Basic Optical Effects Limit Image Quality

Designers can organize the components of an optical system in only a few ways to obtain high-magnification microscope optics. They face additional limitations because even those few components can distort images. These distortions, which limit image quality, arise from optical flaws, or aberrations. Every optical instrument suffers from aberrations to some extent.

This article, the second in T&MW ’s Basic Microscopy Series, describes some of the monochromatic optical aberrations that can affect the images from microscopes. This information on optics will help you when it comes time to purchase a microscope and you need to judge the quality of the instrument’s optics.

Figure 1 shows a simple lens focusing a collimated beam of light at a point. For clarity, the figure shows only five light rays; in reality, a ray passes through each point on the front surface of the lens. A real, simple lens wouldn’t focus all the rays at the same point. Various rays passing through the lens would miss the focal point by different amounts, depending on how the shape of the lens differs from an ideal lens shape.

Optical engineers call the ideal lens shown in Figure 1 the first-order approximation. It’s also known as the paraxial approximation, meaning “near the axis,” because if you consider only the rays entering very near the optical axis, the errors turn out to be very small. Think of aberrations as small variations from the ideal case.

Figure 1. Theoretically, a perfect lens will gather all rays at a single focus point.

The theory of lenses starts with Snell’s Law, which determines how a light ray will bend depending on the sine of the angle it makes with the air/glass interface at the point of contact. To better understand aberrations, look at the sine of an angle as a power-series epression:

The paraxial approximation uses just the first term, q. The five major monochromatic aberrations—spherical aberration, comatic aberration, astigmatism, field curvature, and distortion—all result from the second term, q³. Because this term raises q to the third power, physicists call the effects attributed to it third-order aberrations. They also call them Seidel aberrations after Philipp Ludwig von Seidel (1821–1896), who first studied them in detail.

The most significant Seidel aberration is called spherical aberration, and it comes about due to the spherical shape of a lens. Most lenses have a spherical surface because it’s easier to grind glass into a spherical shape than into any other shape. (Historically, even opticians who took on the challenge of grinding aspherical shapes started by grinding a spherical shape and modifying it.) Thus, spherical aberration appears in many lenses.

Figure 2. Spherical aberration in a lens causes rays to focus at different points depending on how far from center they enter and pass through a lens.

Figure 3. Coma (or tail) appears in an image because off-axis rays focus at different points.

Figure 2 illustrates spherical aberration. Essentially, the collimated rays passing through the outer part of the lens have a shorter focal length than the paraxial rays. The outer part of the lens appears “stronger” at bending the rays than the part at the center.

When spherical aberration occurs, no clearly defined image plane exists. Rays passing through at a given zone (distance from the center) focus sharply at one point, rays from other zones focus sharply at other points. As a result, the image appears blurry, even at the best focus. This kind of aberration applies only to rays that focus on the optical axis.

Coma Effects Show Comet Tails
The second Seidel aberration is comatic aberration, or simply coma. The name derives from the appearance of off-axis stars in telescopes with simple optics. Coma causes stars to look like small comets—sharp points and diffuse tails that point away from the optical axis. The diffuse tails reminded early astronomers of long hair blowing in the wind; hence they described the optical effect as coma, the Greek word for hair.

Coma appears when a light beam passes through a lens obliquely. Figure 3 illustrates the effect for a positive lens and three rays. The center ray passes through the center of the lens. The other rays—spaced equally from the center ray—contact the lens surface at different angles and are affected differently by the lens. The far ray contacts the lens at a more oblique angle, a, than the near ray, and thus gets bent more than the near ray as it enters the lens. On exit, however, the opposite is true. As a result, the near ray focuses closer than the far ray. The lack of a clear focus produces the coma effect.

The third Seidel aberration is astigmatism. Astigmatism comes about from asymmetry around the lens’ axis. In the simplest form of asymmetry, the lens surface appears oval, curving more in one direction than in another other. As Figure 4 shows, parallel rays in red entering the lens in the more strongly curved, or principal, plane focus more closely than rays in blue coming through the other plane. Rays in intermediate planes focus at intermediate distances.

Figure 4. An astigmatic lens has two principal axes with different focal lengths, so for the lens shown, rays in the vertical plane focus more strongly than those in the horizontal plane.

In a lens with astigmatism, light from a point source will produce a blur, rather than a sharp focus. At the inside-focus point, the “image” looks like a line parallel to the weakest principal plane. At the outside-focus point, the “image” looks like a line parallel to the strongest principal plane. The best focus point will be where the “image” appears equally blurry all around, giving a circular blur instead of an elliptical blur.

Focus Fields Actually Curve
The fourth Sidel aberration is field curvature, also known as Petzval field curvature, named after Jozeph Miksa Petzval (1807–1891), a mathematician. Field curvature appears because the object and image distances are measured from the lens’ center. A given object distance u defines a sphere of radius u centered on the lens’ geometric center (Fig. 5a). All of the points on that sphere will focus on another sphere of radius v, which is also centered on the lens’ geometric center.

Figure 5. Distortion appears when the magnification of a lens depends on how far the object point is from the optical axis. (The effect is exaggerated in this image.)

Figure 6. Barrel distortion occurs when the central parts of an image are magnified more than the outer parts.

Most microscope samples aren’t curved, however. Generally, samples for electronic inspection are flat (such as semiconductor wafers) or exhibit some complex surface relief pattern (like that on an SMT PCB). The effect of field curvature for flat samples makes it impossible to perfectly focus both the center and the edge of the field simultaneously. If you focus on what’s in the center, parts of the sample off axis will be out of focus and vice versa.

How big an effect field curvature has depends on the angular field of view. The angular field of view (expressed in radians) is the ratio of the diameter of that part of the sample you can see through the microscope divided by the objective lens’ working distance. The working distance is just the object distance for the objective lens.

For very flat samples, such as wafers, field-curvature effects may be a problem. For samples showing high relief, though, focus variations caused by surface topography will swamp field-curvature effects. Field curvature is one Seidel aberration that is not exacerbated by high magnifications. Highly magnified images generally have a small field of view, which reduces field curvature effects.

The final Seidel aberration is distortion. It’s not just any random distortion like you see in a warped mirror, however. Specifically, the Seidel distortion refers to magnification as a function of distance from the optical axis.

Suppose you were able to magically flatten the object plane. The object distance would then, as Figure 5b shows, increase for object points increasingly far from the optical axis. Noting that for microscope objectives, u is comparable to but somewhat larger than f, the magnification (the ratio of v to u) will decrease as you get farther from the optical axis.

Figure 6 shows what the image of a square grid would look like under the field-curvature conditions. Early opticians thought the resulting image looked like a barrel, so they called the effect barrel distortion. Other optical situations cause the magnification to increase with increasing distance from the axis, forming so-called pincushion distortion. An image suffering from pincushion distortion shows parallel lines bending inward toward the center—the opposite of barrel distortion in which they bend outward.

Correct Those Aberrations
Opticians generally use three strategies to correct Seidel aberrations. They limit the light entering a lens, use multiple lens surfaces, or use aspheric surfaces on lenses. These strategies either severely limit what you can do with an optical system or they rapidly increase costs.

Early optical engineers had little control over the quality of the lenses they ground. But by limiting the light allowed into the instrument to only paraxial rays, their optics would provide images acceptable for scientific work. The engineers used “stops” to control the light going into and through an optical system. A stop is a flat mask of opaque material with a circular hole, or aperture, centered on the optical axis.

The adjustable “f-stop” in a photographic camera provides an example of how a stop defines the bundle of light rays used to produce an image. By closing the f-stop, or “stopping down” a lens, a photographer can reduce the amount of light that reaches the film, allowing a longer exposure in bright light.

Stopping down a lens cuts off rays that would pass through the lens’ outer zones. Because all the Seidel aberrations come from the lens’ effects on these outer rays, removing them or stopping them improves an image. The larger a lens is in relation to its focal length, the more non-paraxial rays appear. But there’s a limit to how big a lens you can make. Large lenses are very expensive not because of the amount of glass needed, but because of the cost of grinding them precisely enough over a large surface.

Optical engineers can incorporate multiple lenses in an optical system to control aberrations. These multi-element lens designs can precisely control all of the Seidel aberrations. These lenses cost a great deal, though, due to the number of precision optical surfaces that must be ground and held at just the right spacing. Designs may include half a dozen lenses. A lot of expensive-to-grind optical surfaces are needed to get acceptable quality images from a large, short-focus lens.

Finally, if all else fails, a designer can opt to use lenses with one or more aspheric surfaces. Aspheric surfaces, however, are difficult to grind, although high-precision, digitally controlled grinding machines have made producing them easier than before. Although complex optics are still somewhat expensive, optical designers can more frequently include them in optical systems to reduce aberrations. T&MW

C.G. Masi works as a freelance technical journalist. He is the former chief editor of Test & Measurement World. E-mail: tmw@cahners.com.