Light Characteristics Limit Optical Quality

The wavelength of light affects microscope optical-system performance.

C.G. Masi, Contributing Technical Editor -- Test & Measurement World, 6/1/2000

The basic theory of lenses, covered in the first article in this Basic Microscopy Series,¹ works well for calculating image positions and sizes when all the light rays pass through the optical system roughly parallel to, and very near, the optical axis. But when the rays exist too far from the optical axis, or when they move at a significant angle to the optical axis, nonlinear effects become important. These nonlinear effects, called Seidel aberrations, manifest themselves as perturbations in the image.

You probably wonder, “How far is too far from the optical axis?” That depends on the optical quality you need. “Too far” is simply the point at which the magnitude of the aberrations starts to interfere with what you want the optical system to do. Professional microscopy is one of the most demanding optical applications, so microscope designers work hard to eliminate even small aberrations.

In the previous article in this series,² I described the third-order theory that governs Seidel aberrations. This theory does not take into account the wavelengths of light in an image. Wavelength-dependent effects that degrade microscope performance come in two forms: chromatic aberration, which arises from wavelength-dependent properties of optical materials, and diffraction, which arises from the basic wave nature of light.

Chromatic aberration arises because light slows down as it passes through a material. The amount it slows depends on the characteristics of the material and on the wavelength of the light (see “A Bit of Theory,”). Blue light slows more than red light at an interface, so it bends more than red light at a glass-air interface (Fig. 1). This effect is known as dispersion. Although the amount by which blue light slows more than red light differs from material to material, there is no material in which red light slows more than blue light.

Because blue light bends more strongly than red light, it will be focused more strongly when passing through a lens. Thus, the effective focal length for blue light is shorter than for red light. This chromatic aberration makes the effective focal length of a lens depend on the color of light passing through it.

Assume your beam contains red and blue light. When you focus an on-axis point of light using the red light, the blue light will appear out of focus as a blue halo around the central red dot. If you focus on the blue light, the red appears as an out-of-focus red halo around a blue dot (Fig. 2). This effect is called longitudinal chromatic aberration because you observe the effect along the optical axis.

Figure 3 shows the effect of lateral chromatic aberration—commonly called lateral color—on white light passing through a microscope. This effect takes place in the plane perpendicular to the optical axis, thus it’s called lateral dispersion. The image has shifted in the image plane. Lateral chromatic aberration may also produce a rainbow-like effect at hard edges in an image. The blue image appears slightly smaller than the red image due to the shorter effective focal length of the blue light. This visually annoying chromatic aberration is a sign of cheap optics in a microscope.

Figure 1. Upon entering a denser medium, shorter-wavelength (blue) light slows down more, and therefore refracts more, than longer-wavelength (red) light. As a result, the lens’ focal length depends on the wavelength of the light.

Figure 2. Longitudinal chromatic aberration makes it impossible to simultaneously focus all wavelengths of light in a single-element lens system. You can focus only one wavelength at a time.

Figure 3. In an uncorrected lens, lateral chromatic abberation appears because blue light forms a slightly smaller image than red light at a focus midway between that for red and blue light.

Figure 4. Combining positive and negative lenses made of different materials makes it possible to cancel out chromatic aberrations.

Actually, an achromatic doublet offers optical engineers more than four parameters to work with. It’s possible to adjust the positions and curvatures of four lens surfaces as well as the dispersive powers of the two glasses. That may seem like too many parameters, but it’s not. While the engineers correct for chromatic aberration, they must also correct for all of the Seidel aberrations. To do a proper job of correcting all of these aberrations, the engineers often use three or more optical elements in each lens.

With enough ingenuity and effort, skilled optical engineers can correct geometric optical aberrations to any needed level. If you want a better microscopic image, all you have to do is put more engineering effort into a new design. But at a certain point you run into a “wall” beyond which no amount of design tweaking will produce a better image. That wall, called the Rayleigh limit, arises from the wave nature of light.

Light Acts Like a Wave
A full understanding of the ramifications of light’s wave nature requires entering the “Alice in Wonderland” world of diffraction theory and spatial Fourier transforms. That’s one rabbit hole you’ll want to avoid. Instead, you can rely on the results provided by others who have made the trip.

Modern diffraction theory works with spatial frequencies. In this theoretical realm, the inverse wavelength or wave number plays a key part. The inverse wavelength is the number of complete waves that fit in one length unit, usually a meter. Thus, monochromatic light with a wavelength of 1000 nm (near infrared) has a spatial frequency of 10⁶ waves per meter.

Similarly, images also contain spatial frequencies. If you image a set of vertical stripes, each 1 mm wide and spaced 1 mm apart, the image has a fundamental spatial frequency of 0.5 mm^-1, or 0.5/mm. This set of sharp vertical stripes is equivalent to a square wave (Fig. 5).

A square wave represents the sum of an infinite number of sinusoids at odd harmonic frequencies. Likewise, the vertical stripes are made up of an infinite number of spatial sinusoids at odd harmonic frequencies. Diffraction theory works with Fourier transformations between spatial frequencies and actual images, just as spectral analysis of electronic signals works with Fourier transforms between the frequency domain and time domain. Sharp edges in electronic signals imply the existence of high frequency components; high resolution in images implies the existence of high spatial-frequency components.

 To resolve parts of an electronic signal that are very closely spaced in time, you need a wide bandwidth (aperture) in the frequency domain. Similarly, to resolve closely spaced image points, you need a wide aperture (bandwidth) in the spatial-frequency domain. In a lens, the low spatial-frequency information tends to pass through the middle of the lens and the higher spatial frequency information passes farther out. Thus, capturing an image that contains high spatial-frequency components requires a wide physical aperture, or opening, at the lens.

Figure 5. Alternating 1-mm.-wide black and white stripes produce a pattern with a 0.5 mm-1 spatial frequency. Like all square waves, the pattern represents the sum of sinusodial patterns of higher frequencies.

where
Y = the minimum spacing that can be resolved between two objects
l = the wavelength of the light forming the image
N_A = the numerical aperture of the lens

The numerical aperture is actually the sine of the objective lens’ angular width (q) as viewed from the object (Fig. 6) multiplied by the refractive index (n) of the medium in which the object exists:

where
A = the diameter of the aperture
w = the distance between the objective lens and the object under inspection

Figure 6. You use the measurements shown in this figure in Equation 2 to determine the numerical aperture of an objective lens.

Usually, the medium behind the lens is air, which has a refractive index close to 1. The theoretical maximum value for sinu is also 1. Thus, the theoretical maximum value of N_A is also 1, which makes the best resolution possible for any compound microscope 0.61 times the wavelength of the light used to make the image.

Because the shortest wavelength you can see is around 350 nm, the very best resolution possible with a visible light compound microscope is 213 nm, or about a quarter of a micron. This relationship between wavelength and resolution explains why suppliers who make microscopes for semiconductor inspection are rapidly moving to use ultraviolet (UV) light for illumination. The short-wavelength UV radiation below 350 nm provides the only practical way to observe small features on a die using a microscope.

Future developments in microscopes for semiconductor inspection will rely on UV optical elements and techniques. And even existing microscopes must make effective use of illumination sources to present specimens in ways that let you observe small features. Our next installment, scheduled for August 2000, will cover illumination issues. T&MW

FOR FURTHER READING
Born, Max, and Emil Wolf, Principles of Optics, 7th ed., Cambridge University Press, New York, NY, 1999.

Hecht, Eugene, Alfred Zajac, and Karen Guardino, Optics, 3rd ed., Addison Wesley, Reading, MA, 1997.

FOOTNOTES
1. Masi, C.G., “Microscopes Rely on Basic Optical Components,” Test & Measurement World, April 2000. pp. 38–46.

2. Masi, C.G., “Basic Optical Effects Limit Image Quality,” Test & Measurement World, May 2000. pp. 73–78.

3. For information about the historic people mentioned in this article, refer to the short biographies at the University of St. Andrews, Scotland: www-history.mcs.st-and.ac.uk/~history/BiogIndex.html

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