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EDIT: Hack: Fix math markup.
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stefanv committed Apr 21, 2011
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Expand Up @@ -93,7 +93,7 @@ NASA required a test that demonstrated the stability of this
engineering article and which verified that the design and
construction techniques will work to meet the requirements of the
telescope: namely, that it will remain stable to within 68nm from
30-50K (-405 to -370 :math:'\deg'F). They also wished to track the
30-50K (-405 to -370 :math:`^{\circ}\mathrm{F}`). They also wished to track the
structural changes from ambient (~295K) to cryogenic temperatures
(~30K) in order to better understand the accuracy of their structural
models. The primary test equipment, a special interferometer,
Expand Down Expand Up @@ -123,7 +123,7 @@ then recombines the reflected signal with a reference beam split from the same
laser pulse, to create an interferometric map of the surface speckles.
As the structure moves, the speckles change phase reflecting the
change in interference between the incident and reference laser pulses.
However, those phases cycle from :math:'-\pi' to :math:'\pi' and back
However, those phases cycle from :math:`-\pi` to :math:`\pi` and back
as the surface continues to move. This required the use of phase
unwrapping across the surface, spatial phase unwrapping, using an
algorithm developed by the manufacturers of the ESPI.
Expand All @@ -134,20 +134,20 @@ algorithm developed by the manufacturers of the ESPI.
Schematic of ESPI showing how the ESPI measures the change in the object due to thermal or mechanical stress by tracking the speckles' phase change on the surface.

As the surface tilted during the test, it produced fringes where
the phases across the structure would transition from :math:'\pi'
to :math:'-\pi'. This tilt needed to be removed in order to allow
the phases across the structure would transition from :math:`\pi`
to :math:`-\pi`. This tilt needed to be removed in order to allow
us to measure the relative changes from one region of the structure
to another.

.. figure:: ESPI_tilt_schematic.jpg

Schematic showing how bulk tilt introduces multiple :math:'2\pi'
Schematic showing how bulk tilt introduces multiple :math:`2\pi`
variations across the structure and how it gets corrected in processing,
allowing for relative variations to be measured across the surface as
described in "Testing the BSTA".


Since the measured phase is ambiguous by multiples of :math:'2\pi',
Since the measured phase is ambiguous by multiples of :math:`2\pi`,
special techniques are required to remove these ambiguities. One is
to presume that continuous surfaces have continuous phase, and any
discontinuities on continuous surfaces are due to phase wrapping. Thus
Expand All @@ -163,7 +163,7 @@ on spatially contiguous sections. Gross tilts are fit to the largest
contiguous sections, and then the average tilt is removed (as well
as the average displacement). However, there are areas of interest
(the mirror pad supports) which are discontiguous and as a result
possibly several factors of 2pi offset in reality as a result of the
possibly several factors of :math:`2\pi` offset in reality as a result of the
tilt, and thus improperly corrected when tilts are removed. Since
these areas are assumed to change slowly in time, temporal phase
unwrapping is applied to these areas.
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