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Microlithography Systems Laboratory
Microelectronic Engineering Department
Laboratory 1. Fraunhofer Diffraction and the Fourier Series
Introduction
The diffraction of light is responsible for image creation in all optical situations. When a beam of
light encounters the edge of an opaque obstacle, propagation is not rectilinear as might be assumed
based on assumptions of geometrical shadowing. The resulting variation in intensity produced at some
distance from the obstacle is dependent on the coherence of light, its wavelength, and the distance the
light travels before being observed.
For projection lithography, diffraction in the far-field or Fraunhofer region needs to be
considered. No longer is geometric shadowing recognizable; rather, fringing takes over in the resulting
intensity pattern. When light encounters a mask, it is diffracted toward the object lens in the projection
system. Its propagation will determine how an optical system will ultimately perform, depending on the
coherence of the light that illuminates the mask.
The amplitude spectrum of the rectangular wave is shown in below, where A/2sinc(u/2u0)
provides an envelope for the discrete Fraunhofer diffraction pattern. It can be shown that the discrete
interference maxima correspond to d sin =m where m = 0, +/-1, +/-2, +/-3, and so on, and where d is
the mask pitch.
The amplitude spectrum of the rectangular wave can be utilized to decompose the function into a
linear combination of complex exponentials by assigning proper weights to complex-valued coefficients.
This allows harmonic analysis through the Fourier series expansion, utilizing complex exponentials as
basis functions. These exponentials, or sine and cosine functions, allow us to represent the spatial
frequency structure of periodic functions as well as non-periodic functions. When a function is even and
real-valued, the amplitude spectrum can be utilized to decompose m(x) into the cosinusoidal frequency
components:
Microlithography Systems – Laboratory I – Fraunhofer Diffraction and Fourier Series
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These discrete coefficients are the diffraction orders of the Fraunhofer diffraction pattern that are
produced when a diffraction grating is illuminated by coherent illumination. These coefficients,
represented as terms in the harmonic decomposition of m(x) correspond to the discrete orders seen in
above. The zeroth order (centered at u = 0) corresponds to the constant DC term A/2. At either side are
the +/- first orders, where u1 = 1/p. The +/-second orders correspond to u2 = +/-2/p, and so on. It would
follow that if an imaging system was not able to collect all diffracted orders propagating from a mask,
complete reconstruction would not be possible. Furthermore, as higher frequency information is lost,
fine image detail is sacrificed. There is, therefore, a fundamental limitation to resolution for an imaging
system determined by its inability to collect all possible diffraction information.
Laboratory Goals
The goal of this laboratory is to produce, measure, and quantify the Fraunhofer diffraction
patterns from a grating photomask through Fourier analysis. Two grating masks will be utilized at two
laser wavelengths. Mask pitch, duty ratio, and source wavelength will be measured.
Procedure
1. Obtain the following items:
a) HeNe laser (wavelength = 632 nm)
b) Green laser (wavelength to be determined)
c) Screen to block beam at end of setup
d) Two grating masks
e) Ruler
f) Intensity detector (Newport Research Model-818 Photosensor) and amplifier (Newport
Research Model-815).
g) Paper
2. Place one grating mask in the mask holder. With the laser illuminating the grating mask, project the
diffracted pattern onto the screen. Align the laser so that the optical axis is perpendicular to the
screen.
3. Make observations of the transmitted diffraction pattern on the screen by measuring distances:
a) From the mask to the screen
b) From the 0-order to the +/- 1st-order
c) From the 0-order to the +/- 2nd-order
d) From the 0-order to higher orders if possible
4. Using the Newport Research Model-818 Photosensor and Newport Research Model-815 amplifier,
measure the intensity of the undiffracted beam and the 0, +/- 1st, and +/-2nd orders.
5. Use the collected data to calculate the grating pitch, the line/space duty-ratio, and the “leakage” that
exists in the line regions of the mask.
6. Repeat steps 2-5 with the second grating mask
7. Repeat steps 2 and 3 with the green laser for both grating masks. Use this data to calculate the
wavelength of the green laser.
Microlithography Systems – Laboratory I – Fraunhofer Diffraction and Fourier Series