In the Classroom edited by
Tested Demonstrations
Ed Vitz Kutztown University Kutztown, PA 19530
W
Exploring the Diffraction Grating Using a He–Ne Laser and a CD-ROM submitted by:
Joel Tellinghuisen Department of Chemistry, Vanderbilt University, Nashville, TN 37235;
[email protected] checked by:
Carl Salter Department of Chemistry, Moravian College, Bethlehem, PA 18018
Although interferometric and Fourier-transform techniques now dominate infrared and magnetic resonance spectroscopy, most UV–visible spectroscopy is still done with dispersive instruments equipped with reflection diffraction gratings (1). Such gratings are also used as tuning elements in dye lasers (2, 3). A plane grating has a large number of precisely ruled parallel grooves separated by a distance d that is comparable to the wavelength λ of the light to be analyzed. The light is diffracted in a plane perpendicular to the direction of the grooves, in accord with the grating equation, d (sin α + sin β) = n λ
a β1
n = -1
n=1
n = -2
n=2
(1)
where α and β are the angles of the incident and diffracted beams (measured relative to the normal to the grating, using the same sense of rotation for both), and n is the order (0 = reflection, ±1 = first order, ±2 = second order, etc.) (1). For some time I have included as a component of an undergraduate course in spectroscopy an illustration of eq 1 using a He–Ne laser and one or more small research gratings having groove densities (1/d ) ranging from 600 to 1800/mm. I have also included a hands-on exercise for students, the goal of which is the determination of d for an unknown grating from measurements of α and β for various geometrical configurations. Awareness of the cost of research gratings and their sensitivity to damage from handling has always made this component of the exercise a source of trepidation. The now-ubiquitous compact disc (CD) offers a near-zero-cost alternative for this experiment; used properly, it works almost as well as a 600-groove/mm research grating. Background The use of CDs as dispersive elements for spectroscopy has been described in several articles in this Journal (4–6 ). These applications emphasize qualitative observations of intensities and line positions. Cornwall (7 ) described the measurement of d for a CD, but only for normal incidence (α = 0), which neglects important configurations more widely used in spectrometers and dye lasers—and accordingly, also limits students’ appreciation of the full meaning of eq 1. With proper choices of α, β, and n, it is possible to determine d within ~1%, using just a clear plastic protractor to measure the angles. This is good enough to permit measurement of λ for a red laser pointer with enough precision to confirm that its wavelength is clearly greater than the 632.8-nm value for the He–Ne laser.
b n=1 n=0
β2
n=2 α n=3
n = -1
n=4
Figure 1. Geometry of incident and diffracted laser beams of wavelength 632.8 nm on a reflection grating of groove spacing 1.600 µm. (a) The incident beam is normal to the surface of the grating (α = 0), and the diffracted beams are symmetrically disposed about the normal. (b) α = 35.6°, which is just large enough to permit the fourthorder diffracted beam to occur at 90°. The third-order beam lies very close to the incident beam; they coincide when α = β = 36.4°. Note that in b the diffraction angles are negative for orders less than 2.
The groove spacing for both audio CDs and CD-ROMs is 1.6 µm (7), so for normal incidence, only n = 0, ±1, and ±2 occur for the He–Ne laser (Fig. 1). As the grating is rotated (α increased), the third-order beam will appear on the same side of the normal as the incident beam. It first emerges parallel to the grating surface (β = 90°) when α = 11°. With further rotation, the fourth- and then the fifth-order diffracted beams appear in like manner (Fig. 1); however, the last of these is hard to observe with the CD. If the diffracted beam is made to reflect straight back onto the face of the laser, one has α = β, and only one angle need be measured for a given order. This arrangement,
JChemEd.chem.wisc.edu • Vol. 79 No. 6 June 2002 • Journal of Chemical Education
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In the Classroom
called the Littrow configuration, is widely used in tunable lasers and in some spectrometers. The relevant angles are 11.4° (first order), 23.3° (second order), 36.4° (third order), 52.3° (fourth order), and 81.4° (fifth order); here again, the last may not be observable. Equation 1 is symmetrical with respect to exchange of α and β, so in principle the two beams can be switched. However, with the incident beam near 90°, the spot spreads over much of the radius of the disc and the quality of the diffracted beams deteriorates, owing to the sampling of an increasing range of grooves of varying curvature and hence increasing non-parallelness. The same effect is probably responsible for the difficulty of observing the fifth-order beam in the Littrow configuration. Measurements and Discussion For the angle measurements, the CD can be mounted about half a meter from the laser, with the laser beam horizontal in the laboratory. To ensure that the diffracted beams lie in a plane parallel to the tabletop, the laser spot should be directed onto a horizontal radius of the CD. Since the curvature of the grooves is greater near the center of the disc, spot placement near the outer rim will yield better diffracted beam quality. (The diffracted beams become progressively more elongated in the vertical plane as the incident beam spot is moved toward the inner radius of the disc.) Some protractors do not have the radius point flush with the edge, so students may need to be reminded that all angles must be measured along true radii of the protractor.1 Although it can be a test of manual dexterity, proper placement of the plastic protractor will permit direct and simultaneous observation of the various diffracted beams in the clear plastic.2 The exercise also offers instructive examples of statistical error propagation (8). The angles can be measured with a precision (σα, σβ) of ~0.5° (0.0087 rad). If we consider, for example, the Littrow measurement in the third order, we have 2 sin α = 3 λ/d from which Thus which yields
d ᎑1 = (2 sin α)/(3 λ) σ1/d = (2 cos α)/(3 λ)σα σ1/d = 7.4/mm
The relative error is 1.2% and is the same for d as for d ᎑1, so σd = 19 nm. Note that it is not correct to consider α and β as separately uncertain, since only one angle is measured; however, for configurations where α is set and β is measured at a different angle, the error should be propagated as a function of both angles. As was noted earlier, this precision is adequate to verify that red laser pointers operate at slightly longer wavelengths than He–Ne lasers. As Figure 2 illustrates, an even more striking comparison can be achieved with a green laser pointer. These devices are still expensive by laser-pointer standards (>$200), but may be within the grasp of a laboratory teaching budget. (The online supplement accompanying this article illustrates the red laser–green laser comparison in vibrant color.W)
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Figure 2. Diffraction of red (He–Ne) and green laser beams by a CD-ROM. The parallel incident beams (separation ~1 cm) enter from lower left and are normal to the CD surface (α = 0), as in Figure 1a. The camera is positioned above the diffraction plane approximately in the direction of the first-order diffracted red beam. The second-order beams emerge to the right, and the other firstorder beams (which cross) are just discernible to the left of the incident beams. (The beams were made visible by blowing “fog” from a Dewar of liquid nitrogen across the diffraction plane.)
As a demonstration, this exercise can be completed in
