Abstract

We show how to determine the transmittance of short focal length lenses (f19mm and f25mm, in this case) with a combined uncertainty of 3 parts in 104 or better by measuring transmittances of lens pairs of a set of three or more lenses with the same nominal focal length. Uncertainties are minimized by optimizing the radiometric design of the setup and the measurement procedure. The technique is particularly useful in systems where the detector acceptance angle limits the beam geometry to relatively collimated beams.

© 2007 Optical Society of America

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References

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  1. V. E. Anderson, N. P. Fox, and D. H. Nettleton, "Highly stable, monochromatic and tunable optical radiation source and its application to high accuracy spectrophotometry," Appl. Opt. 31, 536-545 (1992).
    [CrossRef] [PubMed]
  2. E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
    [CrossRef]
  3. B. Munro, "Quantum information processing with light and its requirement for detectors," presented at the NIST Single Photon Detection workshop, Gaithersburg, Maryland, USA, 31 March-1 April 2003.
  4. S. V. Polyakov and A. L. Migdall, "High accuracy verification of a correlated-photon-based method for determining photon-counting detection efficiency," Opt. Express 15, 1390-1407 (2007).
    [CrossRef] [PubMed]
  5. N. P. Fox, "Trap detectors and their properties," Metrologia 28, 197-202 (1991).
    [CrossRef]
  6. S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations (Pitman Publishing Limited , 1979).
  7. A. B. Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley , 1974).
  8. J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.
  9. M. Ware and A. Migdall, "Single photon detector characterization using correlated photons: the march from feasibility to metrology," J. Mod. Opt. 51, 1549-1557 (2004).

2007 (1)

2005 (1)

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

2004 (1)

M. Ware and A. Migdall, "Single photon detector characterization using correlated photons: the march from feasibility to metrology," J. Mod. Opt. 51, 1549-1557 (2004).

2003 (1)

B. Munro, "Quantum information processing with light and its requirement for detectors," presented at the NIST Single Photon Detection workshop, Gaithersburg, Maryland, USA, 31 March-1 April 2003.

2000 (1)

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

1992 (1)

V. E. Anderson, N. P. Fox, and D. H. Nettleton, "Highly stable, monochromatic and tunable optical radiation source and its application to high accuracy spectrophotometry," Appl. Opt. 31, 536-545 (1992).
[CrossRef] [PubMed]

1991 (1)

N. P. Fox, "Trap detectors and their properties," Metrologia 28, 197-202 (1991).
[CrossRef]

1979 (1)

S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations (Pitman Publishing Limited , 1979).

1974 (1)

A. B. Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley , 1974).

Anderson, V. E.

V. E. Anderson, N. P. Fox, and D. H. Nettleton, "Highly stable, monochromatic and tunable optical radiation source and its application to high accuracy spectrophotometry," Appl. Opt. 31, 536-545 (1992).
[CrossRef] [PubMed]

Campbell, S. L.

S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations (Pitman Publishing Limited , 1979).

Cheung, J. Y.

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

Chunnilall, C. J.

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

Fox, N. P.

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

V. E. Anderson, N. P. Fox, and D. H. Nettleton, "Highly stable, monochromatic and tunable optical radiation source and its application to high accuracy spectrophotometry," Appl. Opt. 31, 536-545 (1992).
[CrossRef] [PubMed]

N. P. Fox, "Trap detectors and their properties," Metrologia 28, 197-202 (1991).
[CrossRef]

Greville, T. N. E.

A. B. Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley , 1974).

Harrison, N. J.

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

Israel, A. B.

A. B. Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley , 1974).

Meyer, C. D.

S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations (Pitman Publishing Limited , 1979).

Migdall, A.

M. Ware and A. Migdall, "Single photon detector characterization using correlated photons: the march from feasibility to metrology," J. Mod. Opt. 51, 1549-1557 (2004).

Migdall, A. L.

Mountford, J. R. M.

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

Munro, B.

B. Munro, "Quantum information processing with light and its requirement for detectors," presented at the NIST Single Photon Detection workshop, Gaithersburg, Maryland, USA, 31 March-1 April 2003.

Nettleton, D. H.

V. E. Anderson, N. P. Fox, and D. H. Nettleton, "Highly stable, monochromatic and tunable optical radiation source and its application to high accuracy spectrophotometry," Appl. Opt. 31, 536-545 (1992).
[CrossRef] [PubMed]

Pollard, D. F.

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

Polyakov, S. V.

Theocharous, E.

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

Thomas, P. J.

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

Ware, M.

M. Ware and A. Migdall, "Single photon detector characterization using correlated photons: the march from feasibility to metrology," J. Mod. Opt. 51, 1549-1557 (2004).

Woolliams, E. R.

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

Appl. Opt. (1)

V. E. Anderson, N. P. Fox, and D. H. Nettleton, "Highly stable, monochromatic and tunable optical radiation source and its application to high accuracy spectrophotometry," Appl. Opt. 31, 536-545 (1992).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

M. Ware and A. Migdall, "Single photon detector characterization using correlated photons: the march from feasibility to metrology," J. Mod. Opt. 51, 1549-1557 (2004).

Metrologia (2)

E. R. Woolliams, D. F. Pollard, N. J. Harrison, E. Theocharous, and N. P. Fox, "New facility for the high accuracy measurement of lens transmission," Metrologia 37, 603-605 (2000).
[CrossRef]

N. P. Fox, "Trap detectors and their properties," Metrologia 28, 197-202 (1991).
[CrossRef]

Opt. Express (1)

Pitman Publishing Limited (1)

S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations (Pitman Publishing Limited , 1979).

Wiley (1)

A. B. Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Wiley , 1974).

Other (2)

J. Y. Cheung, P. J. Thomas, C. J. Chunnilall, J. R. M. Mountford, and N. P. Fox, "Measurement of quantum efficiency using the correlated photon technique," extended abstract, presented at the NEWRAD 2005, 9th International Conference on Developments and Applications in Optical Radiometry, WRC/PMOD, Davos, Switzerland, 17-19 October 2005.

B. Munro, "Quantum information processing with light and its requirement for detectors," presented at the NIST Single Photon Detection workshop, Gaithersburg, Maryland, USA, 31 March-1 April 2003.

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Figures (7)

Fig. 1
Fig. 1

Lens transmittance measurement technique developed at National Physical Laboratory (NPL) [2].

Fig. 2
Fig. 2

Schematics of dual lens transmittance measurement.

Fig. 3
Fig. 3

Setup of dual lens transmittance measurement.

Fig. 4
Fig. 4

Ratio of the signal and monitor trap detector outputs (with readings taken simultaneously).

Fig. 5
Fig. 5

Typical transmittance data for one run with a lens pair alternately in and out of the beam. Six such graphs are produced for the 6 lens pair combinations constituting one set of data.

Fig. 6
Fig. 6

(Color online) Transmittances (points) of four lenses with f = 19 mm (top) and four lenses with f = 25 mm (bottom) obtained using the overdetermined method (left) and the just-determined method (right). The averages (lines) over 5 sets are shown for each lens with uncertainties determined from the standard deviations of the 5 data sets for each lens type.

Fig. 7
Fig. 7

Comparison of the transmittance results for the 19 mm and 25 mm lenses obtained using the just- and overdetermined methods.

Tables (1)

Tables Icon

Table 1 Average Uncertainty Budget ( k = 1 ) Showing the Effect of Each Uncertainty Component on the Lens Transmittance

Equations (23)

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m i j = t i t j ( 1 i < N , i < j N ) ,
m 12 = t 1 t 2 , m 13 = t 1 t 3 , m 23 = t 2 t 3 .
t 1 = m 12 m 13 / m 23
t 2 = m 12 m 23 / m 13
t 3 = m 13 m 23 / m 12 .
σ t i t i = 1 2 ( σ m 12 m 12 ) 2 + ( σ m 13 m 13 ) 2 + ( σ m 23 m 23 ) 2 .
m = A t ,
m = [ log ( m 12 ) log ( m 13 ) log ( m 1 N ) log ( m 23 ) log ( m ( N 1 ) N ) ] ,   A = [ 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1 ] ,
t = [ log ( t 1 ) log ( t N ) ] .
t 0 = A + m ,
t n = ( i = n   or   j = n m i j ( i n   and   j n m i j ) 1 / ( N 2 ) ) 1 / ( N 1 ) ,
σ t n t n = 1 N 1 i = n   or   j = n ( σ m i j m ¯ i j ) 2 + i n   and   j n ( 1 ( N 2 ) σ m i j m ¯ i j ) 2
σ t n t n σ t t A t 0 m 2 2 ( N 1 ) m 2 ,
σ t t = σ m m 2 N 3 2 ( N 1 ) ( N 2 ) .
Δ = σ m i j σ m 1 j = σ m i j σ m ,
| σ t t | overdetermined = σ m m 2 ( N 2 ) + Δ 2 2 ( N 1 ) ( N 2 ) .
τ t o t = ( N 1 ) ( 1 + N 2 2 Δ 2 ) τ m e a s ,
| σ t t | just-determined = σ m m 9 τ m e a s 4 τ t o t .
Θ ( Δ , N ) = ( σ t t | overdetermined ) 2 / ( σ t t | 3   lens ) 2 .
Δ optimum = 5 ( N 2 ) 2 > 1.
A A + = P R ( A ) , A + A = P R ( A + ) ,
A t - m = A t - A A + m + A A + m - m = ( A t A A + m ) + [ - ( I P R ( A ) ) m ] ,
A t - m 2 = A t - A A + m 2 + A A + m m 2 .

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