# Introduction to lenses for image formation and manipulation

This is a continuation from the previous tutorial - *what is a laser?*

### Basics

This tutorial provides a basic understanding of using lenses for image formation and manipulation. The principles of image formation are reviewed first. The effects of lens shape, index of refraction, magnification, and F-number on the image quality of a singlet lens are discussed in some detail.

Achromatic doublets and more complex lens systems are covered next. A representative variety of lenses is analyzed and discussed. Performance that may be expected of each class of lens is presented. The tutorial concludes with several techniques for rapid estimation of the performance of lenses.

Figure 1 illustrates an image being formed by a simple lens.

The object height is \(h_o\) and the image height is \(h_i\), with \(u_o\) and \(u_i\) being the corresponding slope angles. It follows from the Lagrange invariant that the lateral magnification is defined to be

\[\tag{1}\begin{align}m&\equiv\frac{h_i}{h_o}\\&=\frac{(nu)_o}{(nu)_i}\end{align}\]

where \(n_o\) and \(n_i\) are the refractive indices of the medium in which the object and image lie, respectively. By convention, a height is positive if above the optical axis and a ray angle is positive if its slope angle is positive. Distances are positive if the ray propagates left to right.

Since the Lagrange invariant is applicable for paraxial rays, the angle \(nu\) should be understood to mean \(n\) tan \(u\). This interpretation applies to all paraxial computations. For an aplanatic lens, which is free of spherical aberration and linear coma, the magnification can be shown by the optical sine theorem to be given by

\[\tag{2}\begin{align}m&\equiv\frac{h_i}{h_o}\\&=\frac{n_o\sin{u_o}}{n_i\sin{u_i}}\end{align}\]

If the object is moved a small distance \(\partial{s_o}\) longitudinally, the corresponding displacement of the image \(\partial{s_i}\) can be found by the differential form of the basic imaging equation and leads to an equation analogous to the Lagrange invariant. The longitudinal magnification is then defined as

\[\tag{3}\begin{align}\bar{m}&\equiv\frac{\partial{s_i}}{\partial{s_o}}\\&=\frac{(nu^2)_o}{(nu^2)_i}\\&=m^2\left[\frac{n_i}{n_o}\right]\end{align}\]

The following example will illustrate one application of \(m\) and \(\bar{m}\). Consider that a spherical object of radius \(r_o\) is to be imaged as shown in Fig. 2.

The equation of the object is \(r_o^2=y_o^2+z^2\), where \(z\) is measured along the optical axis and is zero at the object’s center of curvature. Letting the surface sag as measured from the vertex plane of the object be denoted as \(\zeta_o\), the equation of the object becomes \(r_o^2=(r_o-\zeta_o)^2+y_o^2\) since \(z=r_o-\zeta_o\). In the region near the optical axis, \(\zeta_o^2\ll{r_o}^2\), which implies that \(r_o\approx{y_o}^2/2\zeta_o\). The image of the object is expressed in the transverse or lateral direction by \(y_i=my_o\) and in the longitudinal or axial direction by \(\zeta_i=\bar{m}\zeta_o=\zeta_o{m^2}(n_i/n_o)\). In a like manner, the image of the spherical object is expressed as \(r_i\approx(y_i)^2/2\zeta_i\). By substitution, the sag of the image is expressed by

\[\tag{4}\begin{align}r_i&\equiv\frac{n_oy_o^2}{2n_i\zeta_o}\\&=r_o\left[\frac{n_o}{n_i}\right]\end{align}\]

Hence, in the paraxial region about the optical axis, the radius of the image of a spherical object is independent of the magnification and depends only on the ratio of the refractive indices of the object and image spaces.

When an optical system as shown in Fig. 3 images a tilted object, the image will also be tilted. By employing the concept of lateral and longitudinal magnification, it can be easily shown that the intersection height of the object plane with the first principal plane \(P_1\) of the lens must be the same as the intersection height of the image plane with the second principal plane \(P_2\) of the lens. This principle is known as the Scheimpflug condition.

The object-image relationship of a lens system is often described with respect to its cardinal points, which are as follows:

- Principal points: the axial intersection point of conjugate planes related by unit lateral magnification
- Nodal points: conjugate points related by unit angular magnification (\(m=u_i/u_o\))
- Focal points: front (\(f_1\)) and rear (\(f_2\))

The focal length of a lens is related to the power of the lens by

\[\tag{5}\phi=\frac{n_o}{f_o}=\frac{n_i}{f_i}\]

This relationship is important in such optical systems as underwater cameras, cameras in space, etc. For example, it is evident that the field of view is decreased for a camera in water.

The lens law can be expressed in several forms. If \(s_o\) and \(s_i\) are the distance from the object to the first principal point and the distance from the second principal point to the image, then the relationship between the object and the image is given by

\[\tag{6}\phi=\frac{n_i}{s_i}+\frac{n_o}{s_o}\]

Should the distances be measured with respect to the nodal points, the imaging equation becomes

\[\tag{7}\phi=\frac{n_o}{s_i}+\frac{n_i}{s_o}\]

When the distances are measured from the focal points, the image relationship, known as the Newtonian imaging equation, is given by

\[\tag{8}f_1f_2=s_os_i\]

The power of a spherical refracting surface, with curvature \(c\) and \(n\) being the refractive index following the surface, is given by

\[\tag{9}\phi=c(n-n_o)\]

It can be shown that the power of a single thick lens in air is

\[\tag{10}\phi_\text{thick}=\phi_1+\phi_2-\phi_1\phi_2\frac{t}{n}\]

where \(t\) is the thickness of the lens.

The distance from the first principal plane to the first surface is \(-(t/n)\phi_2f_1\) and the distance from the second principal point to the rear surface is \((-t/n)\phi_1f_2\). The power of a thin lens (\(t\rightarrow0\)) in air is given by

\[\tag{11}\phi_\text{thin}=(n-1)(c_1-c_2)\]

### Stops and Pupils

The aperture stop or stop of a lens is the limiting aperture associated with the lens that determines how large an axial beam may pass through the lens. The stop is also called an iris. The marginal ray is the extreme ray from the axial point of the object through the edge of the stop.

The entrance pupil is the image of the stop formed by all lenses preceding it when viewed from object space. The exit pupil is the image of the stop formed by all lenses following it when viewed from image space. These pupils and the stop are all images of one another.

The principal ray is defined as the ray emanating from an off-axis object point that passes through the center of the stop. In the absence of pupil aberrations, the principal ray also passes through the center of the entrance and exit pupils.

As the obliquity angle of the principal ray increases, the defining apertures of the components comprising the lens may limit the passage of some of the rays in the entering beam thereby causing the stop not to be filled with rays.

The failure of an off-axis beam to fill the aperture stop is called vignetting. The ray centered between the upper and lower rays defining the oblique beam is called the chief ray.

When the object moves to large off-axis locations, the entrance pupil often has a highly distorted shape, may be tilted, and/or displaced longitudinally and transversely. Due to the vignetting and pupil aberrations, the chief and principal rays may become displaced from one another. In some cases, the principal ray is vignetted.

The field stop is an aperture that limits the passage of principal rays beyond a certain field angle. The image of the field stop when viewed from object space is called the entrance window and is called the exit window when viewed from image space. The field stop effectively controls the field of view of the lens system. Should the field stop be coincident with an image formed within or by the lens system, the entrance and exit windows will be located at the object and/or image(s).

A telecentric stop is an aperture located such that the entrance and/or exit pupils are located at infinity. This is accomplished by placing the aperture in the focal plane. Consider a stop placed at the front focal plane of a lens. The image is located at infinity and the principal ray exits the lens parallel to the optical axis.

This feature is often used in metrology since the measurement error is reduced when compared to conventional lens systems because the centroid of the blur remains at the same height from the optical axis even as the focus is varied.

### F-Number and Numerical Aperture

The focal ratio or F-number (FN) of a lens is defined as the effective focal length divided by the entrance pupil diameter \(D_\text{ep}\). When the object is not located at infinity, the effective FN is given by

\[\tag{12}\text{FN}_\text{eff}=\text{FN}_\infty(1-m)\]

where \(m\) is the magnification. For example, for a simple positive lens being used at unity magnification (\(m=-1\)), the \(\text{FN}_\text{eff}=2\text{FN}_\infty\). The numerical aperture of a lens is defined as

\[\tag{13}\text{NA}=n_i\sin{U_i}\]

where \(n_i\) is the refractive index in which the image lies and \(U_i\) is the slope angle of the marginal ray exiting the lens. If the lens is aplanatic, then

\[\tag{14}\text{FN}_\text{eff}=\frac{1}{2\text{NA}}\]

### Magnifier or Eye Loupe

The typical magnifying glass, or loupe, comprises a singlet lens and is used to produce an erect but virtual magnified image of an object. The magnifying power of the loupe is stated to be the ratio of the angular size of the image when viewed through the magnifier to the angular size without the magnifier. By using the thin-lens model of the human eye, the magnifying power (MP) can be shown to be given by

\[\tag{15}\text{MP}=\frac{25\text{ cm}}{d_e+d_o-\phi{d_e}d_o}\]

where \(d_o\) is the distance from the object to the loupe, \(d_e\) is the separation of the loupe from the eye, and \(\phi=1/f\) is the power of the magnifier. When \(d_o\) is set to the focal length of the lens, the virtual image is placed at infinity and the magnifying power reduces to

\[\tag{16}\text{MP}=\frac{25\text{ cm}}{f}\]

Should the virtual image be located at the near viewing distance of the eye (about 25 cm), then

\[\tag{17}\text{MP}=\frac{25\text{ cm}}{f}+1\]

Typically simple magnifiers are difficult to make with magnifying powers greater than about \(10\times\).

### Compound Microscopes

For magnifying power greater than that of a simple magnifier, a compound microscope, which comprises an objective lens and an eyepiece, may be used.

The objective forms an aerial image of the object at a distance \(s_{ot}\) from the rear focal point of the objective. The distance \(s_{ot}\) is called the optical tube length and is typically 160 mm. The objective magnification is

\[\tag{18}\text{MP}_{obj}=\frac{s_{ot}}{f_{obj}}\]

The image formed is further magnified by the eyepiece which has a \(\text{MP}_{ep}=250\text{ mm}/f_{ep}\). The total magnifying power of the compound microscope is given by

\[\tag{19}\begin{align}\text{MP}&=\text{MP}_{obj}\text{MP}_{ep}\\&=\frac{160}{f_{obj}}\cdot\frac{250}{f_{ep}}\end{align}\]

Typically, \(f_{ep}=25\) mm, so its \(\text{MP}=10\). Should the objective have a focal length of 10 mm, the total magnifying power of the microscope is \(16\times\) times \(10\times\), or \(160\times\).

### Field and Relay Lenses

Field lenses are placed at (or near) an image location for the purpose of optically relocating the pupil or to increase the field of view of the optical system. For example, a field lens may be used at the image plane of an astronomical telescope such that the field lens images the objective lens onto the eyepiece.

In general , the field lens does not contribute to the aberrations of the system except for distortion and field curvature. Since the field lens must be positive, it adds inward curving Petzval. For systems having a small detector requiring an apparent increase in size, the field lens is a possible solution.

The detector is located beyond the image plane such that it subtends the same angle as the objective lens when viewed from the image point. The field lens images the objective lens onto the detector.

Relay lenses are used to transfer an image from one location to another such as in a submarine periscope or borescope. It is also used as a means to erect an image in many types of telescopes and other such instruments. Often relay lenses are made using two lens groups spaced about a stop, or an image of the system stop, in order to take advantage of the principle of symmetry, thereby minimizing the comatic aberrations and lateral color. The relayed image is frequently magnified.

### Aplanatic Surfaces and Immersion Lenses

Abbe called a lens an aplanat that has an equivalent refractive surface which is a portion of a sphere with a radius \(r\) centered about the focal point. Such a lens satisfies the Abbe sine condition and implies that the lens is free of spherical and coma near the optical axis. Consequently, the maximum possible numerical aperture (NA) of an aplanat is unity, or an FN=0.5. In practice, an FN less than 0.6 is difficult to achieve. For an aplanat,

\[\tag{20}\text{FN}=\frac{1}{2\cdot\text{NA}}\]

It can be shown that three cases exist where the spherical aberration is zero for a spherical surface. These are: (1) the trivial case where the object and image are located at the surface, (2) the object and image are located at the center of curvature of the surface, and (3) the object is located at the aplanatic point.

The third case is of primary interest. If the refractive index preceding the surface is \(n_o\) and following the surface is \(n_i\), then the object is located a distance \(s_o\) from the surface as expressed by

\[\tag{21}s_o=\frac{r(n_o+n_i)}{n_o}\]

and the image is located at

\[\tag{22}s_i=\frac{r(n_o+n_i)}{n_i}\]

An immersion lens or contact lens can be formed from an aplanatic surface and a plano surface. Figure 4 illustrates a hemispherical magnifier that employs the second aplanatic case. The resultant magnification is \(n_i\) if in air or \(n_i/n_o\) otherwise.

A similar magnifier can be constructed by using a hyperhemispherical surface and a plano surface as depicted in Fig. 5. The lateral magnification is \(n_i^2\). This lens, called an Amici lens, is based upon the third aplanatic case. The image is free of all orders of spherical aberration, third-order coma, and third-order astigmatism.

Axial color is also absent from the hemispherical magnifier. These magnifiers are often used as a means to make a detector appear larger and as the first component in microscope objectives.

The next tutorial introduces ** working definitions of cutoff wavelength**.