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Unformatted text preview: Weierstrass Solid Immersion Lens Compact disks may be read by systems with a variety of optical heads. For ﬁne
pitch tracks, these include nearﬁeld optical components such as solid immersion lenses.
In operation, the compact disk tracks are immediately adjacent to the ﬂat surface of the
lens. One type of these lenses is the Weierstrass solid immersion lens as shown in the ﬁg
ure. It is also referred to as a superhemispherical solid immersion lens. The refractive
index of the lens is m. The refractive index of the surrounding medium is n2. The ra—
dius of curvature is R. The Weierstrass solid immersion lens is a truncated sphere with a
thickness of (1 + 1/n1)R. It is known that for paraxial rays the linear magniﬁcation of
the Weierstrass solid immersion lens is (n1 /n2)2. 4:1"... n2 An object on the compact disk is at the ﬂat surface of the lens extending from
the axis of the lens to its tip at height h. Two rays from the tip of this object are shown
in the diagram. Projecting the refracted rays (righthand region) back to the left, the
image may be Visualized. Calculate without approximations and showing all work, an
alytic expressions for the coordinates (sum, gm) of the tip of the image as a function of
the object height (h). Let a: = 0,1} = 0 be located at the center of curvature. Calcu
late, showing all work, an analytic expression for the linear magniﬁcation m. Put your
answers in the spaces provided below. For the case of a lens of radius 1 mm and refractive index 1.5 in air, calculate,
without approximations, ym, mm, and the linear magniﬁcation m for each value of h. Ex—
press xm and ym in pm accurately to the nearest 0.01 pm. Express m accurately to the nearest 0.0001.
Mum) xmmm) ymwm) m
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10 Weierstrass Solid Immersion Lens Equation of a line y 2 mm + b
dy
y — (7558 + yo
andalso
(151: +
m:— a:
dyy 0 Ray 1 (upper ray) h
sin 0:1 2 E for angle of incidence a1
n1 sin a1 2 n2 sin [31 Snell’s law
n
[31 2 sm‘1 (—1)sznal
n2
(1:131
— = —cot a1 — ,81 — 1 slope
M < > / sin — a sin 7T ~—
—(—@1—————2 = A law of sines for x intercept R 3501
sin(7r  31) = sinﬂl
sin
$01 = —_———ﬁ—1R xintercept
Sin(,81  a1) Ray 1 equation of line sinﬂl
sinwl  a1) R 1:1 —cot(ﬁ1 —— a1)y1 + Ray 2 (lower ray) tan a2 = for angle of incidence a2 h
R(1 + 33) n1 sin a2 2 n2 sin ,62 Snell’ s law Ray 2 equation of line 322 —cot,62 yz + R The magniﬁed image occurs at the intersection of the two ray lines. $1=m2=$m and y2=y1=ym sin ﬁl ,—R = —cot m + R
3171(51 — a1) B2 y —C0t(ﬂl — a1)ym + and so
R[1 _ . Sin/Bl ]
y = ”m
m 00tﬁ2 — 0015(31 — 01)
mm = cotﬂ2 ym + R
_ y_m
m ‘ h ForR 2 1mm, m = 1.5, 71221.00 Mum) xmmm) ym(um) m
300 —1608.24 718.93 2.3965
250 4567.06 585.89 2.3436
200 4539.26 461.24 2.3062
150 4520.65 342.02 2.2802
100 4508.76 226.30 2.2629 50 4502.13 112.66 2.2532 10 —1500.08 22.50 2.2501 ...
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 Spring '08
 Gaylord

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