RAY VECTOR FIELDS. Lenses and lens systems used in ophthalmic optics have characteristics that make them different from those opti- cal systems that are ...
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J. Opt. Soc. Am. A/Vol. 11, No. 2/February 1994
Charles E. Campbell
Ray vector fields Charles E. Campbell Humphrey Instruments, Inc., 2992 Alvarado Street, R0. Box 5400, San Leandro, California94577-0700 Received November 30, 1992; revised manuscript received July 29, 1993; accepted August 5, 1993 The concept of treating lenses and other powered optical elements as operators acting on light-ray bundles, with these ray bundles being treated as vector fields, is introduced. A matrix method is developed with this vector field concept to analyze the behavior of optical systems. The method is especially useful for ophthalmic optics.
RAY VECTOR FIELDS Lenses and lens systems used in ophthalmic optics have characteristics that make them different from those optical systems that are usually encountered. Effects such as astigmatism and decentration, which are thought of as aberrations traditionally, are wanted effects in ophthalmic optics and are often quite a significant part of the total optical effect. Because of these unique characteristics, the analysis of ophthalmic optical systems is not so well served by the usual techniques and calls for methods more suited to the particular characteristics of these systems. The theory presented in this paper has been successfully
used, in one form or another, in the design and analysis of ophthalmic instrumentation. It forms a conceptual basis for analyzing almost all optical questions associated with ophthalmic optics. Often in an optical measuring system the refractive characteristics of the object under measurement are the unknowns that must be found from measured deflections and positions. This method gives a straightforward way of finding these refractive characteristics if the position and the deflection of a set of rays are known.
to a ray is a function of its entry position on the lens and of the lens' refractive characteristics. The lens operator and its action on the vector field will be represented in matrix form in the following manner. The field exists in a two-dimensional space, so a 2 2 matrix formalism is used. The lens operator is represented by a 2 X 2 matrix, and the ray position and deflection components are represented by two-component column matrices. Deflection given equally to each ray in the field is represented as a column matrix, as is usual for constants in matrix algebra. The resulting matrix equation giving ray deflection components at a given lens location is S.,
[a b (x"(Py
where , y are deflection vector components at point Q(x, y) and x, y are the coordinates of point Q, usually with the optical center as the origin. The simplest ray vector fields are those that are not a function of position: ax= Px, ay
DEVELOPMENT OF THE THEORY The behavior of a lens or other optical device will be characterized by the way it deflects light rays that pass through it. The light rays entering the lens will be described as a vector field in the following manner. Each ray, as it enters the lens, is represented by a two-
dimensional ray vector created by the components orthogonal to the optical axis of a unit vector directed along the ray's path. Associated with each of these ray vectors is the position at which it enters the lens.
vector field. It is assumed that the deflection components
are added linearly to the entering vector components to create the new vector field. Usually the deflection given 0740-3232/94/020618-05$06.00
PY.e
Such fields describe what is usually called prism in ophthalmic optics or tilt in classical optics. Prism is the constant vector shown in Eq. (1) and has the components P, and Py. The position-dependent matrix can always be subdivided into a sum of matrices, i.e.,
[c d
[0 0]
[0 0]
The totality of
these ray vector position pairs creates the incoming ray vector field. The idea of associating each vector with a position is an essential idea of a vector field. If a set of rays from one field passes through a second surface but at different locations, a different field results just as much as if at a single location the ray vectors had their values changed. The lens is thought of as an operator that acts on the ray vector field, adding a deflection component to each ray as it passes through, thereby creating a new ray
(1)
1 0]
[0
1] (2)
The theory is developed below in terms of these primitive matrices consisting of only 1's and 0's. After prism, the next level of complexity is the case in which the coefficients a, b, c, and d are constants. When this is the case the ray fields will be global in character over the lens. This means that the deflection patterns found in any subarea of the field will be the same, after constant prism terms have been subtracted out. Another way of putting this, to use terms from ophthalmic optics, is to say that the power values of sphere and astigmatism are the same throughout the field. There are only four X 1994 Optical Society of America
Vol. 11, No. 2/February 1994/J. Opt. Soc. Am. A
Charles E. Campbell
not be immediately obvious, but the pattern is created by a cross-cylinder lens (positive cylinder axis, 1800, negative axis, 900). This can be seen by decomposing the primitive matrix as follows:
Y
I
V~!
N1 /z i 1\0Ns\,
_
L
/I
0
]
[1
[1 0
0
0O
x The deflection pattern for the first portion of the decomposed matrix is
'-4
4e
619
0
Fig. 1. Ray vector field for a positive spherical lens.
This is the deflection pattern of a negative (concave) cylinder with axis 90°, as illustrated in Fig. 3. The deflection pattern for the second portion of the decomposed matrix is
Y 8Y)
/
4I
I tP
a
p..
a
-1] (Y
As illustrated in Fig. 4, this is a positive (convex) cylinder at axis 180°. It can be seen that the sum of these two patterns, the C+ pattern, is the effect of a positive and a negative cylinder oriented at 900 from one another, i.e., a cross cylinder. For Cx
X Y1 W
0/4'
4I
[0
1
axy
Fig. 2. Ray vector field for a cross cylinder with the positive axis at 0, and the negative axis at 900.
fundamental fields in this case. Their matrix operators could be given as the operators are given in Eq. (2), but it is more convenient to use these combinations:
X](y
Notice that the deflection pattern shown in Fig. 5 is the same pattern as that for C+, but the C+ axis has been rotated -45°. Hence this is also a cross cylinder, but it is oriented with the positive axis at 450, and the negative axis is oriented at 1350.
Y IS] =S[ [C.]
1
-]
[C+] =C+[
C. C01
[CA
I
I
S
X
CA [-1 -
-
.
z
.
l
.
"
.
S, C+, C., and CA stand for sphere, 90-180° cross cylinder, 45-135° cross cylinder, and circular astigmatism, respec-
S
I
I
0
tively. To illustrate why these names are chosen, the ray fields are drawn for the four primitive matrices. For S Fig. 3. Ray vector field for a negative cylinder lens, axis 90.
(8X)
-
[S]()
S[
0](x)
8
.- Sx,
Y
by = -Sy.
Figure 1 illustrates the deflection pattern of a positive sphere lens. If the coefficient S is a negative number, the deflection pattern will be that of a negative sphere lens. For C+
B., = C+ 1 0 ay 0 -I
X I Y
I I I I I
8 = C+xX,8 , = -C+y.
The origin of the deflection pattern shown in Fig. 2 may
Fig. 4
X
h
17I r
Ray vector field for a positive cylinder lens, axis O
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Charles E. Campbell
J. Opt. Soc. Am. A/Vol. 11, No. 2/February 1994
r
l / y
Z
7-
I(1
Z
l
Z
x
"'10%
When this is done, there is a portion that is the same as Eq. (2) with the constants a, b, c, and d. This portion describes the toric-lens effects. The other terms give nonlinear effects. For an instrument, like a lens meter, which samples in a small area, these nonlinear terms are in general sufficient to describe lenses such as progressive lenses. It must be noted that the overall deflection fields of progressive lenses are even more nonlinear than this first level of complexity above that of the toric case. The nonlinear portion of the matrix is
(ex + fy)[
0 + (gx + hy[0 + (ix + jy)[
Fig. 5. Ray vector field for a cross cylinder with the positive axis at 450 and the negative axis at 135.
] + (k
+
Y)[
ex[
](x)
(e°2)
=
Figure 7 shows this deflection pattern. This behaves like a progressive negative cylinder when x is positive (power increases as a linear function of x), axis 90, and a positive progressive cylinder when x is negative. Figure 8 shows what such a lens might look like. To see an effect more like a true progressive lens, consider the combination
/
fy[
°]0 + ly[ 0
0
Fig. 6. Ray vector field for CA.
For CA
( )
= CA[
;]
1]
To get an idea of what these terms mean, consider
Y
/
1
( )
i8= CAy,
9-0
By = -CAx.
This helical deflection pattern, shown in Fig. 6, cannot be realized by a surface or a lens. However, the concept of such a field has proven useful. For instance, the value of this pattern is known to be zero for all toric lenses. If the value is found to be nonzero in a measurement system designed to measure toric effects, such as that described below, the lens under measurement is known to have properties other than toric. The field is then a type of aberration test. The known absence of the CA field has also been found to be useful in other ways that are beyond the scope of this paper. The above fields describe all lenses that are classed as toric lenses. Such lenses can be parameterized by five constants, P., Py, S, C+, and C. More-complex lenses cannot be fully described by position-independent constants. The next level of complexity is obtained by letting the matrix elements be linear functions of position, i.e., a(x,y) = ex + fy + a,
c(x,y) = ix + jy + c,
b(x,y) = gx + hy + b,
d(x,y) = kx + ly + d.
The coefficients of x and y (e, f g, h, i, j, 1, and m) are constants. As before the matrix can be decomposed.
-
-
vfiel
X
f
Fig. 7. Ray vector field for a progressive cylinder lens.
Fig. 8. Progressive cylinder lens.
Vol. 11, No. 2/February 1994/J. Opt. Soc. Am. A
Charles E. Campbell
_---
are labeled a3g and a5i, where i is the labeling index. Using the four fundamental fields plus the prism terms, we write the following equation for point 1:
Y
Y
i' i
I0-
*
x
l
X
-
( )
+ C(a)
+
(P.+
o](a)
\P,
y
8,, = a[-S + C+ + C + CA] + P,
Fig. 9. Characteristic deflection patterns seen in a progressive spectacle lens. =
C[, o](a) + CA[1
The deflections can be written as
8x
and let f
621
1 so that the combination becomes Ey
1
ay = a[-S - C+ + C, - CA] + Py. In like fashion it is found that the deflections at the other three points can be written as
0]
The primitive matrix has the same form as the S matrix, but instead of a constant factor giving the sphere power, the primitive matrix has a factor that changes linearly with y. The ray field generated by this matrix component is
8x2 =
a[S - C+ + C. + CA] + P.,
ay2
=
a[-S + C+ - C. + CA] + Py,
ax3
=
a[S - C+ - C. - CA] + P,
8y3 =
a[S + C+ - C. + CA] + P, a[-S + C+ - C. - CA] + P.,
Ax4 =
(tfxy) [0
1
y)
a[S + C+ + C. - CA] + Py.
By4 =
The following sum of the deflections is made:
To see how such a field is interpreted by a Humphrey lens analyzer, an automatic lens meter that measures deflection in a square pattern, (x, y) coordinates of (h, h), (-h, h), (-h, h), and (h, -h) are chosen. This gives the deflections illustrated in Fig. 9. The Ax and ay components have been shown separately here. The y deflection pattern looks like Py, y prism, so it does not uniquely identify the progressive characteristic. However, the x deflection pattern is unlike any other pattern yet seen, and it is the pattern that is used to detect the progressive part of a lens. In similar fashion, if e = ,
-axi + 15x2 +
8x3
- 5x4 - 5y1 - ay2 + By3 + Sy4 =
The value a is an instrument constant found during calibration, so this simple sum of deflection values divided by 8a gives the S (sphere) value of the unknown lens. In similar fashion these sums give the other values of the lens: C+ = (8 x1 - 8x2 - 8x3 + 8x4 - 8 y1 C = (1
+
.2 -
-
C+ = ( 1 + 8.2 - 8x3 -
ex[O O + ko
1]
ex[O 1
which is a progressive effect along the x axis, and the y deflection pattern that arises is used to detect progressive lens effect in the x direction.
EXAMPLE OF THE USE OF THE METHOD To illustrate the use of the method a simple example is now given. It is a practical example in that it is the basic method used to measure lens values in an actual automatic lens meter. It shows how measured ray deflections can be used to determine the value of the lens that created them. In this example the ray vector field is sampled in four locations. These sample points are arranged in a square pattern, and the measurement coordinate system has its origin in the center of the pattern. Let the coordinates of the points be (a, a)1, (-a, a) 2, (-a, -a) 3 , and (a, -a) 4 , where the subscripts label the points. The measured values are the deflections created in the rays of an incoming parallel ray field that pass through the measurement points. They
8aS.
a62
P.=
(1
Py =
(8yl + Sy2 + 5y3 +
+
+
3 +
Ax4 +
a4
-
ayl
-
By2
+
6y3
+
ay4 )/8a,
y2 - 8y3 + ay 4 )/8aI
6yl + By2 +
y3 -
y4)/8a,
a54 )/4a,
8y 4 )4a.
COMMENTS ON THE METHOD AND ITS USE When the ray vector field method is used, is it wise to keep in mind certain assumptions implicit in it and certain limitations on its applicability. The first comment is on the size of deflections that the method can accommodate. There is nothing in the formalism to limit the size of the deflections, as there is no limit on the size of the variables x and y. However, there is a real physical limit in that the components of a unit vector can never exceed 1, so there is a real limit on Ax and By, This is never a real limitation in use with actual ophthalmic systems, as the deflections rarely exceed a value of 0.15 (15 prism diopters). In this range the sine and tangent and angles are well approximated by the angles themselves, expressed in radians, so the assumptions used for paraxial optics are justified.
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Charles E. Campbell
J. Opt. Soc. Am. A/Vol. 11, No. 2/February 1994
The second, related comment is on the use of the method when noncollimated light enters the system under analysis. The theory has been developed under the assumption that deflection components can be added directly to the incoming ray vector components. A truly rigorous treatment would account for the fact that refraction is proportional to the sine of the incoming ray with respect to the lens surface. This approximation is valid in the limit that the sine of an angle is equal to the angle expressed in radians. If the deflections meet the limitations mentioned in the first comment, this approximation is certainly valid. The third comment is on the use of the method for lens systems. The deflection operator method assumes that the operator lies in a single plane, so if it is applied to a lens, the method assumes that the lens is a thin lens. Real systems have elements that are spaced apart, and like other matrix methods used in optics this method can be adapted to such systems by inclusion of the transfer operation, which transfers the deflection field exiting one element to the entrance of the next and modifies the field to account for the effect of the axial translation. In this case the operator represents the action at a surface, and the lens operator values are surface powers. An interesting feature of the operation, which is not immediately obvious, is that fields that start as pure cross-cylinder fields not only change their magnitude but also develop sphere fields in the translation process. The development of this feature of the method is beyond the scope of this paper. This method is a matrix method, and those familiar with these methods in optics will undoubtedly see similari-
ties with more traditional approaches. In the more specialized field of ophthalmic optics there have been matrix methods developed that are adapted to use in this field. The reader is directed especially to papers by Long,' Keating,2 3 and Harris, 4 whose matrix methods give results similar to those given in this paper but without the underlying concept of the vector field. A recent paper by Webb et al.5 recognizes the usefulness of the vector field
approach.
ACKNOWLEDGMENT I acknowledge William E. Humphrey, who introduced the concepts of spherocylindrical decomposition and ray deflection analysis to the company that he founded and who has used these concepts extensively in instrument development.
REFERENCES 1. W F. Long, 'A matrix formalism for decentration problems," Am. J. Optom. Physiol. Opt. 11, 118-120 (1976). 2. M. P. Keating, 'An easier method to obtain the sphere, cylinder and axis form an off-axis dioptric power matrix," Am. J. Optom. Physiol. Opt. 57, 734-737 (1980). 3. M. P. Keating, Geometrical, Physical and Visual Optics (Butterworth, Washington, D.C., 1988). 4. W F. Harris, "Solving the matrix form of Prentice's equation for dioptric power," Optom. Vis. Sci. 68, 178-182 (1991). 5. R. H. Webb, C. M. Penney, and K. P. Thompson, "Measurement of ocular local wave-front distortion with a spatially resolved refractometer," Appl. Opt. 31, 3678-3686 (1992).
