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Asymptotics of diffraction corrections in radiometry
Eric Shirley Remote Sensing Group Sensor Science Division, NIST
Presentation Outline: Background 3-element systems (1 aperture) Solar radiometry example Multistage systems Recent innovations Summary
Some prior work diffraction effects in radiometry: Theory: Rayleigh (Phil. Mag. 11, 214, 1881): Rayleigh formula E. Lommel (Abh. Bayer. Akad. 15, 233, 1885): Fresnel diffraction, unfocussed E. Wolf (Proc. Roy. Soc. A 204, 533, 1951) J. Focke (Optica Acta 3, 161, 1956) Radiometry (mostly at NMIs): C.L. Sanders and O.C. Jones (J. Opt. Soc. Am. 52, 731, 1962) N. Ooba (J. Opt. Soc. Am. 54, 357, 1964) W.R. Blevin (Metrologia 6, 39, 1970): effective-wavelength approximation W.H. Steel, M. De, and J.A. Bell (J. Opt. Soc. Am. 62, 1099, 1972): extended source L.P. Boivin (Appl. Opt. 14, 197; 14, 2002; 15, 1204, 16, 377, 1972-1977): general
2 20 11 ( ) ( )J v J v
Consider this optical system:
r r
possible path for light to take
Physical optics: key approximations • Scalar-wave approximation • Kirchhoff diffraction theory, cont’d. • Fresnel-paraxial (Gaussian optics) approx.
Repeated application of approximations gives
P Q
1 1 1
1
1( ) ( , ) ( , ) exp[ ( )]
( )kN N N k kN
A AN
u dx dy dx dy G G i li
Q P x x Q x
Allows for focusing effects
Gustav Kirchhoff
Identification of 3-element subsystems: SAD subsystems: SAD:
source-aperture-detector
1 ( , )BL L v w
2 2
11 ( , ) ( , )B X
LL v w L v w
w w
4
1 ( , )6 (4)
B
AL F A w
4
2 2
11 ( , ) ( , )
6 (4)B X
L AF A w F A w
w w
flux on detector
flux on apertureL
2
geometrically
illuminated
fraction of
detector area
w
2geometrical
value of w
L
v, u, w (& A): depend on geometry, focusing power (& temperature)
Limiting geometries
Non-limiting geometries
Thermal case:
Spectral case:
Spectral case:
Thermal case:
Diffraction Effect in SAD systems (point source)
• Palindromic polynomials at all orders, as well as in LX(v,w) and FX(A,w).
• Integral representation of LB(v,w), LX(v,w), FB(A,w) and FX(A,w)
asymptotic expansions (large v, small A)
exact evaluation by appropriate quadrature (large v, small A)
generalizable to extended sources…
All a-functions involve trivial phase factors and smooth functions of , facilitating interpolation over
wavelengths where integral representations are more practical than exact numerical evaluation!
• In particular, consider small- form for spectral power:
Limiting case: Non-limiting case:
2 4 6 2 4
3 2
8
3 22 52 52
1 20 901 co 20
4( , ) ~
(1
(1 18 )sin(2 )
4 (1 )
2
(1..
s(2 )
(1 )) ).B
w w v
v w
v
v wv
w wL v
w
ww
w
w v
2 4 6 8 8 6 4 2 3 21 1 1
2 5 2
3
52 5 2
2 6log 2 11/ 31 20 90 20 5 112 586 112 5 32 (1 ) 1log log
2 14 (1 ) 6 (1 ) (
4 (3)
(1 )( ,
1 )) ~ e
eB e
w w w w w w w w w w wA A A A
ww w wA w A
wF
• Sample asymptotic results for a point source:
2 2 2 1/21
1 0
{(1 )[(2 ) ]}( , ) ( )
1
x xC dx d L u v L
x
3 4 2 2 2 2 2 2
04 / ( ) (depends on geometry)a s d s dC R R R d d
Extended sources: Generalizations of all previous results are also in hand.
General formula—
W.H. Steel et al., J. Opt. Soc. Am. 62, 1099 (1972) L.P. Boivin, Appl. Opt. 15, 1204 (1976) E.L. Shirley, Appl. Opt. 37, 6581 (1998), JOSA A 21, 1895 (2004); JOSA A 33, 1509 (2016) P. Edwards and M. McCall, Appl. Opt. 42, 5024 (2003)—treats more geometries
Electrical Substitution Radiometer
Principle of measuring total solar irradiance: (1.) Have an aperture of known area (2.) Measure total power passing through that aperture
Not to scale Heat sink
Weak thermal link
Corrected for
0.04% diffraction loss
Corrected for
0.16% diffraction gain
Corrected for
0.12% diffraction gain
0.02% decadal
variation in TSI
relevant to
understanding
climate change!
logarithmic grid points
Sequence of steps & operation count
O(N)
BEFORE FFT TRICK (also, using 10.0 GB memory):
AFTER FFT TRICK* (also, using 1.3 GB memory):
Application of algorithmic speedups to
calculation of diffraction-corrected throughput
of a multi-stage solar radiometer:
LASP SORCE/TIM instrument
* J.S. Rubin, et al., Appl. Opt. 57, 788 (2018)
Beyond the SAD paradigm:
Multistage optics trains
Vignetting effects
One-edge effects:
corrections proportional to or 1/T.
Two-edge effects (most important for source-pinhole-baffle-detector cases):
corrections proportional to 2 or 1/T2.
Example: blackbody calibration in NIST’s Low-background infrared facility:
Some 3-element subsystems can be treated for total or spectral power diffraction effects efficiently. We are moving to treating other effects more automatically.
Light that is diffracted twice…
Light that is diffracted twice…2nd bounce might be on an aperture normally not illuminated
Light that should be diffracted 1x might be reduced by diffraction at BB pinhole aperture
A baffle nominally not illuminated might diffract light after the pinhole diffracts it on the baffle edge…
In another context, having a blackbody recessed from the S of a SAD combination can also have a secondary effect
At high temperature, one can show*
2
1 2
0,
( )1 ( )
( )a a R
1 2
2
0
1 ( )T
b bR T
T T
neglected for ( )
computed for
m
m
R
3
3( ) ~ at large (usually)T
bR T T
T
Remainder plotted for m=4 mm, 10 mm, 15 mm; consistency indicates accuracy
At small wavelength, one can show*
*Based on an extended boundary-diffraction-wave formulation; E. L. Shirley, J. Mod. Opt. 54, 515 (2007)
(a1,a2)=analytic
(b1,b2)=analytic
Treatment of a multi-stage system, in practice:
Effect of toothing of aperture:
- reduction in diffraction effect because of phase cancellations in uBDW(rd).
diff. effect for same for toothed versions of same apertures
calculated
meas.
Comparison of measured and calculated diffraction effects:
Statistical analysis suggests ~2 % systematic error in calculated diffraction corrections.
L.P. Boivin, (Appl. Opt.
17, 3323, 1978)
Summary of results: • Diffraction affects radiometry
• Largely understood
• Much work has been done
• We have programs that are efficient and pull out
asymptotic trends, easing or obviating calculations
• Codes are ever more efficient, and are available