Single Element Lens

This is a continuation from the previous tutorial - atomic energy levels and spontaneous emission.

It is well known that the spherical aberration of a lens is a function of its shape factor or bending. Although several definitions for the shape factor have been suggested, a useful formulation is

$\tag{23}\mathscr{K}=\frac{c_1}{c_1-c_2}$

where $$c_1$$ and $$c_2$$ are the curvatures of the lens with the first surface facing the object. By adjusting the lens bending, the spherical aberration can be seen to have a minimum value.

The power of a thin lens or the reciprocal of its focal length is given by

$\tag{24}\phi=\frac{(n-1)c_1}{\mathscr{K}}$

When the object is located at infinity, the shape factor for minimum spherical aberration can be represented by

$\tag{25}\mathscr{K}=\frac{n(2n+1)}{2(n+2)}$

The resultant third-order spherical aberration of the marginal ray in angular units is

$\tag{26}\text{SA}3=\frac{n^2-(2n+1)\mathscr{K}+(1+2/n)\mathscr{K}^2}{16(n-1)^2(\text{FN})^3}$

or after some algebraic manipulations,

$\tag{27}\text{SA}3=\frac{n(4n-1)}{64(n+2)(n-1)^2(\text{FN})^3}$

where, for a thin lens, the $$\text{FN}$$ is the focal length $$f$$ divided by the lens diameter, which in this case is the same as entrance pupil diameter $$D_\text{ep}$$. Inspection of this equation illustrates that smaller values of spherical aberration are obtained as the refractive index increases.

When the object is located at a finite distance $$s_o$$, the equations for the shape factor and residual spherical aberration are more complex.

Recalling that the magnification $$m$$ is the ratio of the object distance to the image distance and that the object distance is negative if the object lies to the left of the lens, the relationship between the object distance and the magnification is

$\tag{28}\frac{1}{s_o\phi}=\frac{m}{1-m}$

where $$m$$ is negative if the object distance and the lens power have opposite signs. The term $$1/s_o\phi$$ represents the reduced or $$\phi$$-normalized reciprocal object distance $$v$$, i. e., $$s_o$$ is measured in units of focal length $$\phi^{-1}$$.

The shape factor for minimum spherical aberration is given by

$\tag{29}\mathscr{K}=\frac{n(2n+1)}{2(n+2)}+\frac{2(n^2-1)}{n+2}\left(\frac{m}{1-m}\right)$

and the resultant third-order spherical aberration of the marginal ray in angular units is

\tag{30}\begin{align}\text{SA}3&=\frac{1}{16(n-1)^2(\text{FN})^3}\left[n^2-(2n+1)\mathscr{K}+\frac{n+2}{n}\mathscr{K}^2+(3n+1)(n-1)\left(\frac{m}{1-m}\right)\right.\\&\quad\left.-\frac{4(n^2-1)}{n}\left(\frac{m}{1-m}\right)\mathscr{K}+\frac{(3n+2)(n-1)^2}{n}\left(\frac{m}{1-m}\right)^2\right]\end{align}

where $$\text{FN}$$ is the effective focal length of the lens $$f$$ divided by its entrance pupil diameter. When the object is located at infinity, the magnification becomes zero and the above two equations reduce to those previously given.

Figure 6 illustrates the variation in shape factor as a function of $$v$$ for refractive indices of 1.5 – 4 for an $$\text{FN}=1$$.

As can be seen from the figure, lenses have a shape factor of 0.5 regardless of the refractive index when the magnification is $$-1$$ or $$v=-0.5$$.

For this shape factor, all lenses have biconvex surfaces with equal radii. When the object is at infinity and the refractive index is 4, lenses have a meniscus shape towards the image.

For a lens with a refractive index of 1.5, the shape is somewhat biconvex, with the second surface having a radius about 6 times greater than the first surface radius.

Since the minimum-spherical lens shape is selected for a specific magnification, the spherical aberration will vary as the object-image conjugates are adjusted.

For example, a lens having a refractive index of 1.5 and configured for $$m=0$$ exhibits a substantial increase in spherical aberration when the lens is used at a magnification of $$-1$$.

Figure 7 illustrates the variation in the angular spherical aberration as both a function of refractive index and reciprocal object distance $$v$$ when the lens bending is for minimum spherical aberration with the object located at infinity. As can be observed from Fig. 7, the ratio of the spherical aberration, when $$m=-0.5$$ and $$m=0$$, increases as $$n$$ increases.

Figure 8 shows the variation in angular spherical aberration when the lens bending is for minimum spherical aberration at a magnification of $$-1$$.

In a like manner, Fig. 9 presents the variation in angular spherical aberration for a convex-plano lens with the plano side facing the image. The figure can also be used when the lens is reversed by simply replacing the object distance with the image distance.

Figures 7 – 9 may provide useful guidance in setting up experiments when the three forms of lenses are available. The so-called ‘‘off-the-shelf’’ lenses that are readily available from a number of vendors often have the convex-plano, equal-radii biconvex, and minimum spherical shapes.

Figure 10 shows the relationship between the third-order spherical aberration and coma, and the shape factor for a thin lens with a refractive index of 1.5 , stop in contact, and the object at infinity. The coma is near zero at the minimum spherical aberration shape. The shape of the lens as a function of shape factor is shown at the top of the figure.

For certain cases, it is desirable to have a single lens with no spherical aberration. A useful form is the plano-convex, with the plano side facing the object, if the convex side is figured as a conic surface with a conic constant of $$-n^2$$.

Caution should be exercised when using this lens form at other than infinite object distances; however, imaging at finite conjugates can be accomplished by using two lenses with their plano surfaces facing one another and the magnification being determined by the ratio of the focal lengths.

It should be noted that for this lens form, the actual thickness of the lenses is not important and that the inclusion of the conic surface does not alter the focal length.

The off-axis performance of a lens shaped for minimum spherical aberration with the object at infinity can be estimated by using the following equations. Assuming that the stop is in contact with the lens, the third-order angular sagittal coma is given by

$\tag{31}\text{CMA}_s=\frac{\theta}{16(n+2)(\text{FN})^2}$

where the field angle $$\theta$$ is expressed in radians. The tangential coma is three times the sagittal coma or $$\text{CMA}_t=3\cdot\text{CMA}_s$$. The diameter of the angular astigmatic blur formed at best focus is expressed by

$\tag{32}\text{AST}=\frac{\theta^2}{\text{FN}}$

The best focus location lies midway between the sagittal and tangential foci. An estimate of the axial angular chromatic aberration is given by

$\tag{33}\text{AChr}=\frac{1}{2V(\text{FN})}$

where $$V$$ is the Abbe number of the glass and $$V=(n_2-1)/(n_3-n_1)$$, with $$n_1\lt{n_2}\lt{n_3}$$.

If a singlet is made with a conic or fourth-order surface, the spherical aberration is corrected by the aspheric surface, and the bending can be used to remove the coma.

With the stop in contact with the lens, the residual astigmatism and chromatic errors remain as expressed by the preceding equations.

Figure 11 depicts the shapes of such singlets for refractive indices of 1.5, 2, 3, and 4. Each lens has a unity focal length and an $$\text{FN}$$ of 10.

Table 1 presents the prescription of each lens where $$\text{CC}_2$$ is the conic constant of the second surface.

The next tutorial discusses about impact of profile design on macrobending losses