APPARATUS AND DEMONSTRATION NOTES
Jeffrey S. Dunham, Editor
Department of Physics, Middlebury College, Middlebury, Vermont 05753
This department welcomes brief communications reporting new demonstrations, laboratory equipment, techniques, or materials of interest to teachers of physics. Notes on new applications of older
apparatus, measurements supplementing data supplied by manufacturers, information which, while not
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Dunham, Editor.
Revealing the blaze angle: A simple experiment for visualizing diffraction
effects using microscopic and macroscopic gratings
K. G. Vandervoort,a兲 S. L. Adams, and A. M. Hyder
Physics Department, California State Polytechnic University, Pomona, California 91768
共Received 17 December 2005; accepted 31 March 2006兲
We describe a diffraction grating experiment for visible and microwave optics. The groove spacing
for the visible-optics grating is determined through atomic force microscopy, so that an additional
feature, the blaze angle, is revealed. From the blaze angle, students calculate the theoretical shift in
the diffraction pattern and are then able to visualize this shift through measurements of a microwave
diffraction pattern produced using a macroscopic grating. The experiment demonstrates the utility of
an atomic force microscope in an engineering application and allows separation of diffraction
pattern effects associated with slit width and slit spacing. © 2006 American Association of Physics Teachers.
关DOI: 10.1119/1.2198890兴
I. INTRODUCTION
To demonstrate physical optics concepts in introductory
physics courses, typical experiments include measurements
of diffraction or interference patterns from single- to severalslit gratings, as well as diffraction gratings. Diffraction gratings are introduced as a way to produce very narrow interference maxima that can be separated easily by wavelength.
However, while most introductory textbooks correctly describe the theoretical location and sharpness of the diffraction grating patterns, they often ignore the diffraction envelope associated with the slit width that accompanies these
patterns.1 The diffraction envelope’s effect on the reduction
of intensity for lines of higher order is not a minor detail, as
one can readily verify through a simple spectroscopy experiment.
Blazed gratings2 allow the center of the diffraction envelope 共where the intensity is highest兲 to be shifted from its
normal location, at the center of the interference pattern 共zeroth order maximum兲, to any desired higher-order interference maxima. Since all wavelengths exhibit an interference
peak at the same zeroth-order maximum position, higher intensity at this position is undesirable and shifting the center
of the diffraction envelope to higher-order maxima, where
different wavelengths are separated, is the goal.
To address these issues, we have developed an experiment
in which students can measure the blaze angle of a standard
diffraction grating and determine the theoretically shifted
center of the diffraction envelope. They then perform an experiment with a macroscopic diffraction grating and microwaves, and are able to measure directly the intensities of the
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shifted pattern and verify the influence of the blaze angle.
Others3 have demonstrated the use of forked blazed gratings
with groove spacings in the millimeter range. However, the
apparatus employed in those experiments was more specialized and not standard equipment in most undergraduate
physics labs. We describe an experiment that uses readily
available physics-education microwave equipment4 with a
blazed grating with groove spacings on the centimeter scale.
In addition, our experiment allows students to measure patterns with and without blazing that demonstrate the separate
contributions from slit-width and slit-spacing effects. The experiment also introduces students to an atomic force microscope 共AFM兲, allowing them to experience hands-on use of
cutting-edge instrumentation at the introductory level. The
utility of the AFM is revealed in a natural way, in that it
allows measurement of the third dimension that is needed to
determine the blaze angle, a feature that could not be determined using a standard optical microscope.
II. VISIBLE OPTICS EXPERIMENT
To introduce our blaze angle experiment in the proper context, we insert it as a module that follows one of our existing
experiments, in which students use a diffraction grating to
measure the spectra of atomic gas discharge tubes. In the old
experiment, they encounter the reduced intensity of the
higher-order spectral lines, namely, the less intense secondorder maxima compared to the first-order maxima, and
thereby verify the existence of the diffraction envelope. In
the new experiment, the first step is to image the diffraction
grating using our AFM.5 Students obtain a three-dimensional
© 2006 American Association of Physics Teachers
649
Fig. 1. Ray diagram for a blazed transmission grating for light incident
normal to the back or “ungrooved” side of the grating. The blaze angle is
determined so that the direction of the refracted ray transmitted through the
grating is parallel to the direction of one of the mth interference maxima.
The upper incoming ray indicates the subsequent direction of the center of
diffraction envelope, in the direction of the refracted ray. The lower incoming ray indicates the subsequent direction of the interference maxima. For
this example, the grating is blazed to shift the highest intensity to the
m = −1 maximum. The grating produces analogous results for rays incident
on the front of the grating, normal to the groove surface.
image of the surface and use the instrument’s software to
obtain an image cross section that can be used to determine
the blaze angle and spacing between the grooves.6
The diffraction grating we employ is a transmission grating, ruled with 600 lines/ mm and with a specified index of
refraction of 1.51. As shown in Fig. 1, for light of normal
incidence on the back 共“ungrooved” side兲 of the grating, the
center of the diffraction envelope is shifted to the direction of
the refracted ray that exits the grating. However, the interference peaks are not shifted as their directions are determined
by the plane of the grating. To shift the center of the diffraction envelope to the direction of one of these m interference
peaks located at an angle ␪m, it is necessary that2
sin−1共n sin ␪B兲 − ␪B = ␪m ,
共1兲
where n is the index of refraction, ␪B is the blaze angle, and
␪m is measured with respect to the normal of the grating
plane. Our grating is designed to shift the diffraction envelope for light in the middle of the visible spectrum 共␭
= 500 nm兲 to the first order 共m = 1兲 interference peak. Students calculate ␪m from the interference maxima equation,
d sin ␪m = m␭, for m = 1, ␭ = 500 nm, and d = 1 / 600 mm, and
then determine the index of refraction, n, of the glass using
Eq. 共1兲 with their theoretical ␪m and measured blaze angle,
␪B. They usually obtain a value for n within 10% of the
manufacturer’s stated value.
III. MICROWAVE OPTICS EXPERIMENT
To visualize the blaze angle’s influence on the diffraction
envelope, it is useful to demonstrate this effect on a macroscopic scale. Students perform a microwave experiment for a
two-slit interference pattern with and without a blazed macroscopic diffraction grating, as shown in Fig. 2. While not
equivalent to the pattern of a diffraction grating, the two-slit
pattern does show the same qualitative features, that is, the
same diffraction envelope but much broader interference
peaks. The larger scale of the demonstration prohibits the use
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Am. J. Phys., Vol. 74, No. 7, July 2006
Fig. 2. Microwave experiment with reflector plates and diffraction grating.
Distances for the transmitter 共T兲 and receiver 共R兲 are not to scale. Directions
for negative and positive angles are indicated.
of an extended diffraction grating. In addition, even a grating
as small as four slits is somewhat prohibitive. The significant
overall area associated with the slit region would necessitate
a significant crossover from the Fraunhofer to Fresnel diffraction limits,7 considering the short transmitter and receiver distances for this apparatus, requiring analysis which
would be beyond the scope of the course. Our reflector plates
are placed for a slit separation and slit width of 6 and 4 cm,
respectively. For the 10.5-GHz microwave transmitter 共␭
= 2.86 cm兲, this configuration produces theoretical first-order
Fraunhofer interference peaks at ±28.5°, significantly separated from the central maximum, and a diffraction envelope
that provides a significant decrease in intensity for these
first-order maxima 共by a factor 4兲, as compared to the central
maximum.
The diffraction grating is 10 cm thick and made from four
identically machined pieces of 1-in-thick Plexiglas™, glued
together with clear epoxy. The grating is 14 cm long and
3.3 cm across at its widest points. To ensure that microwaves
did not pass over or under the grating, additional reflector
plates were placed above and below the grating. The distances from the transmitter and the receiver to the center of
the grating were 50 and 40 cm, respectively. To determine
the appropriate blaze angle for this grating, so as to shift the
center of the diffraction envelope to one of the first-order
interference maxima at ±28.5°, one can use the measured
value for the index of refraction of the Plexiglas 共determined
from one of our previous microwave experiments to be 1.7兲,
and solve Eq. 共1兲 graphically for ␪B. From this calculation,
we obtain ␪B = 30.1°. The two groove sections of the grating
are cut at this angle, and the alignment of these grooves with
the 4-cm-wide slits, fixes the geometry of the rest of the
grating.
Figure 3 displays the microwave intensity as a function of
angle for the two-slit interference pattern with and without
the diffraction grating. The original two-slit interference pattern demonstrates the strong attenuation of the first-order
peaks as compared to the central maximum, thus emphasizing the desirability of shifting the diffraction envelope. The
interference peaks are moved considerably inward by about
five degrees from their positions calculated using only the
interference maxima equation, d sin ␪m = m␭. The pattern
matches the theoretical predictions for Fraunhofer diffraction
Apparatus and Demonstration Notes
650
tion of those peaks, thereby effectively distinguishing effects
due to slit width from effects due to slit spacing. The experiment introduces students to the use of an AFM to obtain
measurements that could not be made with an optical microscope. A more automated form11 of the microwave experiment could easily be developed that would allow it to be
used as a lecture demonstration.
ACKNOWLEDGMENT
Fig. 3. Data for two-slit interference experiment with 共closed circles兲 and
without 共open circles with connecting line兲 the diffraction grating. The total
integrated intensity over all angles is greater for the data without the diffraction grating than it is with the grating because of the less than perfect
transmission of microwaves through the Plexiglas.
Funding for this project was provided by the National Science Foundation Nanotechnology Undergraduate Education
program, Award No. 0406533.
a兲
Electronic mail: kvandervoort@csupomona.edu
See, for example, D. Halliday, R. Resnick, and J. Walker, Fundamentals
of Physics 共Wiley, New York, 2001兲, 6th ed., p. 904.
2
F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics 共Prentice-Hall,
Englewood Cliffs, NJ, 1987兲, pp. 419–422.
3
G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67共1兲, 55–60
共1999兲.
4
Microwave Optics System, Pasco Scientific, 10101 Foothills Blvd.,
Roseville, CA 95747-7100.
5
Universal Scanning Probe Microscope, Quesant Instruments, 29397 Agoura Rd., Suite 104, Agoura Hills, CA 91301.
6
Students are somewhat surprised that the groove spacing as measured by
the AFM differs by as much as 10% from the spacing listed on the side of
the diffraction grating. It is then explained that, while the AFM is quite
precise 共reproducibility down to the nanometer level兲 it is not necessarily
so accurate, due to the effects of hysteresis and the non-linearity of the
piezoelectric scanning elements.
7
J. M. Blair, “Using 3-cm microwaves for optics laboratory experiments,”
Am. J. Phys. 60共1兲, 63–66 共1992兲.
8
F. Thompson and H. Tsui, “Transmission of normally incident microwave
radiation through parallel plates of material,” Am. J. Phys. 54共8兲, 712–
715 共1986兲.
9
J. Navasquillo, V. Such, and F. Pomer, “The transmission of a bounded
microwave beam through a dielectric slab,” Am. J. Phys. 56共3兲, 257–259
共1989兲.
10
B. S. Perkalskis and J. R. Freeman, “Diffraction for Fresnel zones and
subzones using microwaves,” Am. J. Phys. 64共12兲, 1526–1529 共1996兲.
11
M. R. Gallis, “Automating Microwave Optics Experiments,” Phys. Teach.
40, 217–219 共2002兲.
1
quite well, with only a slight crossover to the Fresnel limit,
as exhibited, for example, by the non-zero minima between
the central and first-order maxima.
The pattern with the diffraction grating demonstrates the
shift of intensity away from the central maximum and to the
first-order maximum near −28.5°. In addition, the intensities
of the central maximum and the first-order maximum at
+28.5° are correspondingly attenuated. The intensity shift
also restores the position of the peak of the negative-angle
first-order maximum, closer to its predicted value using only
the interference maxima equation. Our measured peak,
which is between −26° and −28° indicates that the center of
the diffraction envelope was not shifted quite as far as predicted, and, in fact, the data are better fit assuming an index
of refraction for our Plexiglas of 1.6 rather than 1.7, more
consistent with refractive indices obtained by others on similar materials.8–10
IV. CONCLUSION
In conclusion, we have described an experiment that demonstrates a way to shift the relative intensities of the peaks in
a diffraction pattern without significantly altering the loca-
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