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