Wavefront Theory

In geometric optics, the movement of light obeys Fermat’s principle, which states that light travels between 2 points only along the fastest path (Appendix 1.2). Thus, a stigmatic focus can be achieved only when each of the paths from object to image point requires precisely the same amount of time.

We can construct a spherical surface centered on the image point such that all light moving along image rays must cross the arc of that surface simultaneously to achieve a stigmatic focus. This surface is called the reference sphere (Figure 1-23).

A geometric wavefront is an isochronic (equal-time) surface. We can construct a wavefront anywhere along a group of rays originating from a single object point. All light from a given object point crosses the wavefront simultaneously. If the wavefront that intersects the vertex of the reference sphere is also spherical, then the focus is stigmatic. However, a wavefront is in general irregularly shaped. The difference between the wavefront and the reference sphere is the wave aberration (Figure 1-24).

Figure 1-23 The reference sphere, a spherical arc centered on the image point.

(Courtesy of Edmond H. Thall, MD.)

It is a common misconception that wavefront refers in some way to the wave nature of light. Geometric optics ignores the wave nature of light. (See Appendix 2.1 for a brief discussion of the wave nature of light, an aspect of physical optics.) A geometric wavefront is a surface of equal time, regardless of whether light is a wave or a particle. The term isochrone might be more descriptive, but for historical reasons the term wavefront is entrenched.

The wavefront aberration is a smooth but irregularly shaped surface, typically something like the shape of a potato chip or corn flake. The mathematical description of the wavefront aberration may at first seem daunting, but conceptually it is straightforward.

Consider first a toric surface (see the Quick-Start Guide). A toric surface is the combination of 2 “more fundamental” surfaces: a sphere and a cylinder. We can represent any toric surface as the sum of a certain amount of sphere and a certain amount of cylinder. The fundamental shapes sphere and cylinder never change from one toric surface to another; only the amount of sphere and cylinder (and the cylinder orientation) changes. Sphere and cylinder are the only fundamental shapes required to define any toric surface or for that matter to express any amount of regular astigmatism. The same idea applies to wavefront aberration, except more than 2 fundamental shapes are required. Every wavefront aberration is the sum of the same fundamental shapes. The amount of each shape varies from patient to patient, but the fundamental shapes themselves never change.

The most common set of fundamental reference shapes used for this purpose is known as the Zernike polynomials, which are mathematical functions defined on a disc-shaped region. The first several Zernike polynomials closely resemble simple combinations of the wavefront aberrations that we commonly encounter with simple optical systems. A detailed discussion of this material is far beyond the scope of this text; a brief qualitative discussion follows.

Figure 1-24 The wavefront and wavefront aberration. The actual wavefront is irregularly shaped, and the image is not stigmatic. The difference between the actual wavefront and the reference sphere is the wave aberration.

(Courtesy of Edmond H. Thall, MD.)

The most fundamental aberrations were initially described in the nineteenth century by Philipp Ludwig von Seidel. They are known as Seidel aberrations—spherical aberration, coma, astigmatism, curvature of field, and distortion (Figure 1-25):

• Spherical aberration is a disparity in focal length for rays from a single axial object point that are refracted at different distances from the center of the lens.

• Coma is a disparity in focal length for rays from a single off-axis object point that are refracted at different distances from the center of the lens.

• Astigmatism is the disparity in focal length for rays from a single object point that are incident at different meridians of the lens (see the Quick-Start Guide).

• Curvature of field is a disparity in focal length for objects at different distances from the optic axis.

• Distortion is a disparity in transverse magnification for objects at different distances from the optic axis.

Figure 1-25 Schematic depiction of the classic Seidel aberrations.

(Courtesy of Edmond H. Thall, MD.)

The low-order Zernike polynomials closely mirror basic optical image manipulations and the classic Seidel aberrations. (Figure 1-26 depicts the lowest-order Zernike polynomials.) For example, the first-order Zernike polynomial aberration named tilt is, in essence, equivalent to a prismatic deviation.

The second-order Zernike polynomial aberrations are referred to as defocus, corresponding to myopia and hyperopia, and astigmatism, which corresponds to the potato chip–like waveform surface created by a spherocylindrical lens.

The Seidel aberrations are third-order Zernike polynomial aberrations. These include coma, which corresponds to a lobular asymmetry in the waveform surface.

In myopia, the wavefronts are displaced relative to the reference sphere (Figure 1-27). The wavefront is spherical but has a smaller radius than the reference sphere, and the wave aberration is parabolic.

Figure 1-26 Low-order depictions of the first several Zernike polynomials, labeled according to standard notation. Top row: “piston” aberration affects only the phase of light waves and is not ordinarily detectable. Second row: “tilt” corresponds to misdirection of wave propagation, akin to prismatic deviations, shown as vertical or horizontal. Third row: the outer functions show astigmatic aberrations; the middle function shows variations in focal point with circular symmetry, akin to defocus and/or spherical aberration. Bottom row: the middle 2 functions are lobular wavefront errors, akin to classical coma; the outer functions (“trefoil”) show a threefold symmetry sometimes seen as a complication following refractive surgery procedures, such as LASIK.

(Illustration by Wikipedia user Zom-B via Creative Commons.)

Figure 1-27 Myopia. In pure myopia, the focus is still stigmatic. Therefore, the actual wavefront is also a sphere but with a smaller radius than the reference sphere. The wave aberration has a parabolic shape.

(Courtesy of Edmond H. Thall, MD.)

Figure 1-28 Positive spherical aberration. The peripheral rays focus anterior to the paraxial rays. The wave aberration has a bowl shape—flat centrally, then strongly curved toward the edge.

(Courtesy of Edmond H. Thall, MD.)

In positive spherical aberration, rays at the edge of the wavefront focus anteriorly to central (paraxial) rays (Figure 1-28). The wave aberration is bowl shaped. Spherical aberration shifts the position of best focus anteriorly. This aberration is strongly pupil dependent: as the pupil of a patient with positive spherical aberration dilates, the shift in best focus renders the patient more myopic. The condition is known as night myopia and can be treated by prescribing spectacles with an extra −0.50 D in the distance correction for use at night or in low light.

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.