Abstract

This paper revises the broadband-filter method originally proposed by Wyszecki to evaluate the relative spectral responsivity of photodetectors. The new mathematical procedure consists of representing the responsivity curve by a spline function to reduce the number of free parameters and to preserve the approximation accuracy. Consequently, performance of the calculated spectral responsivity vs experimental errors turns out to be satisfactory. Extended simulated examples show the capabilities of this new approach.

© 1983 Optical Society of America

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References

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  1. “The Measurement of Spectral Responsivity Functions,” CIE Technical Report, First Draft, Sept.1982.
  2. G. Wyszecki, J. Opt. Soc. Am. 50, 992 (1960).
    [CrossRef]
  3. N. Mori, J. Opt. Soc. Am. 51, 1015 (1961).
    [CrossRef]
  4. H. E. Fleming, D. Q. Wark, Appl. Opt. 4, 337 (1965).
    [CrossRef]
  5. D. Hahn, J. Weidemann, PTB Mitt. 4, 323 (1965).
  6. A. C. R. Newbery, in Handbook of Applied Mathematics, C. E. Pearson, Ed. (Van Nostrand Reinhold, New York, 1974), Chap. 18.
  7. J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).
  8. V. Falletti, A. Premoli, M. L. Rastello, Appl. Opt. 21, 4345 (1982).
    [CrossRef] [PubMed]

1982 (2)

“The Measurement of Spectral Responsivity Functions,” CIE Technical Report, First Draft, Sept.1982.

V. Falletti, A. Premoli, M. L. Rastello, Appl. Opt. 21, 4345 (1982).
[CrossRef] [PubMed]

1965 (2)

D. Hahn, J. Weidemann, PTB Mitt. 4, 323 (1965).

H. E. Fleming, D. Q. Wark, Appl. Opt. 4, 337 (1965).
[CrossRef]

1961 (1)

1960 (1)

Ahlberg, J. H.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Falletti, V.

Fleming, H. E.

Hahn, D.

D. Hahn, J. Weidemann, PTB Mitt. 4, 323 (1965).

Mori, N.

Newbery, A. C. R.

A. C. R. Newbery, in Handbook of Applied Mathematics, C. E. Pearson, Ed. (Van Nostrand Reinhold, New York, 1974), Chap. 18.

Nilson, E. N.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Premoli, A.

Rastello, M. L.

Walsh, J. L.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Wark, D. Q.

Weidemann, J.

D. Hahn, J. Weidemann, PTB Mitt. 4, 323 (1965).

Wyszecki, G.

Appl. Opt. (2)

CIE Technical Report, First Draft (1)

“The Measurement of Spectral Responsivity Functions,” CIE Technical Report, First Draft, Sept.1982.

J. Opt. Soc. Am. (2)

PTB Mitt. (1)

D. Hahn, J. Weidemann, PTB Mitt. 4, 323 (1965).

Other (2)

A. C. R. Newbery, in Handbook of Applied Mathematics, C. E. Pearson, Ed. (Van Nostrand Reinhold, New York, 1974), Chap. 18.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

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Figures (7)

Fig. 1
Fig. 1

Complete set of cardinal splines γ1(λ),γ2(λ),…, γ6(λ) defined over a mesh of six joints.

Fig. 2
Fig. 2

Spectral transmittance τi(λ) of an ideal bell-shaped filter.

Fig. 3
Fig. 3

Set of ideal bell-shaped filters.

Fig. 4
Fig. 4

Spectral responsivity of an S20 photocathode.

Fig. 5
Fig. 5

Diagram of σe(λ), −σe(λ), and { S ( λ ) S 0 ( λ ) } for n = 17, m = 8, and β = 50 nm for an S20 photocathode.

Fig. 6
Fig. 6

Diagram of σe(λ), −σe(λ), and { S ( λ ) S 0 ( λ ) } for a V(λ)-corrected photodetector.

Fig. 7
Fig. 7

Diagram of S 0 ( λ ) , S ( λ ) , S ( λ ) ± σ e ( λ ) for a thermopile model FT 16.

Tables (3)

Tables Icon

Table I Results for an S20 Photocathode with n = 17 and m Varying from 5 to 13 and β = 30, 50, and 100 nm

Tables Icon

Table II Results for an S20 Photocathode with m = 8, β = 50 nm, and Different Values of n

Tables Icon

Table III Results for an S20 Photocathode with β = 50 nm and Some Different Values of n = m

Equations (51)

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r 0 i = k λ a λ b P λ ( λ ) τ i ( λ ) S 0 ( λ ) d λ + e i i = 1 , 2 , , n ,
γ j ( λ S k ) = δ k j k = 1 , 2 , , m ,
S ( λ ) = s T Γ ( λ ) = j = 1 m S j γ j ( λ ) .
λ S 1 λ S m [ d 2 f ( λ ) / d λ 2 ] 2 d λ
r i = j = 1 m S j [ λ a λ b P λ ( λ ) τ i ( λ ) γ j ( λ ) d λ ] i = 1 , 2 , , n .
r = A s ,
a i j = λ a λ b P λ ( λ ) τ i ( λ ) γ j ( λ ) d λ ,
ρ 2 = i = 1 n [ ( r i r 0 i ) / ( σ i n ) ] 2 ,
ρ 2 = ( r 0 A s ) T W ( r 0 A s ) ,
s = B r 0 ,
B = ( A T W A ) 1 A T W .
S ( λ ) = r 0 T B T Γ ( λ ) ,
ρ min 2 = r 0 T W ( I n n AB ) r 0 ,
S 0 ( λ ) = s 0 T Φ ( λ ) ,
r 0 = D s 0 + e ,
d i k = λ a λ b P λ ( λ ) τ i ( λ ) φ k ( λ ) d λ .
Δ S ( λ ) = S ( λ ) S 0 ( λ ) = r 0 T B T Γ ( λ ) s 0 T Φ ( λ ) = s 0 T [ D T B T Γ ( λ ) Φ ( λ ) ] + e T B T Γ ( λ ) .
{ Δ S ( λ ) } = s 0 T [ D T B T Γ ( λ ) Φ ( λ ) ] .
[ σ e ( λ ) ] 2 = { Δ S ( λ ) [ Δ S ( λ ) ] } 2 = { e T B T Γ ( λ ) Γ T ( λ ) B e } .
[ σ e ( λ ) ] 2 = i = 1 n { σ i 2 [ j = 1 m b j i γ j ( λ ) ] 2 } ,
ζ e 2 = λ υ λ z [ σ e ( λ ) ] 2 d λ / ( λ z λ υ ) ,
ζ 0 2 = λ υ λ z { Δ S ( λ ) } 2 d λ / ( λ z λ υ ) .
ζ e 2 = λ υ λ z [ 100 σ e ( λ ) / S ( λ ) ] 2 d λ / ( λ z λ υ ) ,
ζ 0 2 = λ υ λ z [ 100 { Δ S ( λ ) } / S ( λ ) ] 2 d λ / ( λ z λ υ ) .
ϑ i j = λ a λ b γ i ( λ ) γ j ( λ ) d λ ,
τ i ( λ ) = { cos 2 [ π ( λ λ c i ) / ( 2 β ) ] for | λ c i λ | β , 0 otherwise .
λ c i = λ c , i 1 + β / α i = 2 , 3 , , n .
α = β ( n 1 ) / ( λ c n λ c 1 )
λ a = λ c 1 β ,
λ b = λ c n + β .
λ c 1 λ υ ,
λ c n λ z .
C y = G y .
c j j = 2 ( h j 1 + h j ) j = 1 , 2 , , m ,
c j , j + 1 = c j + 1 , j = h j j = 1 , 2 , , m 1 ,
g j j = 3 ( h j 1 2 h j 2 ) j = 1 , 2 , , m ,
g j , j + 1 = g j + 1 , j = 3 h j 2 j = 1 , 2 , , m 1 ,
h 0 = h m = 0 ,
h j = 1 / ( x j + 1 x j ) j = 1 , 2 , , m 1
C = Z T Z ,
z j j = 2 ( h j + h j 1 ) h j 1 2 / z j 1 , j 1 2 j = 2 , 3 , , m ,
z j , j + 1 = h j / z j j j = 1 , 2 , , m 1 ,
P 1 ( x ) = 2 x 3 3 x 2 + 1 ,
P 2 ( x ) = x 3 2 x 2 + x ,
P 3 ( x ) = 2 x 3 + 3 x 2 ,
P 4 ( x ) = x 3 x 2
P 1 ( 0 ) = 1 , P 1 ( 0 ) = P 1 ( 1 ) = P 1 ( 1 ) = 0 ,
P 2 ( 0 ) = 1 , P 2 ( 0 ) = P 2 ( 1 ) = P 2 ( 1 ) = 0 ,
P 3 ( 1 ) = 1 , P 3 ( 0 ) = P 3 ( 0 ) = P 3 ( 1 ) = 0 ,
P 4 ( 1 ) = 1 , P 4 ( 0 ) = P 4 ( 0 ) = P 4 ( 1 ) = 0 .
P ( x ) = y j P 1 [ h j ( x x j ) ] + y j P 2 [ h j ( x x j ) ] / h j + y j + 1 P 3 [ h j ( x x j ) ] + y j + 1 P 4 [ h j ( x x j ) ] / h j ,

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