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| Schlieren— or shadow imaging— allows
visualization of small changes in the refractive index of a material caused by thermal
heating. For backside imaging, the material is the semiconductor substrate. Silicon has a
refractive index change of ~ 5x10-5 per 0C. GaAs has a coefficient
of ~ 3x10-4 per 0C. Although small, these changes can be imaged due
to the extraordinary sensitivity Schlieren imaging offers. Topside Schlieren imaging
requires a suitable coating. Classical Schlieren imaging, as
shown in the figure below, uses a knife-edge to partially block the optical beam on its
way to the image plane.
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In the upper illustration in the figure below, Lens 1
focuses a beam of rays from the source onto a transparent object. Lens 2 re-collimates the
beam, which is then re-focused onto the image plane by Lens 3. A knife-edge is placed in
the focal plane of Lens 2, where an image of the source is formed, such that it partially
blocks the source image. The lower illustration in the
figure identical to the upper illustration except that the object causes the rays passing
through it to be deviated by a small angle, e . This angular deviation in the object plane results in
a positional shift of the source image at the knife-edge. The intensity in the image plane
of the Schlieren arrangement is thus proportional to the angular deviation caused by the
object. |
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Ray
traces of Schlieren imaging arrangement using ZEMAX® optical analysis software.
Upper and lower traces indicate effect of deviation of the optical beam by the object. |
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Angular deviations can be caused by geometric
variations in the object. The first historical use of a Schlieren imaging system was the
Focault knife-edge test, which allowed observation of minute geometrical errors in the
surface
of optical components. A non-uniform refractive index also causes angular deviations of
magnitude proportional to the spatial derivative of the refractive index. The angular
sensitivity of a Schlieren system is measured in micro-radians, which translates to
refractive index deviations on the order of 10-6. The
figure above illustrates a transmission-mode Schlieren arrangement. A reflection-mode
arrangement, which is needed to examine an integrated circuit (IC), is obtained by folding
the transmission system about the object plane (Lens 1 and Lens 2 merge) and adding a beam
splitter. The sensitivity of a Schlieren system is proportional
to the source intensity while being inversely proportional to the source size. This
combination requires a high source irradiance that suggests the high intrinsic brightness of a laser source. In addition, the source wavelength
must be matched to the transmission characteristics of the test material. This requirement
suggests a wavelength of approximately 1060 nanometers for silicon.
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A high-resolution imaging detector is
also needed with a computer interface for image acquisition and processing. Digital CCD
cameras offer reasonable sensitivity at the required wavelength
with high-resolution imaging formats of 1024 ´ 1024 and 2048 ´ 2048 pixels coupled with 10- and 12- bit intensity resolution.
A useful Schlieren imaging system does not simply
consist of the optics. The figure below is a schematic overview of a complete system,
including digital frame grabbing, timing circuitry for controlling the state of the device
in coincidence with the image acquisition, staging and navigation, and overall computer
control of the data acquisition and processing.
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| Some example imagery is shown in the next column. The
top figure shows a standard reflection image of a metal film target. The dark areas are
the places where the metal film was removed. The
Schlieren filter in the image in the bottom figure in the next column was set to block all
the light from the flat portions of the target. This background is suppressed, leaving
only light that was deflected from the film edges.
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Reflected-light image of metal film
resolution target
Schlieren image of same target. Slopes at
metal edges that deviate the optical beam are enhanced while the flat background is
suppressed.
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| Scaling |
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A significant advantage of
the Schlieren approach to phase-contrast imaging is how it scales with the size of the
target.
For small changes in the refractive index, the angular deviation of the optical
path within the substrate is given by (1)
where z is the optical axis, x is a transverse direction, and n is
assumed to be the sum of the substrate refractive index, n0, and
a small perturbation, Dn.
The external angular deviation relates to the internal deviation via Snell’s law,
which for small angles near normal incidence, yields
 . (2)
As a target is scaled down, the x and z components will scale
proportionately, which leaves the angular deviation constant.
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As an example, a cylindrically symmetric target with a
radial distribution given by  , (3)
is assumed where Dn is the
refractive index change in the target, with n0 being the peak value, r
is the radial distance, and a is the effective radius of the filament.
Equation 3 yields
 (4)
with a peak deviation given by
 (5)
The peak deviation, which sets the sensitivity limit, is independent of the radius, a.
Other phase-contrast techniques, for example interferometry, have sensitivity that falls
off at least linearly with the scale size. The independence of Schlieren imaging with
scale size represents a significant advantage as semiconductor device size decreases.
OptoMetrix’s complete Schlieren Thermal Mapper (STM) produces both reflected-light
images (knife edge removed) and thermal images that are perfectly registered. The thermal
images are the result of processing two images, one with the heat source on, and one with
the heat source off, which removes the background image of the device.
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