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Unformatted text preview: General Questions
Interference Filter (20)
(22) Name
(Please Print) Rectangular Aperture (22)
Grating Beamsplitter (18) Fourier Hologram (18)
E C E 4500 Third Hour Examination May 5, 2005 Rules for exam: LIIka ooqu 10.
11.
12.
13. 14. 15. . The time allowed is 60 minutes. The test will start at 3:00pm (rather than at 2:50pm).
Answer all questions. The value of each question is given in parentheses by the
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 six new singlesided 8 1/2" x 11" information sheets that you have prepared in your own handwriting. Your information sheets may not include
photocopied material. You may also use the six single—sided information‘sheets that
you prepared for the ﬁrst examination and the six single—sided information sheets
that you prepared for the second examination. Some physical constants are given below. . You may use one "book of math tables" or calculus textbook.
. You may use a "pocket calculator" that can be put in a normalsize pocket and requires no external electrical power. Graphing calculators and programmable
calculators are acceptable. You may not use portable, handheld, laptop, 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, pray'er 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 board. Check the 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! h = 6.6260755 X10'34joulesec
c = 299792458 X108meter/sec
e = 1.6021773349x 101966111
k = 1.38065812x10'23joule/K General Optical Engineering Questions 1. An asymmetric slab waveguide has thickness h, cover index n6, ﬁlm index n f, and
substrate index n8. Light lof freespace wavelength /\ is propagating in this waveguide
both as a TEo mode and as a T E1 mode. It is desired to make the TEl mode become
cutoff. However, it is desired to retain the TEo mode as a propagating mode. How can this be accomplished? (Circle one best answer.)
a) increase A b) decrease h c) increase ns (1) increase 120 e) decrease n f
f) all of the above
g) none of the above 2. An asymmetric slab waveguide has thickness h, cover index nc, ﬁlm index n f, and
substrate index ns. Light of freespace wavelength A is propagating in this waveguide both as a TED mode and as a TEl mode. For these modes, (Circle one best answer.)
a) group velocity of TE; mode < phase velocity of TED mode b) phase velocity of TEO mode < phase velocity of TE1 mode c) The TE1 mode is less tightly bound than the TED mode. (1) The propagation constant ﬂ is smaller for the TE1 mode than for the TEO mode.
e) all of the above f) none of the above 3. A symmetric slab waveguide has a film of refractive index nf and cover and substrate
of refractive index of he : n3. The thickness of the ﬁlm is h. For a T E0 mode excited in this waveguide at a freespace wavelength of A, the thickness at cutoff for the T E0 mode is (Circle one best answer.) a) 0 b) A/nf c) A/ns d) A/nc e) (n; — min/(n3: — n3) 0 (mic/mam; — nan/(n3 — mg)
g) 00 4. A thin slab in air of thickness d and refractive index n with ﬂat and parallel surfaces will strongly reﬂect normally incident light of freespace wavelength A for m being an in teger if a) 2nd 2 m/\ b) 2nd 2 (m — %)/\
c)nd = mA yd) nd 2 (m — %)/\
e) 4nd 2 mA f) 4nd = (m — §),\
g) all of the above
h) none of the above Interference Filter It is desired to use a “one half wavelength” thick interference ﬁlter in air to scan
the freespace wavelength range around A0. The interference ﬁlter is speciﬁed to pass A,
at normal incidence. The speciﬁed wavelength As > /\0. The onehalfwavelength layer
in the interference ﬁlter has a refractive index n. This layer is surrounded by glass of refractive index 711 as shown in the ﬁgure. Calculate, showing all work, the angle of incidence in air, a()\0), needed to pass
freespace wavelength A0. Express your answer as a function of A0, A3, and n only (elimi— nate any angles of refraction). Put your analytic expression in the space provided. (1 (A0) : (analytic expression) Calculate, showing all work, the angular tuning rate of the ﬁlter (the rate of
change of the wavelength transmitted per unit change in the angle of incidence a) at wavelength A0. Put your analytic expression in the space provided. Angular tuning rate : (analytic expression) If a particular interference ﬁlter in air has a speciﬁed wavelength of As = 520 nm,
n z 1.4, and n1 2 1.5, calculate, showing all work, the required angle of incidence
to pass light of 514.5 nm. Express your answer in degrees. For this same case, calcu
late, showing all work, the angular tuning rate of the ﬁlter. Express your answer in
nm/ degree. Express your answers accurately to four signiﬁcant ﬁgures. Put your an— swers in the spaces provided.
0: (A0 : 514.5 nm) 2 degree Angular tuning rate 2 nm/ degree Diffraction by a Rectangular Aperture ' A rectangular aperture has a width of LI and a height of Ly. The amplitude transmittance of the rectangular aperture is given by t(a:,y) = rect(m/L$,y/Ly)
where 'rect(x / Lw, y / Ly) is the two—dimensional rectangular function for which rect(:I:/Lm,y/Ly) = l for —Lx/2S$S+Lm/2 and—Ly/ZSySlLy/Z
= O for x<—L$/2, x>+Lx/2, yg< —Ly/2, y>+Ly/2 This rectangular aperture is coherently illuminated in air at normal incidence by light
of freespace wavelength A. Develop, showing all work, the radiant intensity (analytic ex—
pression) in the far ﬁeld. Express your answer as a function of A, Lx, Ly, 0x (the far—
ﬁeld angle in the x—direction), ﬂy (the far—ﬁeld angle in the ydirection), and I (0) (the
radiant intensity for 0 2 0y 2 0) only (eliminate all other variables). Put your ﬁnal answer in the space provided. I(0z,0y) ll (analytic expression) DVD Grating Beamsplitter Fora DVD tracking system it is desired to split a single laser beam of freespace
wavelength 840 nm into three beams of the same wavelength. One of the beams is to be
undeviated. The other two beams are to be at an angle of 5 degrees on either side of the
undeviated beam. This is to be accomplished by having the original beam normally inci—
dent upon a grating. The grating consists of a periodic series of transparent and opaque stripes (or “line pairs”). Calculate, showing all work, the spatial frequency of the needed grating. Express
your answer in “line pairs per millimeter.” Express your answer accurately to the, near est 0.1 line pairs per millimeter. Put your ﬁnal answer in the space provided. Spatial frequency of grating 2 line pairs/ mm Making a Fourier Transform Hologram A transparency of amplitude transmittance A(a:, y) is coherently illuminated with
collimated light of a freespace wavelength A as shown in the diagram. It is desired to
make a Fourier transtrm hologram of A(:I:, y). Fourier transform lens of focal length
fl is available to use. Draw on the diagram the optical components and light beams re
quired to make a Fourier transform hologram. Label all components. Label important dimensions. Label the Fourier transform hologram. Object Reconstructing a Fourier Transform Hologram It is desired to reconstruct A(a;, y) from the Fourier transform hologram of A(m, y)
as recorded in the previous question. A Fourier transform lensof focal length f2 is avail
A able to use for the reconstruction. Draw on the diagram below the optical components
and light beams required to reconstruct the Fourier transform hologram. Label all com— ponents. Label important dimensions. Label the Fourier transform hologram. Label the reconstruction of A(a:, General Optical Engineering Questions 1. An asymmetric slab waveguide has thickness h, cover index nc, ﬁlm index n f, and
substrate index n3. Light of freespace wavelength A is propagating in this waveguide
both as a TEO mode and as a TE1 mode. It is desired to make the TE1 mode become cutoff. However, it is desired to retain the TEo mode as a propagating mode. How can this be accomplished?
f) all of the above 2. 'An asymmetric slab waveguide has thickness h, cover index nc, ﬁlm index nf, and
substrate index ns. Light of freespace wavelength A is propagating in this waveguide both as a TEo mode and as a TE1 mode. For these modes,
e) all of the above 3. A symmetric slab waveguide has a ﬁlm of refractive index nf and cover and substrate
 of refractive index of 71C 2 n3. The thickness of the ﬁlm is h. For a TEo mode excited in
this waveguide at a freespace wavelength of A, the thickness at cutoff for the TED mode
is a) 0 4. A thin slab in air of thickness d and refractive index n with ﬂat and parallel surfaces will strongly reﬂect normally incident light of freespace wavelength /\ for m being an in
teger if
b) 2nd 2 (m — %))\ Interference Filter 2ndcosﬂ = mA
For [3 = 0 and m = 1, then 2nd 2 /\s Therefore As 003,8 = A0 From Snell’s law sina = nsinﬁ and so cosﬂ = (1—sinzﬂ)1/2
and so As(1 _ saga 1/2 = A0
or <1— — sina : n[1 — ﬁlzr/z
a = a()\0) = SM 1{n [1 — ﬁlial/2} or alternatively
a = sin—1 (n sin ,6) a = a()\0) = sin_1{nsin[(cos“1(§3)]} Differentiate to obtain the tuning rate 2 Ag (1 — Si::a)_1/2(—2 3:17;“ Cosoz) _)\s sinicosa/(l _ Vsig2a)1/2 3L? 3? 3L?
 __ A sin2a 12
) 2n (n2 — sinza
or alternatively dAQ _ _ A  _ _ A 
da — ﬁﬁmacosa) — mszn2a For numerical example A0 = 514.5 nm, As = 520nm, n = 1.4
a()\o) = 11.72° “fl—M = —53.33 nm/md O! m da ——0.931 nm/ degree Diffraction by a Rectangular Aperture Amplitude transmittance t(a:, y) = rect(a:/L$,y/VLy)
t(m,y) = rect(a:/Lx)rect(y/Ly) Twodimensional Fourier transform  +00 +00
F(k,;, kg) = / / t(:1:, y) e_j2"(k’ $+kywdw dy
V w y =~—OO =~OO for even function +00 +00
F(kz, kg) = f / rect(x/L$)  motel/Ly) cos (kzrc) cos(k,,y)dx dy
$=—oo yz—oo . / and separable in a: and y F(k$,ky) = foo 122—00 +00 rect(:z: / L1) cos (kxaz) da:  / 'rect(y/Ly) cos(kyy) dy y=—oo and so DVD Grating Bearnsplitter A = 840nm
OIL] : +50
3:1 z ’50 The forward—diffracted—order grating equation n1 sinﬁ’ — n3 sinﬂg’ = Form=1,ns=1,0’=0,z‘=+1, and 11=—5° 1 _ 31211071
K — _ A
1
Spatial frequency of grating = —— = 000010376 line pairs/mm = 103.76 line pairs/mm A Making. a Fourier Transform Hologram A transparency of amplitude transmittance A(:I:, y) is coherently illuminated with
collimated light of a freespace wavelength A as shown in the diagram. It is desired to
make a Fourier transform hologram of A(:z:, y). A Fourier transform lens of focal length
fl is availableto use. Draw on the diagram the optical components and light beams re—
quired to make a Fourier transform hologram. Label all components. Label important dimensions. Label the Fourier transform hologram. Fourier
Transform Fourier
Transform Reconstructing a Fourier Transform Hologram It is desired to reconstruct A(:c, y) from the Fourier transform hologram of A(:r, y)
as recorded in the previous question. A Fourier transform lens of focal length f2 is avail
able to use for the reconstruction. Draw on the diagram below the optical components
and light beams required to reconstruct the Fourier transform hologram. Label all com— ponents. Label important dimensions. Label the Fourier transform hologram. Label the reconstruction of A(a:, y). Fourier
. r
Fourier Twat..st m
Transform Reconstruction Hologram ...
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 Spring '08
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