Lenses
10
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. con- trolling 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 n i which are incident on a curved interface separating medium n t . From refraction we know that when part of the original wavefront enters medium n t it bends so as to maintain the constant phase. If medium n t >n i 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: n i (SA)+ n t (AD)= Const. (1) where an arbitrary path between S and the plane DD passing through A was chosen. From this equa- tion, we can estimate the shape of the interface region by, which is expressed here in terms of the distance SA as: SA +( n t n i )AD = Const. (2) Equation 2, is the equation of a surface whose shape depends on the ratio e = n t /n i 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 n t >n i : Here the spherical waves can be converted into plane waves. 2. e< 1 or n t <n i : 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. 1
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Lenses
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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. con-trolling the geometry of the wavefront. We have already seen in the chapter on Light that the shape of awavefront can be controlled by the geometry of the source. In this chapter, we look at interaction of lightwith the curved surfaces that make up the most common optical element, a lens. We can begin by asking animportant question: What surface shape is required to create plane waves from spherical waves? Considerg. 1 where a spherical sourceSgenerates spherical wavefronts in mediumniwhich are incident on acurved interface separating medium nt. From refraction we know that when part of the original wavefrontenters medium nt it bends so as to maintain the constant phase. If medium nt>ni then the waves slowdown upon entering the medium. The shape of the new wavefront can be obtained as always, i.e. by joiningthe points having the same phase - or nding the locus of all points with the same phase. Since we areinterested 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 samephase, their optical path length (OPL) must be the identical. Therefore, in Fig. 1(c), any ray travelling fromS 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 equa-tion, we can estimate the shape of the interface region by, which is expressed here in terms of the distanceSA as:SA + (ntni)AD = Const. (2)Equation 2, is the equation of a surface whose shape depends on the ratio e = nt/ni and is similar to theequation 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
