Curved surfaces and lenses

Curved surfaces and lenses (Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 5)

One of the important challenges pertaining to the practical aspects of optics is wave shaping, i.e. controlling the geometry of the wavefront. We have already seen in the chapter on Light that the shape of a wavefront can be controlled by the geometry of the source. In this chapter, we look at interaction of light with the curved surfaces that make up the most common optical element, a lens. We can begin by asking an important question: What surface shape is required to create plane waves from spherical waves? Consider fig. 1 where a spherical source S generates spherical wavefronts in medium ni which are incident on a curved interface separating medium nt . From refraction we know that when part of the original wavefront enters medium nt it bends so as to maintain the constant phase. If medium nt > ni then the waves slow down upon entering the medium. The shape of the new wavefront can be obtained as always, i.e. by joining the points having the same phase - or finding the locus of all points with the same phase. Since we are interested here in creating plane waves, let us hypothetically mark a region of constant phase shown as DD! that is planar and perpendicular to the wavefronts. Now from our earlier discussion of the optical path length (Ch. on Light-Matter interaction) in order for waves leaving S and arriving on plane DD! to have the same phase, their optical path length (OPL) must be the identical. Therefore, in Fig. 1(c), any ray travelling from S to D must have the same or constant OPL phase and this condition can be expressed generally as: ni (SA) + nt (AD) = Const.

(1)

where an arbitrary path between S and the plane DD! passing through A was chosen. From this equation, we can estimate the shape of the interface region by, which is expressed here in terms of the distance SA as: SA + (

nt )AD = Const. ni

(2)

Equation 2, is the equation of a surface whose shape depends on the ratio e = nt /ni and is similar to the equation of a hyperbola with e representing the eccentricity. There are some typical cases of e of interest: 1. e > 1 or nt > ni : Here the spherical waves can be converted into plane waves. 2. e < 1 or nt < ni : Here plane waves can be converted into spherical waves, as can be understood by reversing the path of light in Fig. 1 Some general scenarios representing the above two situations are depicted in Fig. 2.

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Figure 1: Interaction of light with spherical wavefronts a hyperbolic curved surface. The surface here transforms a spherical wavefront into a planar wavefront. Source: Hecht, Ch 5.

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Image formation at a spherical refracting surface

Consider a refracting surface that is part of a sphere of radius of curvature R, as shown in Fig. 3. The light source or object S emits spherical waves from a distance so . Any ray after refraction at the interface is refracted towards the normal (because n2 < n1 , as for example SA is refracted towards AP which lies on an axis of symmetry called the Central or Optic axis. From symmetry, all rays incident at the same angle θi , which will line on a cone with apex at S, will be refracted through the same angle and will all intersect at point P forming the image of S. The distance of this point from the surface is known as the image distance si . To understand the quantitative behavior of light travelling between two points, it is useful to look at Fermat’s principle.

1.1

Fermat’s principle and Image-object spacing relation for spherical surfaces

According to Fermat, a light ray travelling between two points follows a path (compared to nearby paths) where the time required is a minimum (or maximum) or stationary (i.e. mathematically the derivative with respect to a position variable is zero). Using this principle, various laws of optics can be derived, including the laws of reflection and refraction. Going back to the spherical surface of Fig. 3, Fermat’s principle can be used to determine the true OPL for any ray by minimizing its path length with respect to a position variable. Consider the ray travelling 2

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Figure 2: Interaction between light and surfaces of various elliptical shapes that leads to a modification of the wavefront. Source: Hecht, ch 5.

Figure 3: A spherical refracting surface showing various distances and angles used to determine the imageobject relations. Source: Hecht, ch 5. the path SAP whose OP L = n1 l0 + n2 li . In terms of the object and image distance and the the angle φ subtended by point A with respect to the center of the sphere at C, the OP L can be written as: OP L = l = n1 [R2 + (so + R)2 − 2R(so + R)cosφ]1/2 + n2 [R2 + (si − R)2 + 2R(si − R)cosφ]1/2 (3) here the position variable can be taken as φ and so to apply Fermat’s principle we solve for which gives us:

n1 n2 (n2 − n1 ) + = so si R

dl dφ

= 0, (4)

The above result is based on the small angle approximation, i.e. cosφ ∼ 1 and so lo ∼ so and li ∼ si .

This equation give the condition for image formation under First-Order Paraxial or Gaussian Optics. i.e. the angle formed by the incident and refracted beams is very small with respect to to the Optic axis and passes close to the center of the surface.

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1.1.1 Special cases of the Image (I)- Object (O) distances 1. When the image is formed at ∞, i.e. si = ∞, then the refracted rays travel parallel to the optic axis and to each other. From eq. 4, this gives:

so =

n2 R = fo n2 − n1

(5)

This special distance fo is known as the First or Object focal length. 2. When the object is at ∞, i.e. so = ∞, i.e. plane waves are incident on the surface then, from eq. 4: si =

n2 R = fi n2 − n1

(6)

This special distance fi is known as the Second or Image focal length. The two special I/O distances are together known as the conjugate focii.

1.2

Real vs. Virtual Images

So far the cases considered correspond to formation of Real images where light is actually incident at the image position. i.e. a light meter positioned at the location would register the presence of photons. However in many situations an image (or object) position may not have photons associated with it and in these situations the positions are known as virtual positions. Consider the situation in Fig. 4 in which the object is positioned close to the refracting surface such that rays make large incident angles. After refraction, the rays do not converge to the optic axis and in fact diverge away from the optic axis. To an observer, like a human eye, these diverging rays will appear to come from position I. However, there is no light actually incident at or from I. This type of image is a virtual image.

1.3

Convex vs Concave surfaces

A convex surface is defined as one whose refracting surface is thicker at the center (Fig. 3) while for a concave surface the curved refracting surface is thinner at the center (Fig. 5). The behavior of light incident on a concave surface can be understood by using Eq. 4, refraction and the nature of the wavefronts. Consider two situations, with reference to Fig. 5: 1. When the incident light on a concave surface is parallel to the optic axis (Fig. 5(a)): Since light bends towards the normal, it diverges at a concave surface forming a virtual image. Using eq. 4 and noting that so = ∞, we get:

si = −fi fi = −(

n2 )R n2 − n1

(7)

The negative sign arises from the sign convention which states that when the image the image is on the same side as the object it is negative. 4

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Figure 4: Image-Object relation for various conditions at a spherical refracting surface. (a) General case of spherical wavefront from S being focused at P. (b) special case of image being formed at ∞ for the object positioned at a distance of fo . (c) A virtual image formed at P ! when the rays from S are incident such that the refracted rays diverge from the optic axis. Source: Hecht, Ch 5. 2. When the image is formed at ∞, i.e. the refracted light is parallel to the optic axis (Fig. 5(b)): For

this condition to occur, the incident light must be converging on the concave surface. Therefore, the object position will appear to be at position so on the right side of the concave surface as shown in the figure. This object is a virtual object and by sign convention is negative and eq. 4 gives: fo = −(

1.4

n2 )R n1 − n2

(8)

Sign convention

Table 5.1, p 154 in Hecht summarizes the sign convention for refractive surfaces. The convention assumes that light is incident on the refracting surface from the left hand side as so: • so , fo are positive if they are on the left side of the surface • si , fi are positive if they are on the right side of the surface • R is positive if the center of the spherical surface is to the right • yo and yi , which designate object and image heights with respect to the optic axis are both positive if they are above the optic axis.

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Figure 5: Light interaction at a concave surface. (a) Incident light parallel to the optic axis after refraction diverges in medium n2 . A virtual image is formed at position si . (b) Converging light after refraction travels parallel to the optic axis. The virtual object is located at so .

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Spherical thin lenses

From the previous discussion, the image and object distances for a single spherical surface are related by eq. 4. The I and O positions are also symmetric as they are related to each other by the principle of reversibility, i.e. an object at P will be imaged at S (fig. 2). These two points which are this related are conjugate points. Now consider the situation shown in Fig. 6 where two spherical surfaces (not necessarily having the same radius of curvature R) combine to form a lens of refractive index nl . We will describe now the behavior of light traversing from medium with refractive index nm through the two refracting surfaces making up the lens. • From eq. 4, for a fixed value of

nl −nm R

we can easily verify the following:

– As so increases, si decreases and vice-versa – At so = fo , si = ∞

– For so < fo , si = negative, i.e. a virtual image is formed • In Fig. 6, consider the image formed by the left surface when the object S is at so1 < fo1 . Here a virtual image is formed at P ! at a distance si1 away from surface 1. From eq. 4: nm nl nl − nm + = so1 si1 R1

(9)

• Now the second surface sees rays coming effectively from P ! with an object distance | so2 |=| si1 | +d. Using the sign conventions:

– so2 = positive because it is on the left of surface 2 6

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– si1 = negative because it is on the left of surface 1 we get: so2 = −si1 + d. Therefore for the second surface (# 20): nm nm − nl nl + = −si1 + d si2 R2

(10)

Adding equations 9 and 10 gives: nm nm 1 1 nl d + = (nl − nm )( − )+ so1 si2 R1 R2 (si1 − d)si1

(11)

The above is the general form of the lens equation which can be used to solve most lens related problems. However, some useful simplifications of this equation are commonly used: 1. The thin lens approximation: In your lab, A1, you will be required to use the thin lens form of the lens equation which is obtained by setting d ∼0 giving: 1 1 1 1 + = (nl − nm )( − ) so si R1 R2

(12)

2. Lens used in air with nm ∼ 1. Here the equation reduces to the classical lens or lensmakers formula: 1 1 1 1 + = (nl − 1)( − ) so si R1 R2

(13)

3. Gaussian lens formula: Another important form of the lens formula is based on the principle of reversibility. Consider the two cases when so = ∞ and si = ∞. In each case, the lens formula given

by eq. 13 reduces to:

fi = (nl − 1)(

1 1 − ) = fo R1 R2

where so = fo when si = ∞ and si = fi when so − ∞. Therefore, the Gaussian lens formula can be

written as:

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1 1 1 1 1 + = = (nl − 1)( − ) so si f R1 R2

(14)

Image formation rules for a lens

Some common image forming rules for a lens can be defined and on the basis of these rules lens properties can be readily obtained. Some of these rules are: 1. A ray travelling parallel through the optic axis pass through the focus 2. A ray passing through the optic center is undeviated. The optic center is defined ads the point near the center of a lens lying on the optic axis though which all rays passing suffer no angular deviation 3. A ray which goes through the focus is parallel to the optic axis after passing through the lens. 7

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Figure 6: Image formation by thin lens showing the various image/object positions for each refracting surface. Source: Hecht, ch 5.

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Newtonian form of lens equation

With reference to Fig. 7, consider an object of height yo at position s1 . Using image formation rules, the resulting image of s1 is located at P1 and is an inverted image of size yi . In similar triangles AOFi and P1 P2 O :

yo f = | yi | (si − f ) = xi

(15)

so yo = | yi | si

(16)

Similarly in triangles S2 S1 O and P2 P1 O

Therefore, combining eq. 15 and 16 we get: 1 1 f 1 so or = = + si si − f f so si Which is identical to the Gaussian lens equation. Also from similar triangles S2 S1 Fo and BOFo f yi = (so − f ) = xo | yi |

(17)

and combining eq.’s 15 and 17 we get: xo xi = f 2 This is known as the Newtonian form of the lens equation.

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(18)

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Figure 7: Figure showing the variables describing the Newtonian form of the lens equation. Source: Hecht, ch 5.

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Thin lens combination

It is often useful to analyze the behavior of a combination of lenses because numerous optical instruments, like telescopes, binoculars and microscopes, make use of a combination of lenses to form images. Here we discuss the simplest case of two thin lenses separated by a distance d which is much smaller than the focal length of either lens (see Fig. 8). For lens 1, the I/O relation gives:

so1 f1 so1 − f 1

(19)

so2 = d − si1

(20)

so2 f2 so2 − f2

(21)

si1 = • Similarly for lens 2: the object is at P ! , i.e.

If | d |