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A spherical refracting surface is the most common surface in optics. A plane surface may be considered a special case of a spherical surface, with an infinite radius of curvature. In a spherical refracting surface like the one shown in Figure 1.12, we define the following parameters:

1. Center of curvature: The center of an imaginary sphere that contains the refracting surface.

2. Radius of curvature: The distance from the refracting surface to the center of curvature.

3. Vertex: A point on the refracting surface, at the center of its free aperture.

This definition assumes that the aperture is circular and centered and that the surface is spherical. More generally, if the surface is not spherical but has rotationally symmetry, the vertex is the point where the axis of sym-metry intersects the optical surface. Even more generally, we may say that the vertex is the local origin of coordinates to which the surface function is referred.

4. Optical axis: An imaginary straight line passing through the center of cur-vature and the vertex. For the case of nonspherical surfaces with rotational symmetry, the optical axis is the axis of symmetry.

Marginal

FIGURE 1.12 Some definitions in a refractive spherical surface.

12 Handbook of Optical Design According to their direction, rays incident on a refractive spherical surface are classified as follows:

1. Meridional ray: Any ray in a common plane with the optical axis, called the meridional plane. In this case, the surface normal and the refracted ray are also contained on the meridional plane.

2. Oblique or skew ray: Any non-meridional ray. In this case, the ray is not in a common plane with the optical axis.

3. Paraxial ray: A meridional or skew ray that has a small angle with respect to the optical axis is a paraxial (from the Greek, near the axis) ray. However, in a more general way, we can say that a paraxial ray is an approximation to a real ray, obtained by assuming valid small-angle approximations.

4. Real or finite ray: Any non-paraxial ray, either meridional or oblique.

In aberration theory, axial, tangential, and sagittal rays are also defined. Axial rays are meridional rays originating in an object point on the optical axis. Tangential rays are meridional rays originating in an off-axis object point. In other words, all tangential rays are meridional rays but not all meridional rays are tangential rays.

The meridional rays from an off-axis object point are contained in a plane called the tangential plane. On the other hand, the sagittal rays are skew rays contained in a single plane, called the sagittal plane, perpendicular to the tangential plane and containing the off-axis object point and the center of the entrance pupil (to be defined later in this chapter) of the optical system. These concepts will become clearer later when the tangential and sagittal planes are defined. To clarify these concepts, the reader is advised to see Figure 6.1 in Chapter 6.

Meridional rays are used to trace rays through a spherical refracting surface. The behavior of meridional rays permits us to obtain many interesting properties of opti-cal systems. Skew rays are mathematiopti-cally more complex than meridional rays and their study is described in Appendix 4.

Figure 1.13 shows a spherical refracting surface and a meridional ray intersecting the surface at the point P. The surface normal at P is N and the curvature center is C.

A convention sign must be defined for all the parameters in Figure 1.13. Such convention has to be consistent with most textbooks and commercial optical design programs. Unfortunately, there are many notations in books, and the most widely used departs from the old definition by Conrady (1957). The sign convention used in this book, assuming that the light travels from left to right, is as follows, where primed quantities are used after refraction on the surface:

1. Radius of curvature r: Positive if the center of curvature is to the right of the vertex and negative otherwise. The curvature c is the inverse of the radius of curvature (c = 1/r).

2. Angles U and Uʹ: In agreement with analytic geometry, they are positive if the slope of the meridional ray is positive and negative otherwise. [Conrady (1957) and Kingslake (1965) use the opposite convention.]

3. Angles I and Iʹ: The angle of incidence I is positive if the ray arrives at the surface from left to right, below the normal, or from right to left above the

normal. This angle is negative otherwise. The angle of refraction Iʹ is posi-tive if the ray leaves from the surface from left to right, above the normal, or from right to left below the normal. This angle is negative otherwise. This sign convention is illustrated in Figure 1.14.

4. Distances L and Lʹ: L is the distance from the vertex of the surface to the intersection of the meridional ray before refraction (object) with the optical axis. It is positive if this object is to the right of the vertex and negative if it is to the left. Lʹ is the distance from the vertex of the surface to the intersec-tion of the meridional ray after refracintersec-tion (image). It is positive if this image is to the right of the vertex, and negative if it is to the left. This rule is valid for the light traveling from left to right, as well as for light traveling from right to left.

P

V C

B A

N n n'

I

I

Y I – I'

I'

L' L

–U' –U

r

FIGURE 1.13 Meridional ray refracted at a spherical surface.

Normal

Normal –I

–I +I

+I' +I'

–I' –I'

+I

(a)

(b)

FIGURE 1.14 Sign convention for the angles of incidence and refraction.

14 Handbook of Optical Design 5. Thickness t: Positive when the next surface in the optical system lies to the

right of the optical surface being considered and negative if it lies to the left of it.

6. Ray height Y: It is positive if the ray crosses the optical surface above the optical axis and negative otherwise.

7. Refractive index n: It is positive if the light travels in this medium from left to right and negative if it travels in the opposite sense. The index of refrac-tion changes its sign at any reflective surface, in order to be able to use the law of refraction on any reflection.

It is interesting to see that according to this convention, for any particular ray, not all three parameters L, Y, and U can be positive at the same time. Observing again Figure 1.13, with a meridional ray and where negative parameters are indicated with a minus sign, we can apply sine law to the triangle PCB:

sinI sin L r

U

− = − r (1.19)

and by using the same law to the triangle PCA:

sin ′ sin

′ −I = − ′ L r

U

r . (1.20)

Since triangles PCB and PCA both share a common angle, and since the sum for the internal angles in both triangles adds up to 180°, it must be true that

I – U = Iʹ – Uʹ (1.21)

and finally we write Snell’s law:

n sin I = nʹ sin Iʹ. (1.22) From these relations, parameters r, n, and nʹ are fixed and known, while L, Lʹ, I, Iʹ, U, and Uʹ are variables. Since we have four equations, all remaining variables can be calculated if any two of the three parameters L, I, and U for the incident ray are specified.

An optical system is generally formed by many optical surfaces, one after the other. We have a centered optical system when the centers of curvature of all the surfaces lie on a common line called the optical axis. In these systems formed by several surfaces, all parameters relating to the next surface are represented by the subscript +1. Then, the transfer equations are

U+1 = Uʹ, (1.23)

n+1 = nʹ, (1.24)

and

L₊₁ = Lʹ − t, (1.25)

where t is the distance from the vertex of the surface under consideration to the ver-tex of the next surface.

1.3.1 m^eridional r^ay t^racinGbythe L–U m^ethod

The equations in the preceding section have been described by Conrady (1957) and may be used to trace rays. This is the so-called L–U method, because the incident and the refracted rays are defined by the distances L and Lʹ and the angles U and Uʹ.

Although these equations are exact, they are never used in actual practice to trace rays because they break down for plane and low curvature surfaces, and L and Lʹ become infinite for rays parallel to the optical axis.

1.3.2 m^eridional r^ay t^racinGbythe Q–U m^ethod

An alternative ray tracing method defines the meridional ray by the angle U and the perpendicular segment Q from the vertex of the surface to the meridional ray, as shown in Figure 1.15. A line from C, perpendicular to the line Q, divides this seg-ment into two parts. Thus, from this figure, we may see that

sin I = Qc + sin U, (1.26)

where the curvature c = 1/r has been used instead of the radius of curvature r. Then, from the refraction law in Equation 1.22, we have

sin ′ = sin I n′

n I (1.27)

and from Equation 1.21,

C I

L I I'

Q Q'

L'

–U' –U

–U

U r

r sin

r sin

FIGURE 1.15 (See color insert.) Meridional ray tracing by the Q–U method.

16 Handbook of Optical Design

Uʹ = U – I + Iʹ. (1.28)

From Equation 1.26, we may obtain an expression for Q and, placing primes on this result, the value of Qʹ is obtained as

′ = ′ − ′

Q I U

c sin sin

. (1.29) In order to obtain the equivalent expression when the surface is flat (c = 0), we may see from Figure 1.15 that Q = –L sin U, and for flat surfaces, tan U = Y/L; thus, it is possible to show that an alternative expression for flat surfaces is

′ = ′

Q Q U

U cos

cos . (1.30)

The transfer equation, as clearly illustrated in Figure 1.16, is

Q₊₁ = Qʹ + t sin Uʹ. (1.31)