Line Spacing Calculations from Diffraction Gratings
By definition, a diffraction grating is composed of a planar substrate containing a parallel series of linear grooves or rulings, which can be transparent, semi-transparent, or opaque. When the spacing between lines on a diffraction grating is similar in size to the wavelength of light, an incident collimated and coherent light beam will be strongly diffracted upon encountering the grating. This interactive tutorial examines the effects of wavelength on the diffraction patterns produced by a virtual periodic line grating of fixed line spacing.
The tutorial initializes with a beam of coherent and collimated purple monochromatic light (400 nanometers) incident on a periodic diffraction grating. Upon passing through the line grating, the light beam is diffracted into a bright central band (zeroth-order) on the detector screen, flanked by several higher-order (1st, 2nd, and 3rd) diffraction bands or maxima. In order to operate the tutorial, use the Wavelength slider to adjust the size of monochromatic light passing through the grating in a range between 400 and 700 nanometers. As the slider is translated to the right, the wavelength of incident light increases, producing a corresponding change to the diffraction pattern observed on the detector screen (the long, horizontal line above the slider). The diffraction bands formed by the higher-order maxima identify the diffraction angles in which wavefronts having the same phase become reinforced as bright areas due to constructive interference. In regions between the diffraction bands, the wavefronts are out of phase and cancel the intensity of each other by destructive interference.
The zeroth-order central maximum band is formed from light waves that do not become diffracted when passing through the diffraction grating, and displays an intensity value only slightly reduced from that of the incident beam. The diffraction angle, which is identified by the symbol q, is the determined by the angle subtended by the zeroth and first-order bands on the detector with respect to the grating. A right triangle containing the diffraction angle at the detector screen is congruent with another triangle at the grating defined by the wavelength of illumination (l) and the spacing between rulings (d) on the grating according to the equation:
As a result, the reinforcement of diffraction bands or spots occurs at locations having an integral number of wavelengths (l, 2l, 3l, etc.) because the diffracted wavefronts arrive at these locations in phase and are able to reinforce each other through constructive interference. If the sine of the diffraction angle is calculated from the distance between diffraction bands on the detector screen and between the screen and the line grating, the spacing (d) between individual rulings on the grating can be determined using the grating equation in the form:
where l is the wavelength of incident light and m is an integral number of diffraction bands. For example, calculations based on the distance between the first and zeroth order diffraction bands require the value of m to be 1, whereas similar calculations between the zeroth and second order diffraction bands have a value of m equal to 2, and so on. According to the equation, the size of the diffraction angle decreases as the line grating space intervals increase. Likewise, longer wavelengths give rise to larger diffraction angles at constant line grating spacings.
The effect of wavelength can be demonstrated by illuminating the line grating with white light (containing a mixture of all colors). Under these conditions, the zeroth-order diffraction band appears white, but higher order bands display an elongated spectrum of colors with blue being closest to the zeroth-order band. Thus, blue light is diffracted to a lesser extent than is green or red light, as illustrated in the tutorial.
Douglas B. Murphy - Department of Cell Biology and Microscope Facility, Johns Hopkins University School of Medicine, 725 N. Wolfe Street, 107 WBSB, Baltimore, Maryland 21205.
Kenneth R. Spring - Scientific Consultant, Lusby, Maryland, 20657.
Matthew J. Parry-Hill and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.
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