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Unformatted text preview: Slab Waveguide (15) Name Etalon (18) (Please Print)
Holograms (15) Fourier Transform (18) Two Slits (19) Spatial Filtering (15) E C E 4500 Third Examination April 29, 2002 Rules for exam: 1. The time allowed is 60 minutes. The test will start at 9:00am and end at 10:00am. 2. Answer all questions. The value of each question is given in parentheses by the was» 9°89 10.
11.
12.
l3. 14. 15. question. There are a total of 100 points possible.
All work must be shown for full credit.
Put your ﬁnal answers in the locations specified. . You may use the six new singlesided 8 1/2" x 11" information sheets that you have prepared. You may also use the twelve single—sided information sheets that you
prepared for the ﬁrst and second examinations. Some physical constants are given below. You may use one "book of math tables." You may use a "pocket calculator" that can be put in a normal—size pocket and
requires no external electrical power. Graphing calculators and programmable
calculators are acceptable. You may not use portable, lap—top, or notebook
computers or wireless or network connections. You may not use any reference materials other than those listed above. Therefore,
you may not use the class notes, any textbooks, homework problems, reprints of papers, journals, prayer books, etc. There is to be no sharing of anything. If excess information is given in a question, ignore the unneeded information. If too little information is given in a question, assume the information needed and
clearly note this with your work. Any changes to the examination will be written on the chalk board. Check the
chalk board periodically during the examination. Any acts of dishonesty will be referred to the Dean of Students without prior
discussion. The ofﬁcial written Institute procedures on academic honesty (entitled
"Maintaining Academic Honesty" and available from the Dean of Students Ofﬁce)
will be followed in all cases. Have a happy exam! = 6.6260755 x 10'34joule—sec
2.99792458 x 108 meter/ sec
1.6021773349 x 10'19cou1 = 1.38065812 x 10'23joule / K h
c
e
k Slab Waveguide A slab waveguide has a refractive index of n f = 2.234 and a thickness of 4.00pm.
It is surrounded by a substrate of refractive index of n, = 2.214 and an air cover of re
fractive index of 11.6 = 1.000. Light of freespace wavelength 1.550pm is launched into the waveguide. Analysis with the slab waveguide program provides the following informa tion: Mode Zig—zag Effective Index Propagation Angle Index Constant
(°) (Inn‘l) TE 0 86.016289 2.228602 9.034014 TE 1 82.461876 2.214693 8.977631 TM 0 85.927162 2.228358 9.033024 TM 1 82.381248 2.214279 8.975950 At A = 1.550pm, of these four modes which mode is farthest from cutoff? That
is, the operating wavelength (1.550um) is the farthest from the cutoff wavelength for
this mode. Calculate, showing all work, the cut—off wavelength of this mode. Express
your answer in ,um accurately to four signiﬁcant ﬁgures. Put your ﬁnal answers in the spaces provided below.
Mode that is farthest from cutoff = Cutoff wavelength of above mode 2 pm Etalon An etalon is to be used in air (72. = 1.000) inside a laser cavity to produce single—
longitudinalmode operation of the laser. The etalon is a slab of glass of refractive index
1.500 with sides that are accurately optically ﬂat and parallel. The glass slab is exactly
1.000 mm thick. It is to be used to pass (transmit) argonion—laser light of A = 488.0 nm.
In use, the angle of incidence should be as close to 5.000° as possible. The etalon is shown in the following ﬁgure. Calculate, showing all work, the angle of incidence closest to 5.000° that will pass
A = 488.0 nm light. Express this angle accurately to within :t 0.0001°. Put your answer in the space provided. Angle of incidence Transmission and Reﬂection Holograms Transmission holograms and reﬂection holograms have many important uses. On
the diagrams below, correctly draw the laser beams used for recording and reconstruct
ing a transmission hologram and for recording and reconstructing a. reﬂection hologram.
Show transmitted beams wherever they exist. On the diagrams below, correctly draw the resulting grating fringes that are produced for the transmission hologram and for the reflection hologram. Transmission Hologram Recording Reconstruction Reflection Hologram Recording Reconstruction Fourier Transform A time signal f (t) consists of an inﬁnite periodic train of delta functions. The
strength of each delta function is unity. The time interval between these delta functions is T. Mathematically, this time signal may be expressed as f(t) = Z 5(tnT), where n is an integer. Graphically, this time signal may be represented as shown in the following ﬁgure. ~m~ Write the exression for the Fourier transform F(jw) of this time signal f Ex
press your answer as a function of w and we where wo = 271' / T. Put your ﬁnal answer in the space provided. F (jw) = Fraunhofer Diffraction by Two Slits A metal plate in air contains two parallel slits, each of width a. The center—to
center distance between the slits is d with d = 2a. The metal slit plate is illuminated at
normal incidence by a plane wave of freespace wavelength A. The transmittance of the slit plate is shown in the diagram below. There is 100% transmittance within a slit and 0% transmittance elsewhere. so»! L—ma—J Calculate, showing all work, the radiant intensity of the farﬁeld (Fraunhofer)
diffraction pattern resulting from the diffraction of the plane wave by these two slits.
Express your answer as a function of Io, a, A, and 0 on1y (eliminate d) where 0 is the an—
gle of propagation in the farﬁeld (as measured from the normal to the surface of the slit
plate) and I0 is the farﬁeld radiant intensity at 0 = 0. Simplify your answer to a single term (containing one or more factors) and put your ﬁnal answer in the space provided. Spatial Filtering A 4f optical system is used to coherently process data. An object is placed at
the input plane and coherently illuminated. At the Fourier transform plane, the light
pattern consists of an array of beams as shown in the ﬁgure on the left. The diameter
of the solid circle is proportional to the intensity of the light. At the Fourier transform
plane, a spatial ﬁlter is added as shown by the ﬁgure on the right. The spatial ﬁlter
blocks all of the light except for the three beams shown. 0
0.0
0....
0......
.00....00
.00....I000
00.000.00.000
00.00.000.00...
.0...0l000000...
...000000000000..I.
....00.000000....
O..000000.00...
0......OOICOO
00.00.0000.
0......00
0......
.00...
.0.
I For this situation, describe accurately the intensity pattern that appears at the
output plane of the 4f optical system. Draw the image that appears at the output plane. Slab Waveguide nc = 1.000
nf = 2.234
ns = 2.214
h 2 4.00pm Mode that is farthest from cutoff = TEo
QTEO = 86.016289° The cut—off frequency ctan'1 aTE fTEo=—
27rh1/n.2f—n§ The cutoff wavelength 27rh1/ngc —n3 ATE =
o tan‘l‘xaTE where
n3  nE
GTE 2 2 2
TLf _ n8 ATEO = 5.27550pm Etalon /\ = 0.488 pm n = 1.500 d = 1000 pm a z 5.0° sina = nsin ﬂ, 3 = sin‘1[(1 sina] = 3.330980°
For transmission maxima 2 n (1 cos ,B = m A and so m = 2ndcosﬁ/A = 6137.155 and so use m = 6137 then g = cos—12171:? = 3.355750° sina = nsinﬂ, a = sin1(nsinﬁ) = 5.03723° Transmission and Reﬂection Holograms Transmission hologram, recording — two beams enter on same side of plate; fringes basi cally perpendicular to surface of plate Transmission hologram, reconstruction — one beam incident; transmitted beam on oppo site side of plate; diffracted beam on opposite side of plate Reﬂection hologram, recording — two beams enter on opposite sides of plate; fringes basi cally parallel to surface of plate Reﬂection hologram, reconstruction — one beam incident; transmitted beam on opposite side of plate; diffracted beam on same side of plate Fourier Transform From optics, an inﬁnite periodic train of delta functions is given by +00
2 6(x — nA) n=—oo where n is an integer and A is the distance (period) between the delta functions. The spatial Fourier transform of this function is given by Multiplying by 27r and using to = 27r f and 0.20 = 27r/ T this may be rewritten as +00
my) = (do 2 6(w—iwo). i=—oo Fraunhofer Diffraction by Two Slits With d = 2a, the transmittance is a 3a
f2(93) — 5 <17 < 7 Alternatively f2($) = f21 — f22 with
—3a 3a
f21 —— l T<$<—é—
—a a
j32 == 1 § <:$‘< 5 The Fourier transforms are obtained using If —) ac, w ——) k3, and 7' —> a and where k2 = galsinﬂ.
 k
szn(3—2=9) 3kza
2 I§1Cjkxto =:3a ' 5:1
panama—“M 2 ) P31 — IE2 " 3a 3ﬁ2: “ a ﬁg:
1 31¢ a , kma
F21 — F22 = ak—Jg [sm( 2” ) — sm( 2 , 3kza __ , 2k$a kza. _ , 2kxa kxa 2kza kxa
sm( 2 ) —— szn( 2 + 2 ) = szn( 2 )cos( 2 ) + cos( 2 )szn( 2
, kxa __ , 2k$a kza _ , 2191a kxa 2k$a kza
sm( 2 ) —— sm( 2 — 2 ) = sm( 2 )cos( 2 ) — cos( 2 )sm( 2)
andso
 k a.
F(jkza) = F21 — F22 aszng) cos(kza)
( 2 )
andso
 2 k a
= [owwngm
(hf) where k1 = gfsinﬂ and Io = 4a2. Alternatively, :79 sin2(5‘2—a) (sin(2kz 0.))2
4 sinchc a) Spatial Filtering The three beams represent the i = —1, i = 0, and i = +1 Fourier components of
the output image. In the output plane of the 4f optical system the intensity is therefore sinusoidal. The “fringes” are oriented vertically. ...
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 Spring '08
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