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  • 2020–2021 BCSC Basic and Clinical Science Course™

    Go to Academy Store Learn more and Purchase.

    3 Clinical Optics

    Chapter 1: Geometric Optics

    General Refracting Systems

    Refracting systems can be built up from multiple combinations of lenses of arbitrary thickness and various compositions of optical media, such as types of glass with different refractive indices. Regardless of the complexity, for paraxial rays, such systems can be characterized by a single pair of principal planes and a single refractive power, P.

    This is a remarkable simplification. We can ignore everything that takes place at the numerous optical interfaces within such an optical system, and proceed by simply determining the object vergence at the first principal plane, adding the power P, and then locating the image by referencing the exiting vergence to the second principal plane (Figure 1-15).

    EXAMPLE 1-4

    Principal planes. Redo Section Exercises 1-3–1-6 for a complex lens system with principal planes separated by 1.0 cm = 0.01 m and net power P = +2.0 D.

    Assume the object distances are referenced to the front principal plane. Then the image distances are the same as before, when referenced to the second principal plane, and are thus 1.0 cm greater than the answers to Section Exercises 1-3–1-6.

    Figure 1-15 References for a compound optical system. Each lens has its own power and pair of principal planes, as indicated by subscripts 11 and 12, 21 and 22, and 31 and 32. The combined system has an equivalent or “net” power and a single pair of principal planes representing the entire system, indicated by H1 and H2.

    (Courtesy of Edmond H. Thall, MD.)

    Section Exercises 1-11–1-14

    Redo Section Exercises 1-3–1-6 for a lens system with net power P = +10 D.

    1-11  u = −4.0 m, P = +10 D

    ν = 1/9.75 D = 0.10256 m from the second principal plane

    1-12  u = −1.0 m, P = + 10 D

    ν = 1/9.00 D = 0.111 m from the second principal plane

    1-13  u = −0.5 m, P = + 10 D

    ν = 1/8.0 D = 0.125 m from the second principal plane

    1-14  u = −0.4 m, P = + 10 D

    ν = 1/7.50 D = 0.1333 m from the second principal plane

    EXAMPLE 1-5

    Principal planes and distant objects. The Gullstrand mathematical model closely approximates the human eye. (The Gullstrand model eye is discussed in Chapter 3.) The net power of the entire refracting system of the model eye is P = +58.64 D; the refractive index of the vitreous humor is 1.336, and the second (posterior) principal plane is located 1.602 mm behind the anterior corneal surface. Locate an image of a distant object as formed by the Gullstrand model eye with relaxed accommodation.

    The image is located according to the vergence equation: 0 + 58.64 D = 1.336/ν, so ν = 0.02278 m, or 1.602 + 22.78 = 24.382 mm from the anterior corneal surface. For comparison, the axial length is 24.40 mm.

    EXAMPLE 1-6

    Effectivity of lenses. Suppose a lens of power P in air places the image of a distant object at a distance u to the right of the lens. To move the lens t meters to the right while preserving the image location, how must you adjust the power of the lens?

    In its new position, the lens will form the image at a distance u2 = u − t to the right of the lens (as v = 0). Thus, the power P2 of the moved lens should be P2 = 1/(u − t) = 1/(1/P − t) = P/(1 − tP). Indeed, the same formula applies for divergent lenses, with the initial image location to the left of the lens, so long as the proper sign conventions are observed. This adjustment is referred to as the correction for the effectivity of lenses.

    The effectivity of lenses comes into play, for example, when we convert a spectacle correction to the corresponding contact lens correction. For example, a hyperopic correction will place the image of a distant object at the far point of the eye (the location conjugate to the fovea with relaxed accommodation—see the Quick-Start Guide). In this case, the far point is behind the retina, and the effectivity calculation requires that the power of the correcting contact lens be greater than the power of the equivalent spectacles. For example, for a spectacle correction of +4.00 D worn at the standard vertex distance of 13.75 mm from the corneal apex, the power of the equivalent contact lens is +4.00 D/[1 − (0.01375 m)(4.00 D)] = +4.232 D, which suggests that the contact lens power should be increased by ¼ D. This correction is negligible for lenses weaker than about 3 D.

    Gaussian Optics

    To calculate the location of the principal planes, we must know the details of the refractive events within the lens system. In practice, this is usually done by means of an elegant formalism known as Gaussian optics (or Gaussian reduction). The details are beyond the scope of this book, but the basic idea is as follows: the paraxial approximation is effectively a linearization of Snell’s law. We can therefore express the approximation in terms of 2 types of “linear operators.” (1) The “translation operator” describes the propagation of a light ray through a medium of refractive index n over a distance t. This is, in essence, equivalent to the step of recalculating the vergence of light from a source object (or image created by a previous refracting surface) as referenced to the next refracting surface in an optical system. (2) The “refraction operator” describes the change in vergence at a curved optical interface of radius r between media of refractive indices n1 and n2. It is the equivalent of adding P to find the vergence of light exiting from a refractive interface. We can represent each of these operators in an appropriate coordinate system by a suitable 2 × 2 matrix. We can then use matrix multiplication to calculate the net action of a sequence of refractive events at the various refractive surfaces in a complex optical system.

    Excerpted from BCSC 2020-2021 series : Section 3 - Clinical Optics. For more information and to purchase the entire series, please visit https://www.aao.org/bcsc.

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