Mirror surface autocovariance functions and their associated visible scattering

R. J. Noll; P. Glenn

doi:10.1364/AO.21.001824

Applied Optics
Vol. 21,
Issue 10,
pp. 1824-1838
(1982)
•https://doi.org/10.1364/AO.21.001824

Mirror surface autocovariance functions and their associated visible scattering

R. J. Noll and P. Glenn

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R. J. Noll and P. Glenn, "Mirror surface autocovariance functions and their associated visible scattering," Appl. Opt. 21, 1824-1838 (1982)

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Abstract

Visible scattering is measured and compared with predictions based upon surface roughness measurements. The results indicate two scales of roughness, a short scale of the order of 1 μm and a long scale ranging from 30 to 150 μm. A study of the relative amount of scattering from the two scales shows that the short scale accounts for all the scattering for scattering angles larger than 3°. Furthermore, the analysis demonstrates the importance of spatial frequency bandwidth limits in optical scattering and finish measurements for a particular class of highly polished mirror surfaces.

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Sample		Length (in.)^a	TIS	σ_TALY2(Å²)	σ_G² (Å²)^b
ULE	Ni	6.5	2.65 ±2.06 × 10⁻⁵	11.6	2.2
ULE	Pt	6.5	4.11 ±3.52 × 10⁻⁵	14.4	3.1
Zdur	Au	4.0	1.46 ±0.30 × 10⁻⁴	17.6	3.6
	Ni	6.5	5.68 ±5.40 × 10⁻⁵	13.0	7.3
	Pt	4.0	7.26 ±3.62 × 10⁻⁵	10.9	5.5
FS	Au	6.5	2.25 ±0.51 × 10⁻⁴	13.7	2.3
	Ni	6.5	6.83 ±5.95 × 10⁻⁵	16.0	5.1
	Pt	6.5	5.79 ±2.77 × 10⁻⁵	13.7	4.1
Be	Ni	6.5	7.16 ±3.88 × 10⁻⁵	134.6	14.5
	Pt	6.5	6.09 ± 1.87 × 10⁻⁵	67.2	13.7
	Ni	4.0	4.82 ±0.86 × 10⁻⁵	161.3	9.6
	Pt	4.0	5.74 ± 1.94 × 10⁻⁵	86.5	10.9

Glass Sample		Short Scale Model						Long Scale Model

		^K_o			$e^{- B x / \sqrt{B x}}$			C	1/D^a	Fit Error^b

		A	1/B^a	Fit Error^b	A	1/B^a	Fit Error^b
ULE	Ni	0.92	1.64	0.184				7.70	50	0.061
					0.63	2.17	0.279	7.70	50	0.061
ULE	Pt	1.43	0.88	0.379				9.18	34.5	0.083
					0.74	2.08	0.263	9.18	34.5	0.083
ZDUR	Au	1.78	0.95	0.141				12.10	147	0.008
					0.50	8.33	0.342	12.10	147	0.008
ZDUR	Ni	3.34	1.30	0.667				3.57	28.6	0.081
					1.56	3.45	0.858	3.57	28.6	0.081
ZDUR	Pt	2.69	1.05	0.345				2.29	50	0.023
					1.15	3.45	0.225	2.29	50	0.023
FS	Au	1.92	0.32	0.093				9.86	145	0.002
					1.41	0.48	0.078	9.86	145	0.002
FS	Ni	2.37	1.22	0.265				9.27	135	0.024
					0.85	5.56	0.475	9.27	135	0.024
FS	Pt	2.20	0.74	0.041				8.11	152	0.011
					1.37	1.32	0.086	8.11	152	0.011

Sample		Short Scale Model	g(0.06)	Measured Mean Square Roughness	Short Scale %	Long Scale %
ULE	Ni	K_o	10.77 Å²	11.56 Å²	28.5	71.5
		$e^{- x / \sqrt{x}}$	11.30		31.9	68.1

ULE	Pt	K_o	13.01	14.4	29.9	70.1
		$e^{- x / \sqrt{x}}$	13.28		30.9	69.1

ZDUR	Au	K_o	17.22	17.6	29.7	70.3
		$e^{- x / \sqrt{x}}$	17.86		32.3	67.7

ZDUR	Ni	K_o	14.15	12.96	74.8	25.2
		$e^{- x / \sqrt{x}}$	15.11		76.4	23.6

ZDUR	Pt	K_o	10.28	10.89	77.7	22.3
		$e^{- x / \sqrt{x}}$	10.82		78.8	21.2

FS	Au	K_o	13.27	13.69	25.7	74.3
		$e^{- x / \sqrt{x}}$	13.38		26.3	73.7

FS	Ni	K_o	16.66	16.0	44.4	55.6
		$e^{- x / \sqrt{x}}$	17.28		46.4	53.6

FS	Pt	K_o	13.90	13.69	41.7	58.3
		$e^{- x / \sqrt{x}}$	14.21		42.9	57.1

Sample			Short scale model						C	Long scale model

			K_o			exp(−Bx)/ Bx

			A	1/B^a	Fit Error	A	1/B^a	Fit Error		1/D^a	E	H	Fit Error
Be	Ni	6.5″	7.82	0.34	12.28				108	26.5	15.39	0.109	1.31
						6.22	0.46	12.24	108	26.5	15.39	0.109	1.31
			7.82	0.34	12.28				127	20.4	-	-	2.35
Be	Pt	6.5″	26.75	0.12	5.0				9.51	18.9	36.2	0.088	0.55
						25.55	0.13	5.2	9.51	18.9	36.2	0.088	0.55
			26.75	0.12	5.0				54.07	8.13	-	-	16.3
Be	Ni	4″	5.92	0.20	32.3				125	34.5	35.6	0.127	2.2
						5.39	0.23	32.4	125	34.5	35.6	0.127	2.2
			5.92	0.20	32.3				169	18.5	-	-	15.8
Be	Pt	4″	7.50	0.29	4.01				65.0	29.4	12.7	0.204	0.43
						6.47	0.35	4.22	65.0	29.4	12.7	0.204	0.43
			7.50	0.29	4.01				72.3	23.8	-	-	26.8

Sample		Model	g(0.06)	Measured mean–squared roughness	Short scale %	Long scale %	Cosine%
Ni	6.5″	g4	138.9 A²	134.6 A²	11.1	77.8	11.1
Pt	6.5″	g4	78.42	67.2	41.8	12.1	46.1
Ni	4.0″	g4	168.3	161.3	4.7	74.2	21.1
Pt	4.0″	g4	91.0	86.5	14.6	71.4	14.0

Probe beam diameter	9 mm
Polarization	Random
Probe beam uniformity	± 3%
$Threshold \frac{dP}{d Ω}$	8 × 10⁻⁷ sr⁻¹
Angular range	4° – 45°
Incident beam angle	3°
Minimum angle from specular	4°
Wavelength	0.6328 μm

Sample		$(\frac{{Scattering}_{MEAS}}{Scattering_{PRED}})$ ^a	TIS_MEAS^b	T1S_G^c	$(\frac{{TIS}_{MEAS}}{{TIS}_{G}})$	TIS_SCAT^d	$(\frac{{TIS}_{SCAT}}{{TIS}_{G}})$
ULE	Ni ^d	5.2	2.65 ±2.06 × 10⁻⁵	8.71 × 10⁻⁶	3.0	9.14 × 10⁻⁵	10.5
ULE	Pt ^d	8.0	4.11 ±3.52 × 10⁻⁵	1.20 × 10⁻⁵	3.4	2.20 × 10⁻⁴	18.3
Zdur	Au ^e	14.6	1.46 ±0.30 × 10⁻⁴	1.43 × 10⁻⁵	10.2	9.79 × 10−4	68.5
Zdur	Ni ^d	2.7	5.68 ±5.40 × 10⁻⁵	2.87 × 10⁻⁵	2.0	1.24 × 10⁴	4.3
Zdur	Pt ^e	7.4	7.26 ±3.62 × 10⁻⁵	2.18 × 10⁻⁵	3.3	2.70 × 10⁻⁴	12,4
FS	Au ^d	31.9	2.25 ± 0.51 × 10⁻⁴	1.61 × 10⁻⁵	14.0	2.61 × 10⁻⁴	16.2
FS	Ni ^d	2.7	6.83 ± 5.95 × 10⁻⁵	2.00 × 10⁻⁵	3.4	1.13 × 10⁻⁴	5.7
FS	Pt ^d	6.0	5.79 ± 2.77 × 10⁻⁵	9.26 × 10⁻⁶	6.3	1.31 × 10⁻⁴	14.1
Be	Ni 6.5 ^e	1.7	7.16 ± 3.88 × 10⁻⁵	5.75 × 10⁻⁵	1.2	1.61 × 10⁻⁴	2.8
Be	Pt 6.5 ^e	2.2	6.09 ± 1.87 × ⁻⁵	5.67 × 10⁻⁵	1.1	1.49 × 10⁻⁴	2.6
Be	Ni 4.0 ^f	1.8	4.82 ± 0.86 × 10⁻⁵	3.82 × 10⁻⁵	1.3	1.71 × 10⁻⁴	4,5
Be	Pt 4.0 ^f	1.8	5.74 ±1.94 × 10⁻⁵	4.34 × 10⁻⁵	1.3	1.49 × 10⁻⁴	3.4

Abstract

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Figures (13)

Tables (7)

Equations (57)

Applied Optics