A model for the diffraction of visible light through a periodic grating is an excellent tool with which to address both the theoretical and practical aspects of image formation in optical microscopy. Light passing through the grating is diffracted according to the wavelength of the incident light beam and the periodicity of the line grating. This interactive tutorial explores the mechanics of periodic diffraction gratings when used to interpret the Abbe theory of image formation in the optical microscope.

In its simplest form, a line or amplitude grating is composed of a linear array of thin opaque strips (or slits) having a periodic spacing and suspended on a solid matrix, usually an optical glass plate. The most convenient and accurate method of forming gratings of this type is through the use of metallic vacuum deposition techniques. The spacing between the centers of two adjacent slits (d) is called the grating period, and the reciprocal of d is termed the spatial frequency, which is measured in the number of slits or periods per unit length.

The tutorial initializes with a grating periodicity of 1000 nanometers (producing a spatial frequency equal to 1000 lines/millimeter) and an incident light beam of 400-nanometer wavelength impacting the grating at a 90-degree angle. Each slit in the grating diffracts light over the entire range of angles covering 180 degrees on the opposite side of the grating. The Spatial Frequency slider is utilized to change the grating periodicity and the Wavelength slider alters the wavelength of the incident light wave.

Individual light waves diffracted from successive grating slits are emitted as concentric spherical wavelets that interfere both constructively and destructively because they are all derived from the same wavefront and are therefore in phase. Wavefronts passing through the grating slits that are parallel to the incident light wave are referred to as zeroth order (undiffracted) or direct light. Diffracted higher-order wavefronts are inclined at an angle (q) according to the equation:

where l is the wavelength of the wavefront, d is the grating slit spacing and M is an integer termed the diffraction order (e.g., M = 0 for direct light, ±1 for first order diffracted light, etc.) of light waves deviated by the grating. The combination of diffraction and interference effects on the light wave passing through the periodic grating produces a diffraction spectrum, which occurs in a symmetrical pattern on both sides of the zero order direct light wave.

If the diffracted light waves produced by the periodic grating are then passed through a convergent lens, they appear as a series of bright spots on the focal plane of the lens. The intensity of these spots decreases as the diffraction order increases, and the number of higher order diffracted waves that can enter the lens is restricted by the size of the lens aperture. Those waves that enter the lens form what is termed a Fraunhofer diffraction spectrum (also called a Fourier spectrum) that can be observed at the focal plane of the lens.

The periodic diffraction grating can now be used to examine Ernst Abbe's theory of image formation in the optical microscope. When the line grating is placed on a microscope stage and illuminated with a parallel beam of light that is restricted in size by the condenser aperture diaphragm, both zero and higher order diffracted light rays enter the front lens of the objective. Direct light that passes through the grating unaltered is imaged in the center of the optical axis on the objective rear focal plane. First and higher order diffracted light rays enter the objective at an angle and are focused at discrete points (a Fraunhofer diffraction pattern) on both sides of the direct light beam at the objective rear focal plane. A linear relationship exists between the position of the diffracted light beams and their corresponding points on the periodic grating.

If the periodic grating placed on the microscope stage is a micrometer or similar grid, then the Fraunhofer diffraction pattern can be observed by removing one of the microscope eyepieces and examining the objective rear focal plane (or by using a phase telescope or Bertrand lens). First, reduce the condenser aperture size to a minimal value then, using a low-power (10x or 20x) objective, focus the bright central spot on the focal plane while viewing through the eyepiece tube. A series of higher-order light spots of diminishing intensity can now be observed flanking the central spot. The diffracted light spots display a spectrum of color with lower wavelengths (blue and purple) nearer the optical axis and higher wavelengths (red) spread on the periphery. Spacing between the light spots is dependent upon the grating interval and the wavelength of light passed through the condenser. Finely spaced gratings and longer wavelengths produce larger spot intervals than do coarse gratings and lower wavelengths.

At the microscope intermediate image plane, coherent light emitted from the diffracted orders at the objective rear aperture undergoes interference to produce an intermediate image of the periodic grating, which is further magnified by the eyepieces. The integrity of the intermediate image depends upon how many diffracted orders produced by the grating pass through the aperture and are captured by the objective front lens. Objectives having a higher numerical aperture are able to gather more of the diffracted light waves and produce clearly better images.

Abbe determined that in order to form a recognizable image, the objective must capture the zeroth order light rays and at least one of the higher order diffracted waves or two adjacent orders. Because the diffraction angle is dependent upon the grid spacing and the wavelength is determined by the refractive index (n) of the medium between the grating and the objective front lens, the diffraction equation (given above) can be rewritten as:

Abbe originally defined the numerical aperture (NA) of the objective as:

so the equation reduces to:

This equation is one of the most fundamental to optical microscopy and demonstrates that an objective's ability to resolve fine details in a specimen, such as a periodic grating, is dependent upon both the wavelength of illuminating light rays and the numerical aperture. Thus, the lower the wavelength or the higher the numerical aperture, the greater the resolving power of the objective.

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