Abstract

Continuing the work of the first paper in this series [Appl. Opt. 50, 4998–5011 (2011)APOPAI0003-6935], we extend our design methods to compound prisms composed of three independent elements. The increased degrees of freedom of these asymmetric prisms allow designers to achieve greatly improved dispersion linearity. They also, however, require a more careful tailoring of the merit function to achieve design targets, and so we present several new operands for manipulating the compound prisms’ design algorithm. We show that with asymmetric triplet prisms, one can linearize the angular dispersion such that the spectral sampling rate varies by no more than 4% across the entire visible spectral range. Doing this, however, requires large prisms and causes beam compression. By adding a beam compression penalty to the merit function, we show that one can compromise between dispersion linearity and beam compression in order to produce practical systems. For prisms that do not deviate the beam, we show that Janssen prisms provide a form that maintains the degrees of freedom of the triplet and that are capable of up to 32° of dispersion across the visible spectral range. Finally, in order to showcase some of the design flexibility of three-element prisms, we also show how to design for higher-order spectral dispersion to create a two-dimensional spectrum.

© 2011 Optical Society of America

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References

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  1. N. Hagen and T. S. Tkaczyk, “Compound prism design principles, I,” Appl. Opt. 50, 4998–5011 (2011).
    [CrossRef]
  2. P. J. C. Janssen, “Note sur trois spectroscopes [Note on three spectroscopes],” C.R. Hebd. Seances Acad. Sci. 55, 576–578 (1862), in French.
  3. N. Hagen and T. S. Tkaczyk, “Compound prism design principles, III: linear-in-wavenumber and OCT prisms,” Appl. Opt. 50, 5023–5030 (2011).
    [CrossRef]
  4. E. L. Dereniak and T. D. Dereniak, Geometric and Trigonometric Optics (Cambridge University, 2008), pp. 347–350.
    [CrossRef]
  5. J. R. M. Barr, “Achromatic beam expanders,” Opt. Commun. 51, 41–46 (1984).
    [CrossRef]
  6. ZEMAX Development Corp.www.zemax.com.
  7. J. M. Mooney, W. S. Ewing, and R. J. Nelson, “Multi-band direct vision prism,” U.S. patent 6,935,757 (30 August 2005).
  8. R. J. Nelson, J. M. Mooney, and W. S. Ewing, “Pseudo imaging,” Proc. SPIE 6233, 62330M (2006).
    [CrossRef]
  9. S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
    [CrossRef]
  10. Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
    [CrossRef]
  11. S. E. Holland, W. F. Kolbe, and C. J. Bebek, “Device design for a 12.3-megapixel, fully depleted, back-illuminated, high-voltage compatible charge-coupled device,” IEEE Trans. Electron Devices 56, 2612–2622 (2009).
    [CrossRef]
  12. http://www.owlnet.rice.edu/~tt3/.

2011 (2)

2009 (1)

S. E. Holland, W. F. Kolbe, and C. J. Bebek, “Device design for a 12.3-megapixel, fully depleted, back-illuminated, high-voltage compatible charge-coupled device,” IEEE Trans. Electron Devices 56, 2612–2622 (2009).
[CrossRef]

2008 (1)

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

2006 (1)

R. J. Nelson, J. M. Mooney, and W. S. Ewing, “Pseudo imaging,” Proc. SPIE 6233, 62330M (2006).
[CrossRef]

2003 (1)

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

1984 (1)

J. R. M. Barr, “Achromatic beam expanders,” Opt. Commun. 51, 41–46 (1984).
[CrossRef]

1862 (1)

P. J. C. Janssen, “Note sur trois spectroscopes [Note on three spectroscopes],” C.R. Hebd. Seances Acad. Sci. 55, 576–578 (1862), in French.

Bai, Y.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Bajaj, J.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Barr, J. R. M.

J. R. M. Barr, “Achromatic beam expanders,” Opt. Commun. 51, 41–46 (1984).
[CrossRef]

Bebek, C. J.

S. E. Holland, W. F. Kolbe, and C. J. Bebek, “Device design for a 12.3-megapixel, fully depleted, back-illuminated, high-voltage compatible charge-coupled device,” IEEE Trans. Electron Devices 56, 2612–2622 (2009).
[CrossRef]

Beletic, J. W.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Dereniak, E. L.

E. L. Dereniak and T. D. Dereniak, Geometric and Trigonometric Optics (Cambridge University, 2008), pp. 347–350.
[CrossRef]

Dereniak, T. D.

E. L. Dereniak and T. D. Dereniak, Geometric and Trigonometric Optics (Cambridge University, 2008), pp. 347–350.
[CrossRef]

Ewing, W. S.

R. J. Nelson, J. M. Mooney, and W. S. Ewing, “Pseudo imaging,” Proc. SPIE 6233, 62330M (2006).
[CrossRef]

J. M. Mooney, W. S. Ewing, and R. J. Nelson, “Multi-band direct vision prism,” U.S. patent 6,935,757 (30 August 2005).

Farris, M. C.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Groom, D. E.

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

Hagen, N.

Holland, S. E.

S. E. Holland, W. F. Kolbe, and C. J. Bebek, “Device design for a 12.3-megapixel, fully depleted, back-illuminated, high-voltage compatible charge-coupled device,” IEEE Trans. Electron Devices 56, 2612–2622 (2009).
[CrossRef]

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

Janssen, P. J. C.

P. J. C. Janssen, “Note sur trois spectroscopes [Note on three spectroscopes],” C.R. Hebd. Seances Acad. Sci. 55, 576–578 (1862), in French.

Joshi, A.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Kolbe, W. F.

S. E. Holland, W. F. Kolbe, and C. J. Bebek, “Device design for a 12.3-megapixel, fully depleted, back-illuminated, high-voltage compatible charge-coupled device,” IEEE Trans. Electron Devices 56, 2612–2622 (2009).
[CrossRef]

Lauxtermann, S.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Mooney, J. M.

R. J. Nelson, J. M. Mooney, and W. S. Ewing, “Pseudo imaging,” Proc. SPIE 6233, 62330M (2006).
[CrossRef]

J. M. Mooney, W. S. Ewing, and R. J. Nelson, “Multi-band direct vision prism,” U.S. patent 6,935,757 (30 August 2005).

Nelson, R. J.

R. J. Nelson, J. M. Mooney, and W. S. Ewing, “Pseudo imaging,” Proc. SPIE 6233, 62330M (2006).
[CrossRef]

J. M. Mooney, W. S. Ewing, and R. J. Nelson, “Multi-band direct vision prism,” U.S. patent 6,935,757 (30 August 2005).

Palaio, N. P.

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

Petersen, A.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Stover, R. J.

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

Tkaczyk, T. S.

Wei, M.

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

Williams, G.

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Appl. Opt. (2)

C.R. Hebd. Seances Acad. Sci. (1)

P. J. C. Janssen, “Note sur trois spectroscopes [Note on three spectroscopes],” C.R. Hebd. Seances Acad. Sci. 55, 576–578 (1862), in French.

IEEE Trans. Electron Devices (2)

S. E. Holland, D. E. Groom, N. P. Palaio, R. J. Stover, and M. Wei, “Fully depleted, back-illuminated charge-coupled devices fabricated on high-resistivity silicon,” IEEE Trans. Electron Devices 50, 225–238 (2003).
[CrossRef]

S. E. Holland, W. F. Kolbe, and C. J. Bebek, “Device design for a 12.3-megapixel, fully depleted, back-illuminated, high-voltage compatible charge-coupled device,” IEEE Trans. Electron Devices 56, 2612–2622 (2009).
[CrossRef]

Opt. Commun. (1)

J. R. M. Barr, “Achromatic beam expanders,” Opt. Commun. 51, 41–46 (1984).
[CrossRef]

Proc. SPIE (2)

R. J. Nelson, J. M. Mooney, and W. S. Ewing, “Pseudo imaging,” Proc. SPIE 6233, 62330M (2006).
[CrossRef]

Y. Bai, J. Bajaj, J. W. Beletic, M. C. Farris, A. Joshi, S. Lauxtermann, A. Petersen, and G. Williams, “Teledyne imaging sensors: silicon CMOS imaging technologies for x-ray, UV, visible and near infrared,” Proc. SPIE 7021, 702102(2008).
[CrossRef]

Other (4)

http://www.owlnet.rice.edu/~tt3/.

E. L. Dereniak and T. D. Dereniak, Geometric and Trigonometric Optics (Cambridge University, 2008), pp. 347–350.
[CrossRef]

ZEMAX Development Corp.www.zemax.com.

J. M. Mooney, W. S. Ewing, and R. J. Nelson, “Multi-band direct vision prism,” U.S. patent 6,935,757 (30 August 2005).

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Figures (8)

Fig. 1
Fig. 1

Ray trace through an example triplet compound prism where the second element is assumed to be oriented symmetrically with respect to the normal to the optical axis. The system shown here has prism apex angles ( α 1 , α 2 , α 3 ) = ( 20 ° , 70 ° , 60 ° ) , indices of refraction ( n 1 , n 2 , n 3 ) = ( 1.45 , 1.80 , 1.60 ) , overall height h, beam displacement Δ y , and axial thicknesses t 1 , t 2 , and t 3 . The input ray has angle θ 0 = 10 ° such that δ = 15 ° .

Fig. 2
Fig. 2

Dispersion gradients d δ / d λ and example layouts of the triplet designs of Table 1. Note that an ideal linear dispersion would produce a horizontal line on these plots and that the three layouts are drawn to the same relative scale.

Fig. 3
Fig. 3

Dispersion gradients d δ / d λ and example layouts of the K-constrained triplet designs in Table 2, together with three example layouts. An ideal linear dispersion would produce a horizontal line on these plots. The layout figures were generated by replicating the design parameters in Zemax [6] and selecting the thicknesses of each prism.

Fig. 4
Fig. 4

Ray trace through an example Janssen prism, where the central element is assumed to be oriented symmetrically with respect to the optical axis normal. The diagram shown here has input ray angle θ 0 = 0 ° and a ray deviation angle δ = 0 ° for clarity. The beam displacement is zero, Δ y = 0 .

Fig. 5
Fig. 5

Dispersion gradients d δ / d λ and example layouts of the Janssen prisms shown in Table 3. An ideal linear dispersion would produce a horizontal line on these plots.

Fig. 6
Fig. 6

Three prism apex angles α 1 , α 2 , and α 3 , together with the central deviation δ ¯ and total angular dispersion Δ achieved by a LAK34/SF66/LITHQ prism. Above Δ = 32.1 ° (indicated by the greyed region of the plot), the algorithm can no longer locate a solution that satisfies the design targets, defining the dispersion limit achievable by this glass set.

Fig. 7
Fig. 7

Dispersion of the sapphire–air–ZnS Mooney prism over the visible and midwavelength IR bands. The prism prescription and layout are shown in boxed insets.[7]

Fig. 8
Fig. 8

Dispersion of prisms 1a–5a and 1b–5b of Table 4.

Tables (4)

Tables Icon

Table 1 Best-Performing Prism Triplets for θ 0 = 0 , Optimized Over the Schott Glass Catalog a

Tables Icon

Table 2 Best-Performing Prism Triplets for θ 0 = 0 , Optimized Over the Schott Glass Catalog a

Tables Icon

Table 3 Bst-Performing Janssen Prisms for δ * = 0 ° and θ 1 = 0 Optimized Over the Schott Glass Catalog a

Tables Icon

Table 4 Best-Performing Triplet Prisms Optimized for Second-Order Dispersion, for θ 0 = 0 , (a) the Schott Glass Catalog, and (b) ZEMAX’s Infrared Glass Catalog

Equations (21)

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δ ¯ δ C = ( δ ¯ 1 + δ ¯ 2 + δ ¯ 3 ) ( δ C , 1 + δ C , 2 + δ C , 3 ) = δ ¯ 1 ( P 1 V 1 ) + δ ¯ 2 ( P 2 V 2 ) + δ ¯ 3 ( P 3 V 3 ) ,
Δ ˜ 2 = δ ¯ 1 P 1 V 1 + δ ¯ 2 P 2 V 2 .
( δ ¯ Δ ˜ Δ ˜ / 2 ) = 1 1 1 1 / V 1 1 / V 2 1 / V 3 P 1 / V 1 P 2 / V 2 P 3 / V 3 δ ¯ 1 δ ¯ 2 δ ¯ 3 = H 3 δ ¯ 1 δ ¯ 2 δ ¯ 3 ,
δ ¯ ^ 1 δ ¯ ^ 2 δ ¯ ^ 3 = H 3 1 δ ¯ Δ ˜ Δ ˜ / 2 ,
α 1 α 2 α 3 = δ ¯ ^ 1 / ( n ¯ 1 1 ) δ ¯ ^ 2 / ( n ¯ 2 1 ) δ ¯ ^ 3 / ( n ¯ 3 1 ) .
θ 1 = θ 0 + β 1 , θ 3 = arcsin ( n 2 n 3 sin θ 3 ) , θ 1 = arcsin ( 1 n 1 sin θ 1 ) , θ 4 = θ 3 α 3 , θ 2 = θ 1 α 1 , θ 4 = arcsin ( n 3 sin θ 4 ) , θ 2 = arcsin ( n 1 n 2 sin θ 2 ) , θ 5 = θ 4 + γ 3 , θ 3 = θ 2 α 2 , } ,
M 0 = ( δ ¯ δ ¯ * ) 2 + ( Δ Δ * ) 2 + Θ ,
Θ = i = 1 I { 0 θ i < θ limit ( θ limit θ i ) 2 θ i > θ limit .
M nl = M 0 + w nl NL ,
NL = | d 2 δ d λ 2 | d λ .
K = n = 1 N | cos θ n | | cos θ n |
θ 1 = θ 0 + β , θ 4 = arcsin ( n 3 n 2 sin θ 4 ) , θ 1 = arcsin ( 1 n 1 sin θ 1 ) , θ 5 = θ 4 α 4 , θ 2 = θ 1 α 1 , θ 5 = arcsin ( n 2 n 1 sin θ 5 ) , θ 2 = arcsin ( n 1 n 2 sin θ 2 ) , θ 6 = θ 5 α 5 , θ 3 = θ 2 α 2 , θ 6 = arcsin ( n 1 sin θ 6 ) , θ 3 = arcsin ( n 2 n 3 sin θ 3 ) , θ 7 = θ 6 + β , θ 4 = θ 3 α 3 , } ,
M H = M 0 + w H ( 1 / NL ) ,
M 0 = ( δ ¯ δ ¯ * ) 2 + ( Δ Δ * ) 2 + Θ .
A = n N α n 2 ,
B = i = 1 I { ( α low | α i | ) 2 α i < α low 0 α i > α low ,
C = | max { δ ( λ ) } min { δ ( λ ) } | ,
K = n = 1 N | cos θ n | | cos θ n | ,
NL = | d 2 δ d λ 2 | d λ ,
Θ = i = 1 I { 0 θ i < θ limit ( θ limit θ i ) 2 θ i > θ limit ,
M = ( δ ¯ δ ¯ * ) 2 + ( Δ Δ * ) 2 + Θ 2 + 0.01 ( K 1 ) 2 + 0.1 NL .

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