Download Lateral shearing interferometry for EUV optical testing

Lateral shearing interferometry for EUV optical testing
Ryan Miyakawa1,2, Patrick P. Naulleau2, and Ken Goldberg2
1
Applied Science and Technology Group, UC Berkeley,
2
Lawrence Berkeley National Laboratory
Contact: [email protected]
SYSTEM ERROR DUE TO GRATING AND DETECTOR TILT
WHY LSI?
As demands increase for high quality optics in the Extreme Ultraviolet
(EUV) and Soft X-ray (SXR) regions for applications in high resolution
microscopy and lithography, it is becoming increasingly important to
have a reliable means for measuring these optics. In Lateral Shearing
Interferometry (LSI), the test wavefront is interfered with a shifted
(sheared) copy of itself. The shear at EUV wavelengths is created by a
low frequency diffraction grating. LSI is an attractive alternative to
other interferometric techniques because of its simple setup and loose
requirements on coherence and photon flux.
LSI can be thought of as a linear system D that maps a wavefront of N Zernikes
to its X and Y derivatives. Since the dimension of the codomain W is twice that
of the domain V, the solution wavefront is overspecified.
• Improper alignment of
the grating or CCD can
exacerbate systematic
errors. Astigmatism in
particular is very
sensitive to grating tilt
Where Dim{V} = N, and Dim{W} = 2N
Since Rank{D} ≤ N, there exists a set F = W \ D{V}, where F = D{V},
such that vectors fi Є F are not mapped by D and thus represent forbidden
solutions (solutions that cannot come from a real wavefront).
Astigmatism (waves)
0
-0.2
-0.4
We can then define an error function to determine the validity of the data w:
-0.6
-0.8
-1
-1.2
-6
-4
-2
0
2
4
Grating tilt angle (mrad)
•Object beam diffracts off of diffraction grating
• Multiple orders propagate at slightly different angles and interfere
on the CCD.
• Optical pathlength difference is the function f(x) = W(x+s) – W(x),
which approximates the discrete derivative of the wavefront W(x)
• Relaxed coherence requirements
• Can operate without large amounts of photon flux
• Robust against mechanical vibrations
D {v} =
D: V W
Measured astigmatism vs. grating tilt angle
• Errors can be
determined analytically by
treating the grating as a
hologram and performing
a coordinate transform to
the grating coordinates
LSI SETUP FOR EUV AND SXR WAVELENGTHS
MONITORING ALIGNMENT USING A LINEAR SYSTEMS ANALYSIS
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If E is non-zero, then the data vector w has a nonzero projection onto F, which
means that the system has drifted from proper alignment. This technique allows
for a real-time method to asses the health of the LSI system.
SYSTEMATIC METHOD TO ALIGN THE LSI SETUP
PRECISION OF THE LSI ALIGNMENT
Although LSI is sensitive to grating and detector tilt angles, it can
leverage this sensitivity to ensure its own proper alignment. The
alignment process is briefly outlined:
Since LSI is so drastically affected by grating alignment it is important to know
how effective the alignment procedure will be in the presence of mechanical
uncertainties. The following plots show the relationship between these
mechanical uncertainties and the eventual aberrations in the fringes from the
resulting calibration errors. For clarity we show only astigmatism, which is the
largest systematic error.
• Align CCD to be perpendicular to the optical axis by finding the CCD
angle that has the highest density of fringes.
GRATING DISTORTION TO THE SPHERICAL WAVE CARRIER
• Collect interferograms for many (~30) different grating tilt angles and
measure the resulting tilt and astigmatism in the fringes. Repeat for at
least 3 different grating translations.
Spatial frequencies incident upon a periodic grating will diffract based on
the grating equation:
• Align grating to the tilt of its translation stage axis by noting that
aberrations will be independent of grating displacement at this tilt angle
Astigmatism error vs.
grating tilt precision error
fout = fin + mfg
Astigmatism (waves)
0.06
Tilt
Since spatial frequencies are related to the sin of the angle, the angle of
the outgoing wave will experience a different shift for each incoming
angle. A complex waveform that is composed of many spatial
frequencies will thus not preserve its shape when it is diffracted into a
given order.
• Plane wave illumination diffracts into plane waves
0.05
0.04
0.03
0.02
0.01
0
0
50
100
150
200
250
300
Grating tilt uncertainty (mrad)
• Grating distorts spherical wave illumination in diffracted orders
Astigmatism error vs.
grating translation parallelism uncertainty
• Perform a polynomial fit of tilt and astigmatism as a function of grating
tilt angle using the translation stage tilt as the origin:
Astigmatism (waves)
0.09
T(θ) = Σci xn, A(θ) = Σdi xn
+1 order diffracted spherical wave decomposition:
=
+
astig
+
+
+…
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
coma
0
0.1
0.2
0.3
0.4
0.5
Uncertainty in grating translation parallelism (µm)
The astigmatism and coma aberrations due to the grating distortion
means that the sheared beams are not identical. These errors
need to be corrected for when performing the LSI analysis by
subtracting out the null interferogram aberrations.
• Either match the resulting linear term c1 or d1 of the fit to its analytic
model depending on the slope of the model around recovered angle
Translation axis
0.6