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represents the algebraic sum of the asphericity sags of the two surfaces.

Since the power of a lens is K = (n - 1) (c1 - c2), after substitution the condition is

from where, given the power of the lens, the three available parameters are the cambrure (or bending) c2/c1 and the two conic constants. For instance, this allows removing Sphe 3 and Coma 3 simultaneously.

• Stigmatic aspherical lens: If the power is positive and Sphe 3 only removed, and if c2 = k2 = 0 and n = 3/2, then c1 > 0 and the amount of asphericity Zasp = 8 K1c\r4 must be set by K = -n3/(n +1)2 = -27/50 = -0.54. The other solution with a plane surface, c1 = k1 = 0, is with c2 < 0 and k2 = -27/50 which, of course, is the same in the Gaussian theory of thin lenses.

Aplanatic aspherical lens: If Sphe and Coma 3 are removed, then SI = SII = 0 where the second condition entails c2/c1 as given by (9.15). Hence, for n = 3/2, we obtain c2/c1 = -1/9 from (9.16), and (9.18c) writes

showing, of course, that there is an infinity of aplanatic lenses with two aspherical surfaces. When only one surface is aspherical, the two solutions are K1 = 0, k2 = -540 with c2 negative, and K1 = -20/27, k2 = 0 with c1 positive. For an object at infinity and a positive lens with the entrance pupil on it, the conic constants in the above results are all negative. This can be easily generalized whatever the refractive index.

^ In the third-order theory, whether a stigmatic or an aplanatic lens, and whether one or both surfaces are aspherized, a single lens must be thickened towards its edge.

### 9.1.6 Power of a Two-Lens System

From formula (9.1) expressing the power of a thin lens, one shows that the resulting power K of a system with two thin lenses which are axially air-spaced by a distance d is

For two lenses which are in contact, the resulting power simply reduces to the sum of the individual powers.

It is clear that the thin lens concept gives rigorous analytical results for a lens whose central thickness is zero which is manifestly impossible to achieve in practice. However, in a first stage, the concept allows accurate predictions and studies of the basic properties of any lens system in the first- and third-order theories of dioptrics and aberrations.

9.2 Thin Lens Elastically Bent by Uniform Load 9.2.1 Equilibrium Equation of the Thin Plate Theory

The flexure of an axisymmetric lens by a uniform load applied over all its surface with a reaction at the circular edge involves taking into account its non-uniform thickness t (r). The rigidity of the lens is defined by

where E and v are the Young modulus and Poisson's is ratio of the material.

The equilibrium equation of the thin plate theory, (3.3) in Sect. 3.2, applies to the case of a variable thickness plate. For a lens with an axisymmetric thickness this equation reduces to

dr where the radial and tangential bending moments are, from (3.1a) and (3.1b),

r dr dr2 r dr and the slope of the flexure is

At a circle of radius r, the shearing force per unit length is

where q is the intensity of the uniform load. As for all other chapters,

^ the load q is positive when directed towards the positive z direction.

Substituting the expressions of the bending moments and shearing force into (9.21), the equilibrium equation may be written d2m (1 3 dt\drn (1 3v dt\ 2. qr

where the unknown is the slope 2 of the flexure. The two coefficients in parentheses on the left side are determined as input data from the geometry of the lens.

9.2.2 Lens Deformation and Parabolic Thickness Distribution

References on studies of the flexure of variable thickness plates by various authors can be found in Timoshenko's book [9]. However, none of them directly represent the geometry of a lens, thus we developed the following analysis for this purpose [10].

Approximating the spherical surfaces of a lens by the curvature term of their power expansion, the thickness distribution is then represented by the addition of a constant thickness plate and a paraboloid, t(r)=to - 1 (ci - C2) r2, (9.25)

where t0 = t(0) is the thickness at its center.

The thin plate theory of elasticity assumes that the middle surface of the plate is flat, and then does not take into account the cambrure of the lens. Hence for a noticeable cambrure, say |B| > 2 as defined in Sect. 9.1.1, a more accurate analysis would require using the theory of shells. We adopt for the lens curvatures the same sign convention as in the previous sections. The quantity c1 - c2 is proportional to the optical power, and thus is positive for a lens with positive power K [cf. (9.1)]. A simple configuration for the aspherization of a positive lens by stress polishing consists of bonding its edge into a thin L-square ring absorbing the reaction and allowing it to tangentially rotate when the load is applied (Fig. 9.7).

Let us introduce dimensionless quantities with respect to the outer radius r = a of the lens, as p = -, T = -, To = - and m = ± a(c1 - C2) (9.26)

where the sign of the pseudo-power m is m > 0 for a positive lens(K > 0), m < 0 for a negative lens(K < 0).

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Fig. 9.7 Lens bent by a uniform load q in reaction with simply supported edge. Left: Positive lens and positive load. Right: Negative lens and negative load grease

Fig. 9.7 Lens bent by a uniform load q in reaction with simply supported edge. Left: Positive lens and positive load. Right: Negative lens and negative load

The dimensionless thickness is

and substituting these quantities into (9.24), we obtain

^ + (( i + = 6( 1 — v2)q p dp^np fp -ip?;*=6(1 - v2)e. (9.28)

This can be simplified by defining the quantities

Hence the differential equation of the slope-flexure of a lens, bent by a uniform load in reaction at its edge of diameter 2a, and its associated reduced thickness T = t/a assumed of quadratic shape, take the form

T = m(S -p2) with T0 = mS> 0 and 0 < p < 1. (9.30b)

One can easily show that the particular solution cpo of (9.30a) is of the same form as the left side term but only differs by from a multiplicative factor. After identification of this factor, the particular solution is

For deriving the solution (p1 without left side, let us denote | such as 2(5 - 3v) = 4(4 - |) i.e.

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