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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

    Two-Sided Lenses

    More typical than the 1-sided lenses we have discussed are 2-sided lenses, with front and back refractive surfaces. These surfaces separate the denser medium of the lens from the surrounding medium, usually air or a watery tissue such as aqueous or vitreous humor. In many cases, if the distance between the front and back surfaces is small, we may treat the lens as a single refracting object with power P = P1 + P2, where P1 and P2 are the (signed!) powers of the front and back lens surfaces. (Notice that, for example, for a typical biconvex lens in air, both the front surface and the back surface have positive power, as the both the numerator and the denominator in the lensmaker’s equation for the back surface are negative numbers.) This is referred to as the thin-lens approximation.

    In some cases, such as the crystalline lens of the normal human eye or intraocular lens implants, we cannot ignore the separation between the front and back surfaces of a lens. The power of such a “thick lens” for paraxial rays is given, to a first approximation, by the formula

    where P1 and P2 are the powers of the front and back surfaces of the lens, respectively, t is the distance between them (ie, the thickness of the lens) in meters, and n is the refractive index of the lens material. This formula reduces to the thin-lens approximation as the lens thickness approaches 0.

    EXAMPLE 1-3

    Derive the “thick-lens formula” for a 2-sided lens in air:

    Suppose the front surface has power P1, the lens has refractive index n and thickness t, and the back surface has power P2 (Figure 1-E3). At the front surface, a paraxial ray with zero vergence is refracted and intersects the optic axis at ν, where 0 + P1 = n/ν, or ν = n/P1. The reduced vergence of the refracted ray as it reaches the back surface is n/(ν − t) = n/[(n/P1) − t]. After the ray is refracted at the back surface, it meets the optic axis at distance ν2 from the back surface of the lens, and the exiting vergence is

    For small values of t, this approaches P = P1 + P2 − (t/n)P1P2.

    Figure 1-E3 Calculating the power of a “thick lens.” Here v is the distance from the front lens surface to the point where the ray refracted by the front lens surface intersects the optic axis. The powers of the front and back surfaces are P1 and P2, respectively.

    (Courtesy of Scott E. Brodie, MD, PhD.)

    Principal Planes

    The thin-lens approximation treats the front and back lens surfaces as if they coincide, and we use their common location as the position of the lens to calculate the object and image vergences, as we did in the Quick-Start Guide. Unlike the thin-lens case, we do not immediately know how to measure the distance from a source object to a thick lens (or from the lens to the image location) in order to apply the vergence equation. It is helpful to consider a diagram (Figure 1-10). Recall from the Quick-Start Guide that the focal point F of a lens is the location of the image of light from a very distant object (at a distance f = 1/P).

    Here, rays from a distant object at left are brought to a focal point F2 on the right side of the lens. But careful inspection shows that the refraction takes place in 2 steps: (1) as the rays cross the front surface of the lens, and (2) as they cross the back surface. If we extend the incident rays and the exiting rays (shown by the dotted lines in Figure 1-10), they cross at an apparent single refracting surface—the second, or “back,” principal plane—located within the lens substance. That plane, indicated by the vertical line, passes through the optic axis at H2. Similarly, rays that originate from the front focal point, F, and emerge from the lens as parallel rays appear to have been refracted at a different internal plane—the first, or “front,” principal plane)—which passes through the optic axis at H1 (Figure 1-11).

    These apparent refracting planes provide the appropriate locations for the calculation of object and image vergence, respectively. The intersections H1 and H2 of the principal planes with the optic axis are known as the first and second (or “front” and “back”) principal points.

    The location of the principal planes depends on the lens design and on the refractive indices of the material in the object space, lens, and image space. For “meniscus lenses,” such as those used for spectacles, with 1 convex surface and 1 concave surface, the principal planes may lie outside the lens itself (Figure 1-12).

    Figure 1-10 Ray tracing for parallel incident light rays striking a thick convex lens. Notice that the rays appear to bend at an internal plane at H2.

    (Redrawn from Creative Commons illustration by Bob Mellish.)

    Figure 1-11 Ray tracing for rays emerging from the first focal point, F1, striking a thick convex lens. Notice that the rays appear to be refracted at a different plane than the incident parallel rays shown in Figure 1-1.

    (Redrawn from Creative Commons illustration by Bob Mellish.)

    Figure 1-12 Ray tracing for a meniscus lens in air. Notice that the principal planes H1 and H2 lie outside the lens.

    Calculations of the location of the principal planes are beyond the scope of this book. Fortunately, for most purposes involving ophthalmic lenses in air, such as spectacle corrections and contact lenses, the thin-lens approximation is adequate. However, most calculations of refraction within the eye, where the thickness of the lens is a large fraction of the axial length, may require more detailed treatment.

    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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