Abstract

Hemi-ellipsoidal mirrors are used in reflection-based measurements due to their ability to collect light scattered from one focal point at the other. In this paper, a radiometric model of this energy transfer is derived for arbitrary mirror and detector geometries. This model is used to examine the imaging characteristics of the mirror away from focus for both diffuse and specular light. The radiometric model is applied to several detector geometries for measuring the Directional Hemispherical Reflectance for both diffuse and specular samples. The angular absorption characteristics of the detector are then applied to the measurement to address measurement accuracy for diffuse and specular samples. Examining different detector configurations shows the effectiveness of flat detectors at angles ranging from normal to 50°, and that multifaceted detectors can function from normal incidence to grazing angles.

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  1. W. M. Brandenberg, “Focusing properties of hemispherical and ellipsoidal mirror reflectometers,” J. Opt. Soc. Am.54, 1235–1237 (1964).
    [CrossRef]
  2. J. Workman and A. Springsteen, Applied Spectroscopy: A Compact Reference for Practitioners (Access Online via Elsevier, 1998).
  3. J. Lorincik and J. Fine, “Focusing properties of hemispherical mirrors for total integrating scatter instruments,” Appl. Optics36, 8270–8274 (1997).
    [CrossRef]
  4. R. P. Heinisch and E. M. Sparrow, “Efficiency characteristics of hemi-ellipsoidal and hemispherical collectors of thermal radiation,” Int. J. Heat Mass Transfer14, 1275–1284 (1970).
    [CrossRef]
  5. R. P. Heinisch, F. J. Bradac, and D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Optics9, 483–487 (1970).
    [CrossRef]
  6. E. Hecht, Optics (Addison-Wesley, 2002).
  7. R. W. Boyd, Radiometry and the Detection of Optical Radiation (John Wiley and Sons, 1983).
  8. K. A. Snail and L. M. Hanssen, “Magnification of conic mirror reflectometers,” Appl. optics37, 4143–4149 (1998).
    [CrossRef]
  9. K. A. Snail, “Reflectometer design using nonimaging optics,” Appl. Optics26, 5326–5332 (1987).
    [CrossRef]
  10. B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
    [CrossRef]
  11. M. R. Benson, M. A. Marciniak, and J. W. Burks, “Characterization and measurements collected from infrared grazing angle reflectometer,” in “Proc. SPIE,” vol. 8154 (2011), vol. 8154, p. 81541B.
    [CrossRef]
  12. M. R. Benson, M. A. Marciniak, and J. W. Burks, “Measuring grazing-angle dhr with the infrared grazing angle reflectometer,” in “Proc. SPIE,” (International Society for Optics and Photonics, 2012), pp. 84950R–84950R.
    [CrossRef]
  13. W. R. Blevin and W. J. Brown, “An infra-red reflectometer with a spheroidal mirror,” J. Sci. Instrum.42, 385–389 (1965).
    [CrossRef]

2011 (1)

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Characterization and measurements collected from infrared grazing angle reflectometer,” in “Proc. SPIE,” vol. 8154 (2011), vol. 8154, p. 81541B.
[CrossRef]

1998 (1)

K. A. Snail and L. M. Hanssen, “Magnification of conic mirror reflectometers,” Appl. optics37, 4143–4149 (1998).
[CrossRef]

1997 (1)

J. Lorincik and J. Fine, “Focusing properties of hemispherical mirrors for total integrating scatter instruments,” Appl. Optics36, 8270–8274 (1997).
[CrossRef]

1987 (1)

K. A. Snail, “Reflectometer design using nonimaging optics,” Appl. Optics26, 5326–5332 (1987).
[CrossRef]

1976 (1)

B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
[CrossRef]

1970 (2)

R. P. Heinisch and E. M. Sparrow, “Efficiency characteristics of hemi-ellipsoidal and hemispherical collectors of thermal radiation,” Int. J. Heat Mass Transfer14, 1275–1284 (1970).
[CrossRef]

R. P. Heinisch, F. J. Bradac, and D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Optics9, 483–487 (1970).
[CrossRef]

1965 (1)

W. R. Blevin and W. J. Brown, “An infra-red reflectometer with a spheroidal mirror,” J. Sci. Instrum.42, 385–389 (1965).
[CrossRef]

1964 (1)

Benson, M. R.

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Characterization and measurements collected from infrared grazing angle reflectometer,” in “Proc. SPIE,” vol. 8154 (2011), vol. 8154, p. 81541B.
[CrossRef]

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Measuring grazing-angle dhr with the infrared grazing angle reflectometer,” in “Proc. SPIE,” (International Society for Optics and Photonics, 2012), pp. 84950R–84950R.
[CrossRef]

Blevin, W. R.

W. R. Blevin and W. J. Brown, “An infra-red reflectometer with a spheroidal mirror,” J. Sci. Instrum.42, 385–389 (1965).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (John Wiley and Sons, 1983).

Bradac, F. J.

R. P. Heinisch, F. J. Bradac, and D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Optics9, 483–487 (1970).
[CrossRef]

Brandenberg, W. M.

Brown, W. J.

W. R. Blevin and W. J. Brown, “An infra-red reflectometer with a spheroidal mirror,” J. Sci. Instrum.42, 385–389 (1965).
[CrossRef]

Burks, J. W.

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Characterization and measurements collected from infrared grazing angle reflectometer,” in “Proc. SPIE,” vol. 8154 (2011), vol. 8154, p. 81541B.
[CrossRef]

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Measuring grazing-angle dhr with the infrared grazing angle reflectometer,” in “Proc. SPIE,” (International Society for Optics and Photonics, 2012), pp. 84950R–84950R.
[CrossRef]

Fine, J.

J. Lorincik and J. Fine, “Focusing properties of hemispherical mirrors for total integrating scatter instruments,” Appl. Optics36, 8270–8274 (1997).
[CrossRef]

Hanssen, L. M.

K. A. Snail and L. M. Hanssen, “Magnification of conic mirror reflectometers,” Appl. optics37, 4143–4149 (1998).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Heinisch, R. P.

R. P. Heinisch, F. J. Bradac, and D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Optics9, 483–487 (1970).
[CrossRef]

R. P. Heinisch and E. M. Sparrow, “Efficiency characteristics of hemi-ellipsoidal and hemispherical collectors of thermal radiation,” Int. J. Heat Mass Transfer14, 1275–1284 (1970).
[CrossRef]

Lorincik, J.

J. Lorincik and J. Fine, “Focusing properties of hemispherical mirrors for total integrating scatter instruments,” Appl. Optics36, 8270–8274 (1997).
[CrossRef]

Marciniak, M. A.

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Characterization and measurements collected from infrared grazing angle reflectometer,” in “Proc. SPIE,” vol. 8154 (2011), vol. 8154, p. 81541B.
[CrossRef]

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Measuring grazing-angle dhr with the infrared grazing angle reflectometer,” in “Proc. SPIE,” (International Society for Optics and Photonics, 2012), pp. 84950R–84950R.
[CrossRef]

Perlick, D. B.

R. P. Heinisch, F. J. Bradac, and D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Optics9, 483–487 (1970).
[CrossRef]

Pipes, J. G.

B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
[CrossRef]

Roux, J. A.

B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
[CrossRef]

Smith, A. M.

B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
[CrossRef]

Snail, K. A.

K. A. Snail and L. M. Hanssen, “Magnification of conic mirror reflectometers,” Appl. optics37, 4143–4149 (1998).
[CrossRef]

K. A. Snail, “Reflectometer design using nonimaging optics,” Appl. Optics26, 5326–5332 (1987).
[CrossRef]

Sparrow, E. M.

R. P. Heinisch and E. M. Sparrow, “Efficiency characteristics of hemi-ellipsoidal and hemispherical collectors of thermal radiation,” Int. J. Heat Mass Transfer14, 1275–1284 (1970).
[CrossRef]

Springsteen, A.

J. Workman and A. Springsteen, Applied Spectroscopy: A Compact Reference for Practitioners (Access Online via Elsevier, 1998).

Wood, B. E.

B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
[CrossRef]

Workman, J.

J. Workman and A. Springsteen, Applied Spectroscopy: A Compact Reference for Practitioners (Access Online via Elsevier, 1998).

Appl. Optics (4)

J. Lorincik and J. Fine, “Focusing properties of hemispherical mirrors for total integrating scatter instruments,” Appl. Optics36, 8270–8274 (1997).
[CrossRef]

K. A. Snail and L. M. Hanssen, “Magnification of conic mirror reflectometers,” Appl. optics37, 4143–4149 (1998).
[CrossRef]

K. A. Snail, “Reflectometer design using nonimaging optics,” Appl. Optics26, 5326–5332 (1987).
[CrossRef]

B. E. Wood, J. G. Pipes, A. M. Smith, and J. A. Roux, “Hemi-ellipsoidal mirror infrared reflectometer: development and operation,” Appl. Optics15, 940–950 (1976).
[CrossRef]

R. P. Heinisch, F. J. Bradac, and D. B. Perlick, “On the fabrication and evaluation of an integrating hemiellipsoid,” Appl. Optics9, 483–487 (1970).
[CrossRef]

Int. J. Heat Mass Transfer (1)

R. P. Heinisch and E. M. Sparrow, “Efficiency characteristics of hemi-ellipsoidal and hemispherical collectors of thermal radiation,” Int. J. Heat Mass Transfer14, 1275–1284 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

W. R. Blevin and W. J. Brown, “An infra-red reflectometer with a spheroidal mirror,” J. Sci. Instrum.42, 385–389 (1965).
[CrossRef]

Proc. SPIE (1)

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Characterization and measurements collected from infrared grazing angle reflectometer,” in “Proc. SPIE,” vol. 8154 (2011), vol. 8154, p. 81541B.
[CrossRef]

Other (4)

M. R. Benson, M. A. Marciniak, and J. W. Burks, “Measuring grazing-angle dhr with the infrared grazing angle reflectometer,” in “Proc. SPIE,” (International Society for Optics and Photonics, 2012), pp. 84950R–84950R.
[CrossRef]

J. Workman and A. Springsteen, Applied Spectroscopy: A Compact Reference for Practitioners (Access Online via Elsevier, 1998).

E. Hecht, Optics (Addison-Wesley, 2002).

R. W. Boyd, Radiometry and the Detection of Optical Radiation (John Wiley and Sons, 1983).

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Figures (21)

Fig. 1
Fig. 1

Diagram of a hemi-ellipsoidal mirror used for making reflectance measurements. The sample is placed at the upper focus, in object space, while the detector is located at the lower focus, in image space.

Fig. 2
Fig. 2

Given two vectors p1, which is the object point, and 1, which is the direction of the ray, there are two values of c1 which place the vector R on the surface of the ellipsoid. 2 is found by reflecting 1 across the normal at R. In this case, c2 is found by solving for where p2 has a zero component.

Fig. 3
Fig. 3

The radiance theorem shows that the radiance emitted in object space will be equal to that reflected by the mirror into image space. Because the mirror deforms the differential area, it is easier to perform calculations in object space. Imaging is used to calculate the new location of the differential area. If this differential area is within the geometry of the detector, this portion of the radiance emitted in object space is detected.

Fig. 4
Fig. 4

Schematic of five points with Lambertian radiance located near the object space focal point which are imaged into image space.

Fig. 5
Fig. 5

The irradiance of five point sources in object space (as shown in Fig. 4) are imaged into image space. The middle point (c) is the point located at the focus, and is best imaged. Note that both points located off focus on the principle axis, (b) and (d), are blurred only along that axis. The points located perpendicular to this axis, (a) and (e), are blurred significantly more, and are symmetrical about the principle axis.

Fig. 6
Fig. 6

Schematic of a rectangularly shaped object uniformly illuminated in object space using light that is incident along the vectors (a)–(e). These rays are reflected off the object towards the mirror, which reflects them into image space along the dashed vectors (a′)–(e′). The image point is defined as the point where these rays intersect the plane of the object in image space. Note that vectors (a),(b),(d) and (e) are oriented at θ = 45°, while (c) is oriented at normal.

Fig. 7
Fig. 7

A perfectly specular sample (shown on the left of each plot) is imaged using light incident parallel to the rays detailed in Fig. 6. Note that (a′) is the reflection of (a) off the sample, then the mirror.

Fig. 8
Fig. 8

Detectable irradiance map for a circular detector with radius 1.596 units (surface area = 8 units-squared) and a mirror with the dimensions given in Section 3. The white cross is the location of the object-space focal point.

Fig. 9
Fig. 9

Detector efficiencies for the detector of Fig. 8. This detector works very well at angles close to normal incidence, but is inefficient towards grazing angles. Note that performance is limited by the diffuse case for normal incidence, as the spot size is overfilling the detectable area.

Fig. 10
Fig. 10

Detectable irradiance map for a elliptical detector with major radius 3.19 units and minor radius 0.798 units (surface area = 8 units-squared). The white cross is the location of the object-space focal point.

Fig. 11
Fig. 11

Detector efficiencies for the detector of Fig. 10. This detector functions better at higher angles of incidence, but has poor performance closer to normal incidence.

Fig. 12
Fig. 12

Detectable irradiance map for a rectangular detector with dimensions 4 units long by 2 units wide (surface area = 8 units-squared). The white cross is the location of the object-space focal point.

Fig. 13
Fig. 13

Detector efficiencies for the detector of Fig. 12. This detector functions slightly better than the elliptical detector at all but the highest angles of incidence.

Fig. 14
Fig. 14

Maximum δ as a function of mirror eccentricity for the circular detector (in green), the rectangular detector (in blue) and the elliptical detector (in red).

Fig. 15
Fig. 15

Detectable irradiance maps for a circular detector with radius 1.618 units (surface area = 8 units-squared), taking absorption into account. The white cross is the location of the object-space focal point.

Fig. 16
Fig. 16

Detector efficiencies for the detector of Fig. 15. Note that the minimum error in the measurement is significantly higher than in the perfect absorber case (3.2% here, as opposed to the 1% error presented in Fig. 9). The limiting case cannot make measurements past 60°.

Fig. 17
Fig. 17

Detectable irradiance maps for a rectangular detector with 4 units long and 2 units wide, (Surface area = 8 units-squared) taking absorption into account. The white cross is the location of the object-space focal point.

Fig. 18
Fig. 18

Detector efficiencies for the detector of Fig. 17. This detector has a larger usable area in sample space than the circular detector, but still functions poorly.

Fig. 19
Fig. 19

Detectable irradiance maps for a parallel-piped detector with dimensions 1 unit tall by 1 unit wide by 2 units long (surface area = 8 units-squared), taking absorption into account. The white cross is the location of the object-space focal point.

Fig. 20
Fig. 20

Detector efficiencies for the detector of Fig. 19. This detector has much smaller errors than the flat detectors examined, and can be used to make accurate grazing-angle measurements.

Fig. 21
Fig. 21

Maximum δ as a function of mirror eccentricity for the case of a parallel-piped detector. Omitting the normal incidence case, the overall trends are similar, but not the same, as with the perfectly absorbing detector.

Tables (2)

Tables Icon

Table 1 Maximum spot size for the three detector configurations examined. All spot sizes give less than 1% error over the range of measurements for both specular and diffuse samples. The orange entries represent the largest spot size for the given measurement.

Tables Icon

Table 2 Maximum spot size for the three detector configurations examined. The green cells indicate the measurement has < 1% error. The yellow cells indicate < 3.5% error, and the red cells indicate that the error is too high to make a measurement.

Equations (9)

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[ ( p 1 + c 1 d ^ 1 ) x ^ ] 2 r major 2 + [ ( p 1 + c 1 d ^ 1 ) y ^ ] 2 r minor 2 + [ ( p 1 + c 1 d ^ 1 ) z ^ ] 2 r minor 2 = 1 .
p 2 = p 1 + c 1 d ^ 1 p 1 z ^ + c 1 ( d ^ 1 z ^ ) 2 ( d ^ 1 n ^ ) ( n ^ z ^ ) d ^ 1 z ^ [ 2 ( d ^ 1 n ^ ) n ^ d ^ 1 ]
Φ = A o Ω o L ( d ^ i ; p 1 , d ^ 1 ) ( d 1 z ^ ) d Ω o d A o ,
L ( d ^ i ; p 1 , d ^ 1 ) = E 0 ( p 1 ) f ( d ^ i ; d ^ 1 )
Φ d = A o Ω o E o ( p 1 ) f ( d ^ i ; d ^ 1 ) ( d ^ 1 z ^ ) D [ p 2 ( p 1 , d ^ 1 ) ] d Ω o d A o = A o E o ( p 1 ) E d ( d ^ i , p 1 ) d A o
E d ( d ^ i , p 1 ) Ω o f ( d ^ i ; d ^ 1 ) ( d ^ 1 z ^ ) D [ p 2 ( p 1 , d ^ 1 ) ] d Ω o .
A ( θ , ϕ ) = 1 R ( θ , ϕ )
Φ d = A o Ω o E 0 ( p 1 ) f ( d ^ i , d ^ 1 ) ( d ^ 1 z ^ ) D [ p 2 ( p 1 , d ^ 1 ) ] A [ d ^ 2 ( p 1 , d ^ 1 ) ] d Ω o d A o = A o E 0 ( p 1 ) E A ( p 1 , d ^ 1 ) d A o
E A ( p 1 , d ^ 1 ) Ω o f ( d ^ i , d ^ 1 ) ( d ^ 1 z ^ ) D [ p 2 ( p 1 , d ^ 1 ) ] A [ d ^ 2 ( p 1 , d ^ 1 ) ] d Ω o

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