5 .lV= i. [rohl ~Q.B:f
~ B. e1S =- \ \7. 8 .AV== D ~ 7. B:: 0 s v
S.io k e ~ lh e. b r-e.. V\i\
J, B·il ~ )\7x8·JA=µol"'f[µD(J+g:Q),dA ) A A ~
=-/ 7 x B ::: ,µD ( "f + ~)
s To Ile~ ~or~~
f - - tJf- - - t:Ja EI e/-e = . v x E . d.A = - -A d-:r
ff J B -- - . dA Q
A
{ H 2.
2. Calculate the electrical force of attraction between a positive and negative charge separated by a distance of 5 .3 x io-11 rn (the average distance between an electron in its ground state and the nucleus). How does this force compare to the gravitational force an electron would experience at sea level?
--n~ 1 V"a. v t t tt. tt ()" $\~ I
~:::. ~ 't
f I
L 1-1 2..
3. Show that the function E = E 0 e-i(kzz-wr) is a solution to Maxwell's wave equation (eqn. 2.26), assuming that c = 1/(µ 0 .s 0 )
112.
4. Consider two waves, having slightly different angular frequencies w1 and cv 2 and wave numbers k 1 and k 2 , of the form
Ei = Ei°e-i (k1z- w1t) and E2
= Ege-i (kH. - w2 t),
where Ei°_ = Ef. Assume their "difference" wave numbers and frequencies can be written as
6k = k1 - k2 and 6w = W1 - w2.
Show that the sum of these two waves leads to a slowly varying amplitude function associated with the difference frequencies and wave numbers multiplied by a term associated with the actual frequencies and wave numbers. Hint: Reduce the sum to the product of two cosine functions.
0 -.A ( ... ~ - w' ~) E, -::. E1 e.
, '
,,h_ I -:-_ ,(..,_ -t" ~ '2-
I.) - I~ .__ AW. v-'l- vv ~
- "2_ i; e - j ( ~ ~ -W ~) l.-o.S [{ ( t. ~ °i- - ./.\. u.J ~
\..__-~--.. - . A•~-'-
5{~w'1 vo.v-~1'.-i7 a~p/,'frtde
5. Show that the group velocity and the phase velocity are equal when there is no dispersion and are different when dispersion is present. Hint: Begin with the definitions of group and phase velocity.
Vp ~ ol ~ - -cut -
vb~ ~w n-
w w ~ 4<.. = -- uJ ---~
I d'%w-
cl Y\ ~W-=. D
Vp
V\.. - ..+ c..
Vcr = %_-=- Vp
-C./rt
. .......,.,. ~•Ollil"'1M
w el Y\... - dW l
I-+ d; s p e.f"-S ; ~ "I. Tl._e. VI. ~ ~ * 0
a~~ Vb- -:/ Vr;;,
V\_W c_
c f..t z.
6. Obtain (2.101) from (2.90) and (2.95).
s lA k s:t,· r~ n 117 : .
(j /1) 2
~ /: (1 + ~e~ b~6~-~~-/~wJ) 11_'\ I + [ \ .... ........ -....... ]
w ~-·'- w"" - ,I "w b .
l 5j • -- Y- r
:net•-~·-'·~~
7. Express (2.102) and (2.103) in terms of frequency (instead of angular frequency) and r (instead of y).
fl) -- 27/ v -~ --,) - Jd_ vv -/ r - '21/-
8. For a medium (such as a gas) of low density and with a single resonance frequency, show that , if the imaginary part K of the complex index of refraction is significantly smaller than the real part TJ, then for values of w far from w 0 the expressions for TJ and K can be expressed as follows:
Ne2
( 1 ) . Ne2
( yw ) TJ = 1 + ? and K = . . 2mso w 0 - w 2 2mso (w6 - w2) 2
2, "L rv- 'L (.,,{.) - w j) >> Cl := w
)( <_ <_ lU J 6 pt1 ~_,.j f Y4-1l/41fC!es
~ "' V\)_ - k "2 S' V\_ ~ I+
f""\J [1 + V\_ -/\le l ( I \l t/~ Vit 4 w,}· - uiU
2J :r. y ,/ ,-.J 10 -/I) l'lft.tl
(,,l ~ ,,, V\ f 'f' t) ~ ~ bo t.rt..
9. Assume that a dielectric material has a single resonance frequency at v 0
3 x 1014 Hz, the polarization decay time is r = 2 x 10-7 s, and the density of polariiable charges is 5 x 1026 m-3 . Determine the full width at half maximum (FWHM) of the ~bsorption resonance in the material and determine the maximum numerical value of a.
ex. -=. "2. w I< .. c.
\c..l-=. - I
c, '2. -($'-w '- =!
c 2 Lf ~ r w ""'i
+~
(),_ ~e-1 l.. V\ \< :: .fJ_ ¥W
'-/ ( I + I<.'°) l<=-l"' ~>.,_w .._
l tf 2 c, -
o v- le. +-- /t::. - '-/ r 7 w 'l. 0
I + l \ + C1 '4.. ~ ·- ___ ..
- °2 - l. 'll'J...W .._
L# 2
10 (Uivit)
2.2
2
1.8
1.'6
Y\ 1.4
1.2
1
'0.8 I . I
0.6
0-4
0.2 0.92 0.94 0.96 0.98 1 1.02 1.04 ':1:()6 1.0:8 1.1
W/w/)
1.8
1.6
1.4
1.2
K 1
0.8
0.6
0.4
0.2 / 01.--~-1-~-d:=====:i::::::....~_L_~-"'~-~-======-~.l--~....L~_J 0.92 0.94 0.% 0.98 1 1.02 1.04 1.06 1.08 1.1
W/U/c
LH 2..
10. A species of atomic weight 60 is doped into an Ah 0 3 crystal at a concentration of 0.1 o/o by weight (cf. Section 5.3). The combined material is found to have an absorbing feature that peaks at 750 nm and a damping constant of 1013 s-1. Assume that each atom of the species contributes to the macroscopic polarization associated with that absorbing feature . Make a plot of TJ and K versus wavelength in the wavelength region of the absorbing feature.
)... =-7 _s--~ ~ IM..
¥ =: It/~
,, \'- lli ~ 3,Cf9xtoi>~-~ XI~ - !.
c, -~~ - "' >tur.J1 ~. ~s-xio- 1'
:: j,£.7)(10 ~: -
l'} t if_xto_ . . ,
11. Show that, for a low-density medium such as a gas (as in Problem 8), the maximum and minimum values of r; are located at frequencies that correspond to the half-maximum values of K. Hint: Assume that in this frequency region y 2w 2 >> Cw6 - w2 ) 2 near resonance.
' ~ Lis:.? ( \ \ ~ (_ '<""\ + A k) = I +- V1A t-Q W/ - w ~ - ..., 'r w) ~3 Al,J!l.. 2.~
. F & v- ~ 1 tA.. $ N ~ I /) Y~ l '=? ~· rv 3 XI tJ
' \ '2.. 2. .;:!/ t;t c.i\..-J.. W '2 """-' (!- rr X I D 1 'I/ ~ t./ X I D
{_/-1 2.
1-1-t."-tO. \ > '? ~ ~: ( w. •-1~ "..'. ::rvw} .for ,,,,...d x.
' • •
~
/ I
, I t
" "'T": t • r, ' x )11'- --;"\,,,. I +- -k x +- · · • · . (;\. ~ I"'"{ ' "-·ti 0 f' exptt~S I ()"\ L' -r ~
\/'\ ·+ .) II"" ::::: l + .l. t::.J. ~, .. ~ I A. I .. t: ' v \ r I -.... - . '2... 1-"\et ~() \_ W0 - W -A O WJ
= I + i ~e ~ [Lw0 -w)~;fw)::::i d'~
12. In regions of the spectrum where the absorption is small, TJ can be expressed as
TJ2 = 1 + Ne2 ( w6 - w2 ) mco Cw6 - w2)2 + y2w2 .
Convert this equation to wavelength A. instead of angular frequency w. Show that it is of the form
Use the following index data for crown glass in the expression just displayed to plot (TJ 2 - 1)-1 versus A.-2 and obtain the values of C, N, and A. 0 .
L.. Ne..... [ tvu '-- w L... 1 Wavelength Index of Y\ : t + ~eo ~,,, "l_ w'l...)"+ '('w'-j (nm) refraction
Y; J l,lo\ v--l~ ( '1'-\. W ~- 0.. h S. ~ v--p 1t tn1 I-~ $ ~t.t. 728.135 1.5346 ' 706.519 1.5352 (!3!,/- ~ W 'I..) 2.. > '/ '( 1
W L. ~~ a...~!.t.tiC4. uJ )Wo
667.815 1.53629 I :J ..,., .. i.. t\, Ne.). [ • .. -- .
587.562 1.53954 I"-"""'- \r\ - \ ";:: - - 161 't - L-tJ "a... 1 ~ M.l. t;) CN' cl .
504.774 501.567
492.193 471.314 447.148 438.793 414.376
412.086 401.619 388.865
1.54417 1.54473
1.54528
1.54624 1.54943 1.55026 1.55374
1.55402 1.55530 .. 1.55767
- - Ne" f \ . . . 1 - ~l~~~)2.-~~!:)LJ
\....
w ~ (_-::: ~ &ol7;1fC.) IV e ..._
u ~ clct-t'"t 1'b o i-ir a.. ~lo\. C) N > 1\ u
C = 7.7f; x 10-'~11'\Al..
f.-0
-== ltOIS-x.10- 7V1'A.
N ':> /,'-('( X /02."retfa*$/'M'J
C. # 2.
13. Two electromagnetic waves in the blue spectral region ( 450 nm) are separated in frequency by 1 MHz and in wavelength by 1 pm. What is the group velocity of · this wave combination?
W= 27TV
- - Rl'Tl ... ~
. . - -----c_ II ~ 14. Two electromagnetic waves of nearly the same (600-nm) wavelength are mea-
sured on a spectrum analyzer to have a frequency difference of 1.0 x 108 Hz. While traveling through a dispersive medium, an interferometer is used to mea- ·_ sure their w.avelength difference of 0.0010 nm. What is the group velocity of the wave combination in that medium?
A :::: ~ 00 f\ V\,;\
A. v -.:. ltJ ~ Ht
\I '".: A W - 'L1T A /..) Ak= 3J[' AA V& Ak. - ~k A'l-
v & : )_ '- ~ _ ((,., OOXll.l-"1,,._f /O % AA . {)r
0
1JC/ XI~_.,,.,;;-
c. fl 2-
15. Determine what emission frequency width would be required to have a temporal coherence length of 10 m at a source wavelength of 488 nm.
)\-;;. 1..-t q, Cl V\ 11\A.._
>-I..._
.ci,\
/0 ~
CHL 16. If a photographic film has a minimum resolution of 10 µ.,m, what minimum feature
size could be observed at a distance of 2 m without observing coherent effects? (Assume the minimum feature size is equal to the minimum resolution.)
a. s. s t.it. "'~ A :::. S" o o "'- !;I.A
y-::: "L w-.,.
