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Unformatted text preview: THIN LENSES Using geometrical optics concepts, a lens can frequently be approximated as a
“thin lens.” Such a lens is described by two refractions at spherical surfaces. The dis
tance between the spherical surfaces is treated as negligible compared to the object and
image distances. In many practical cases, this is a good approxiation. Thus the lens
thickness is taken as approaching zero. Thus for a thin lens there is negligible displace
ment of the rays passing through it. Rays are solely deviated by the lens. For the anal—
ysis of thin lenses, a standard conﬁguration and set of sign conventions is used here sim
ilar to that used for refractive surfaces. The region surrounding the lens has a refractive
index of m. The thin lens has a refractive index of 722. The left—hand refracting surface
has a radius of R1 and the right—hand surface has a radius R2. The focal length of the
thin lens is given by f. The conventions used here are as follows:
1) The incident light propagates from left to right toward the lens. 2) The object distance is s. It is positive to the left of the lens and negative to the right of the lens. 3) The image distance is s’. It is positive for images to the right of the lens. A positive
3 ’ indicates a real image. It is negative for images to the left of the lens. A negative 3’ indicates a virtual image. 4) The radius R1 or R2 is positive for convex refractive surfaces. In this case the center
of curvature C is to the right of the lens. The radius R1 or R2 is negative for a concave refractive surface. The center of curvature C is to the left of the lens. 5) The focal length f is positive for a converging lens and negative for a diverging lens. In the paraxial ray approximation, for a thin lens, the object and image distances are interrelated by For parallel rays, the focal length f is given by }= (Bil7%), which is known as the lens maker’s equation since it predicts the focal length of a lens for given radii of curvature and refractive index. Combining these gives 1 1
7:" 1+
s f’ S which is known as the thin lens equation. The linear magniﬁcation m is given by THIN LENS EQUATION FROM TWO REFRACTIONS Imaging by a thin lens may be described as a refraction by the ﬁrst spherical sur—
face followed by a refraction due to the second spherical surface. The two refractions
combine to produce the same effect‘as the thin lens. Since the lens is approximated as
being “thin,” the two refractive surfaces are treated as though they are at the same lo—
cation. The paraxial ray analysis of refraction by a single spherical surface, as presented
previously, is used. All of the conventions used there, apply to the thin lens case. The
output image distance calculated for the ﬁrst surface is then used as input image dis—
tance for the second surface. The lefthand incident region (region 1) has a refractive
index of m; the lens (region 2) has a refractive index n2; the righthand transmitted re gion (region 3) has an index of n3. This conﬁguration is shown in Fig. 1 Fig. 1. A thin lens of refractive index 712 with surfaces of radii R1 and R2. The refrac— tive indices of the incident and transmitted regions are n1 and n3 respectively. For the ﬁrst spherical refractive surface of radius R1 (which may be convex or
concave), the refraction is described, in the paraxial ray approximation, by
n1 n2 n2 — n1 __ _=_.._ 1
81+51’ R1 , where 31 and 31 ’ are the object distance and the image distance associated with the ﬁrst surface. For the second refractive surface of radius R2 (which may be convex or concave), the refraction is described by _+—= 1 (2) where 32 and 32 ’ are the object distance and the image distance associated with the sec—
ond surface. The object distance for the second surface is the same as the image dis— tance from the ﬁrst surface. That is, 82 = ‘31, Adding Eqs. (1) and (2) and using the fact that n2/82 2 —n2 / 31 ’ gives 711 n3 n2 — n1 n3 — n2 81 82' R1 + R2 ( )
which is the general thin lens equation for differing incident and transmitted regions.
The linear magniﬁcation m is given by n1 82 I
m = _ — —_ 5
M 81 ( ) For parallel on—axis rays propagating from left to right (set 31 = 00), the focal length f3 is given by
713 “2 — 711 7L3 — n2
_ I 73— — R1 R2 (6) For parallel on—axis rays propagating from right to left (set 32 ’ = 00), the focal length fl is given by
n1 n2 — m + n3 — n2 , f1 R1 R2 I For the commonly occurring case of 723 2 n1, Eq. (4) reduces to the standard thin lens equation 1 1 _ (HQ—77,1) 1 1
81 + 82/ n1 (R1 R2), and the magniﬁcation is ...
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 Spring '08
 Gaylord
 7%, Geometrical optics, 5 m

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