Optometric texts define back-vertex power in two equivalent ways. Some define it as the reciprocal of the back-vertex focal length or the reduced back-vertex length if the medium behind the system is not air.^{1â€“4} Others define it as the vergence leaving the system when the incident rays are parallel.^{5â€“7} The two approaches are equivalent because vergence is the reciprocal of the back-vertex focal length. Front-vertex power (also called the neutralizing power) is defined similarly except that there is an asymmetry in the definition in terms of vergence: the front-vertex power is the negative of the vergence entering when the emergent rays are parallel.^{5,7}

Although they can be applied to astigmatic systems (separately in each principal meridian, so that the system has two principal back-vertex powers for example), these definitions are primarily for systems that are not astigmatic. The purpose of this note is to offer modified definitions, which lead to very simple derivations of Keating's^{8,9} elegant formulas for front- and back-surface powers as dioptric power matrices. They are general and hold for systems that may have astigmatic and decentered refracting elements. These definitions have the advantage of clarity, of directly relating the concepts to measurement by manual neutralization, and of not being asymmetric in the sense mentioned above.

Modified definitions of back- and front-vertex power for a general system are offered below. The definitions lead directly to general expressions for these powers. The general expressions are then applied to the special case of a thick bitoric lens in which the principal meridians on the two surfaces are not necessarily aligned. Finally, the familiar equations for thick lens with spherical surfaces are shown as special cases of the general expressions.

Fig. 1A, B illustrates the traditional vergence representation of back- and front-vertex powers, respectively. In Fig. 1A, parallel rays enter an optical system S. The back-vertex power of the system is the vergence L of the pencil emerging from the system. In Fig. 1B, rays emerge parallel and the front-vertex power of the system is the negative of the vergence L_{0} at incidence.

Fig. 1C, D shows a modified viewpoint. Thin lenses are placed immediately after (Fig. 1C) and immediately before (Fig. 1D) system S. They can be described as post- and prejuxtaposed to S, respectively. The thin lenses are such that the combination of system S and juxtaposed thin lens is afocal in each case. The juxtaposed thin lens is negating or neutralizing the focusing power of the lens, and it is doing so at the back or front vertex of the system. Its power, therefore, is the negative of the back- or front-vertex power. The discussion above suggests the following definitions for optical systems in general.

*Definitions:* Consider a thin lens of power **F**_{b} postjuxtaposed to system S. If the combination is afocal, then

is the back-vertex power of system S. Similarly, if the combination of a thin lens of power **F**_{f} prejuxtaposed to S renders the combination afocal, then

is the front-vertex power of S.

Although the powers mentioned so far could be the familiar scalar powers of Gaussian optics, the boldface symbols are intended to represent dioptric power matrices independently described by Fick^{10} and Long.^{11} Such matrices are 2 Ã— 2 and symmetric and readily calculated from the familiar representation of astigmatic power as sphere, cylinder, and axis. The definitions, therefore, hold to first order for all systems whether astigmatic or not and whether refracting elements are relatively decentered or not.

One notes the symmetry in these definitions that contrasts with the asymmetry in the definitions based on vergence. A greater advantage, however, is that these definitions lead directly to elegant formulas for vertex powers for optical systems in general. This is shown below.

System S has transference^{12,13}

**C** is simply related to the dioptric power matrix **F** by^{8, 14}

and **o** is the 2 Ã— 1 null matrix and **o**^{T} is its matrix transpose.

Suppose that, for system S, **F** = **O**, where **O** is the 2 Ã— 2 null matrix. Then S is an afocal system.

A thin lens has transference^{12,13}

where **I** is the 2 Ã— 2 identity matrix. When postjuxtaposed, as in Fig. 1C, the combination of S and the thin lens has transference obtained by multiplying the transferences in reverse order,^{12,13} that is,

This becomes

where â€¢ represents a submatrix that is not needed for what follows. The combination is rendered afocal by setting

Hence

and, because of Eq. 1, the back-vertex power is as follows:

Equation 1 Image Tools |
Equation (Uncited) Image Tools |

a result originally obtained by Keating^{8, 9} by a different method.

Similarly, suppose a thin lens of power **F**_{f} is prejuxtaposed to S as in Fig. 1D, then the transference of the combination is as follows:

or

The combination is afocal if

Hence

and, because of Eq. 2, the front-vertex power is

Equation 2 Image Tools |
Equation (Uncited) Image Tools |

a result also obtained by Keating.^{8, 9}

From Eqs. 6 and 7, it follows immediately that

Equation 6 Image Tools |
Equation 7 Image Tools |

This shows neatly how back- and front-vertex power both reduce simply to power in the case of thin systems. (For thin system, both **D** and **A** reduce to **I**, as shown by Eq. 5.)

Equation 5 Image Tools |
Equation (Uncited) Image Tools |

Equations 6 and 7 give the back- and front-vertex powers of a system in terms of its transference.

As a particular example, consider a thick bitoric lens or reduced thickness Ï„. The principal meridians of the front and back surfaces may or may not be matching. Multiplying the transferences of the component elementary systems backward, one obtains

or

Hence, applying Eqs. 6 and 7, one obtains the back- and front-vertex powers of the thick lens as follows:

and

respectively.

For a thin bitoric lens, one sets Ï„ = 0 and obtains

as one would expect.

If the two surfaces of the thick lens are stigmatic, then the surface powers take the form

Substitution into Eq. 10 results in

Equation 10 Image Tools |
Equation (Uncited) Image Tools |

Equating coefficients of **I** on the left and right one obtains the familiar scalar equation^{15} for the back-vertex power of the stigmatic thick lens,

Similarly Eq. 11 leads to the scalar equation^{15} for front-vertex power

Equation 11 Image Tools |
Equation (Uncited) Image Tools |

From a system's transference, one can calculate its back- and front-vertex powers via Eqs. 6 and 7. Given the structure of the system, one can calculate the transference as illustrated above in the case of a thick bitoric lens. When **A** is singular, the inverse in Eq. 6 does not exist, and the back-vertex power is undefined; this occurs when the second principal focal point or a second principal focal line falls on the back surface. Similarly, when **D** is singular, the front-vertex power (Eq. 7) does not exist; the first principal focal point or a first principal focal line falls on the front surface of the lens.

#### ACKNOWLEDGMENTS

I thank Dr R.D. van Gool for commenting on the manuscript and the National Research Foundation for funding.

William F. Harris

Department of Optometry

University of Johannesburg

PO Box 524

Auckland Park 2006, South Africa

e-mail: wharris@uj.ac.za