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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23378–23384
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Broadband astigmatism-corrected Czerny–Turner spectrometer

Kye-Sung Lee, Kevin P. Thompson, and Jannick P. Rolland  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23378-23384 (2010)
http://dx.doi.org/10.1364/OE.18.023378


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Abstract

We report an optical design for a low-cost optics, broadband, astigmatism-corrected practical spectrometer. An off-the-shelf cylindrical lens is used to remove astigmatism over the full bandwidth. Results show that better than 0.1 nm spectral resolution and more than 50% throughput were achieved over a bandwidth of 400 nm centered at 800 nm.

© 2010 OSA

1. Introduction

Two-beam spectral interferometry (SI) has enabled the rapid development of such techniques as spectrometer-based frequency-domain optical coherence tomography (FD-OCT) [1

1. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995). [CrossRef]

3

3. S. Murali, P. Meemon, K. S. Lee, W. P. Kuhn, K. P. Thompson, and J. P. Rolland, “Assessment of a liquid lens enabled in vivo optical coherence microscope,” Appl. Opt. 49(16), D145–D156 (2010). [CrossRef] [PubMed]

] and femtosecond pulse characterization [4

4. A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31(12), 1914–1916 (2006). [CrossRef] [PubMed]

]. The Czerny-Turner spectrometer is a commonly used instrument in SI because by definition, only two spherical mirrors and a plane grating can be configured in a coma-free geometry where, together with a low numerical aperture, spherical aberration can be avoided [5

5. M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11-12), 792–797 (1930). [CrossRef]

]. However, this solution simply does not provide resolution better than 0.1 nm. The spectral resolution 0.1 nm is necessary for the application of FD-OCT to get 1 mm imaging depth in skin with submicron axial resolution, which is state-of-the-art in OCT. Czerny and Turner first showed that the coma aberration introduced by the off-axis reflection from a spherical mirror can be corrected by a symmetrical, but oppositely oriented, spherical mirror for spectrometer design [5

5. M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11-12), 792–797 (1930). [CrossRef]

]. After that, Shafer showed that the coma aberration can be still corrected in the Czerny-Turner spectrometer, even though its symmetry is broken, if the geometry parameters satisfy a condition that is known as the Shafer equation [6

6. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny-Turner spectrometer,” J. Opt. Soc. Am. 54(7), 879–887 (1964). [CrossRef]

,7

7. Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009). [CrossRef]

]. However, astigmatism remains in both configurations yielding different focal lengths in the tangential and sagittal planes. The astigmatism can be ignored in 1D spectroscopy by locating an exit slit at the tangential focal plane. However, some applications sensitive to power efficiency such as high speed spectrometer-based FD-OCT are required to collect most power with high speed line CCD or CMOS cameras where the width of the detector area is limited to maximize signal to noise ratio. In those applications, the uncorrected astigmatism does result in degraded performance. Some methods to reduce or remove the limiting astigmatism have been investigated such as, using additional convex mirrors [8

8. G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. III. Some applications,” J. Opt. Soc. Am. 52(4), 412–415 (1962). [CrossRef]

], placing compensating optics before the entrance slit [9

9. M. Goto and S. Morita, “Spatial distribution measurement of atomic radiation with an astigmatism-corrected Czerny–Turner-type spectrometer in the Large Helical Device,” Rev. Sci. Instrum. 77, 10F124 (2006). [CrossRef]

], using toroidal mirrors [6

6. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny-Turner spectrometer,” J. Opt. Soc. Am. 54(7), 879–887 (1964). [CrossRef]

,10

10. A. B. Shafer, “Correcting for astigmatism in the czerny-turner spectrometer and spectrograph,” Appl. Opt. 6(1), 159–160 (1967). [CrossRef] [PubMed]

], using a cylindrical grating [11

11. M. L. Dalton Jr., “Astigmatism compensation in the Czerny-turner spectrometer,” Appl. Opt. 5(7), 1121–1123 (1966). [CrossRef] [PubMed]

], or introducing divergent illumination [12

12. B. Bates, M. McDowell, and A. C. Newton, “Correction of astigmatism in a Czerny–Turner spectrograph using a plane grating in divergent illumination,” J. Phys. E 3(3), 206–210 (1970). [CrossRef]

14

14. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef] [PubMed]

]. Some techniques were pursued with a goal of providing extended spectral range. However, these methods need to satisfy a condition of the parameters of the Czerny-Turner spectrometer to compensate the astigmatism of the spectrometer [7

7. Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009). [CrossRef]

,14

14. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef] [PubMed]

], which may limit the spectrometer design for various specifications. Here we propose a novel solution also based on a modified Czerny-Turner geometry using only off-the-shelf, low-cost optical components without requiring the restrictive condition. Specifically, the solution includes all-spherical mirrors together with a cylindrical lens to compensate the astigmatism over a broadband width. This method enables low cost construction using off-shelf optics in place of components such a toroidal mirror or a cylindrical grating, both more complex elements to fabricate. We present how the cylindrical lens can remove the astigmatism over a broadband width and then show the design of the broadband astigmatism-corrected Czerny-Turner spectrometer and its result.

2. Broadband astigmatism compensation using a cylindrical lens

The difference in astigmatic foci generated by the two off-axis spherical mirrors of a Czerny-Turner can be written as [11

11. M. L. Dalton Jr., “Astigmatism compensation in the Czerny-turner spectrometer,” Appl. Opt. 5(7), 1121–1123 (1966). [CrossRef] [PubMed]

]
Δz=(R1/2)(sinα1tanα1)+(R2/2)(sinα2tanα2) ,
(1)
where R 1 is the radius of the first spherical mirror, R 2 is the radius of the second spherical mirror, α 1 is the off-axis incident angle on the first mirror, and α 2 is the off-axis incident angle on the second mirror. In our preferred design, a cylindrical lens is located near the detector as shown in Fig. 1
Fig. 1 Astigmatism correction by a cylindrical lens in (a) tangential view (b) sagittal view after the second mirror in a Czerny-Turner spectrometer.
that displays both in (a) the tangential view and in (b) the sagittal view. The change of the sagittal focus scs – s’cs illustrated in Fig. 1(b) is calculated by the sagittal lens equation of the cylindrical lens s’cs −1 equal fcs −1 + scs −1, which yields
scss'cs=scs[fcsscs/(fcs+scs)] ,
(2)
where the subscript c stands for cylindrical, scs is the sagittal object distance, s’cs is the sagittal image distance, and fcs is the sagittal focal length of the cylindrical lens. The change of the tangential focus introduced by the cylindrical lens (i.e. plane parallel plates in the tangential view) is given simply by
s'ctsct=[(n1)/n]t0 ,
(3)
where n is the refractive index and t0 is the central thickness of the cylindrical lens.

The astigmatism is removed when
Δz=scss'cs+s'ctsct ,
(4)
as illustrated in Fig. 1. Substituting the Eq. (2) and Eq. (3) into Eq. (4) gives

scs2[Δz((n1)/n) t0]scsfcs[Δz((n1)/n) t0]=0,        scs&fcs>0.
(5)

Solving for scs yields
scs=[P+P2+4Pfcs]/2 ,
(6)
where P=Δzt0(n1)/n  . The position of the cylindrical lens is then determined as the distance from the second mirror Lc as shown in Fig. 1(b) given by
Lc=fsscst0 ,
(7)
where fs=R2/2cosα2  .

The off-axis angle α 2 incident to the second spherical mirror varies according to the diffracted wavelength from the grating as shown in Fig. 2
Fig. 2 Broadband astigmatism correction by a tilted cylindrical lens in the tangential view of chief ray-tracing for the different wavelengths in the part of the spectrometer, i is the incident angle to the grating.
. The variation in α 2 across the second mirror results in a variation in astigmatism Δz across the detector. The Δz ascends (R 2 > LGF, where LGF is the distance between the grating and the second mirror) or descends (R 2 < LGF) when the second mirror is rotated counter clockwise by α¯2 from the perpendicular position at the central wavelengthλ¯. The ascending or descending astigmatism can be overall compensated by tilting of the cylindrical lens with an optimized angle. The differentiation dscs/dΔ is calculated from Eq. (6) as

dscs/dΔz=(1/2)+[(P+2fcs)/(2P2+4Pfcs)]  .
(8)

The tilt angle ξ of the cylindrical lens with respect to the sagittal focal plane is shown in Fig. 2 and can be estimated with the first order polynomial fitting at the center (i.e. H = 0) as
tanξ(dscs/dH)|H=0=(dscs/dΔz)|Δz=Δ¯z(dΔz/dλ)|λ=λ¯(dλ/dH)|H=0 ,
(9)
where dscs/dΔz|Δz=Δ¯z is obtained from Eq. (8) as
(dscs/dΔz)|Δz=Δ¯z=(1/2)+[(P¯+2fcs)/(2P¯2+4P¯fcs)] ,
(10)
whereP¯=Δ¯zt0(n1)/n and according to Eq. (1)

Δ¯z=(R1/2)(sinα1tanα1)+(R2/2)(sinα¯2tanα¯2)  .
(11)

The second term dΔz/dλ|λ=λ¯ in Eq. (9) is derived as
(dΔz/dλ)|λ=λ¯=(dΔz/dα2)|α2=α¯2(dα2/dθ)|θ=θ¯(dθ/dλ)λ=λ¯=[(R2/2)sinα¯2(1+sec2α¯2)][1(L¯GF/R2cosα¯2)][1/dcosθ¯] ,
(12)
where the first bracket on the right side of Eq. (12) is obtained from Eq. (1), the second bracket is derived from geometry [14

14. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef] [PubMed]

], and the third bracket is the angular dispersion of the grating with θ¯being the diffraction angle at the central wavelengthλ¯, d is the groove spacing of the grating. The last term in Eq. (9) is the reciprocal of the linear dispersion of the grating with the second mirror R 2 at the central wavelength, which is given as

(dλ/dH)|H=0=2dcosθ¯/R2  .
(13)

The tilt angle δ of the cylindrical lens with respect to the plane perpendicular to the optical axis as designated in Fig. 2 can be calculated as
δ=ξ[tan1((dfs/dH)|fs=f¯s)+α¯2] ,
(14)
where (dfs/dH)|fs=f¯s=(dfs/dα2)|α2=α¯2(dα2/dθ)|θ=θ¯(dθ/dλ)|λ=λ¯(dλ/dH)|H=0 , given by (dfs/dα2)|α2=α¯2=(Rsinα¯2/2cos2α¯2) , while the other three terms are given in Eqs. (12) and (13).

3. Design procedure for broadband astigmatism-corrected Czerny-Turner spectrometer

 Δθ= (dθ/dλ)|λ=λ¯Δλ=(Δλ/dcosθ¯)  .
(15)

The groove spacing d was calculated using the relation of detector length L, the focal length f 2 and the angular spread Δθ of the full bandwidth 400 nm.
d=(Δλ/cosθ¯)(f2/L) ,
(16)
where θ¯ was set to 45° to secure sufficient free spectral range as shown in Fig. 3
Fig. 3 (a) Optimized layout of the broadband astigmatism-corrected Czerny-Turner spectrometer (b) The pixel function in a red dotted line and the LSF in a blue solid line (c) the convolution over the pixel (d) the 80% dip in the separation of the two convolutions.
. d was calculated to be 1.1 μm. We selected an off-shelf value of 1.2 μm, which corresponded to 833 lines/mm. The resolving power of the grating was checked to satisfy the condition (Φ/d) ≥ (λ0/SR) [15

15. M. Born, and E. Wolf, Principles of Optics, (Cambridge University Press, 2002), Chap. 8.6.

]. If it was not, Φ would be increased by adjusting the focal length of the first mirror. The tilt of the first mirror, α 1, was selected to be near the smallest angle to avoid obscuration of the rays by the secondary mirror. We chose 12°. The sagittal focal length of the cylindrical lens was chosen to secure some margin between the detector and the cylindrical lens. We chose an off-shelf cylindrical lens with fcs equal 100 mm, n equal 1.5, and t equal 5.2 mm. We optimized the system parameters using the ray-tracing software CODE V to get the best balance of coma. The initial and final parameters are shown in Table. 1. The fixed parameters are shown in Table 2

Table 2. Fixed parameters

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. These values correspond to off-shelf components.

Before optimization with the cylinder lens, the astigmatism Δz at 800nm was calculated to be 21.5 mm using Eq. (1), where α 1 and α¯2 are given in Table 1

Table 1. Initial and optimized parameters

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. The position Lc of the cylindrical lens was calculated to be 100.7 mm using Eq. (6) and (7), which is in close agreement with the optimized value 102.5 mm. The tilt angle δ of the cylindrical lens was calculated to be 6.4° from Eq. (9) and (14), which is close to the optimized value 9.1°. The optimized layout is shown in Fig. 3(a).

4. Analysis

The spectrometer performance was evaluated to predict the spectral resolution along the length of the detector and power collection. The spectral resolution is defined as the spectral bandwidth that one pixel subtends. The wavelength distribution detected by one pixel can be derived by the convolution of the pixel and the line spread function (LSF) for each pixel. The pixel represents a rectangular function with 10 μm width and the LSF was computed from the ray tracing software. The pixel function and the LSF are shown as a red dotted line and blue solid line in Fig. 3(b), respectively. The convolution was done with respect to the distance along the pixels and then the corresponding convolution with respect to the wavelength was obtained considering the dispersion of the grating. The result is shown in Fig. 3(c). The spectral resolution was measured using two metrics; the full width at half maximum and second, the 80% dip in the separation of the two convolutions as shown in Fig. 3(c) and 3(d).

To see how much the performance is improved, we compared the new design with recent attempts at astigmatism management and reduction; specifically with a divergent illumination method [14

14. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef] [PubMed]

] and a toroidal mirror method [7

7. Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009). [CrossRef]

], respectively, in the same NA condition and using the detector size (i.e. NAi = 0.05, L = 80 mm, pixel size 10 μm) and grating dispersion. This comparison is more relevant than a comparison with the original implementation as it puts the corrections in the context of other technically viable options for improvement. As seen in Fig. 4
Fig. 4 (a) spectral resolutions over the full bandwidth and (b) power efficiencies collected by the pixel array with 10 µm width for three different methods, the new method reported here (blue), a method based on a divergent wavefront [14] (red), and a method that replaces the spherical mirrors of a traditional Czerny-Turner with toroidal mirrors [7] (green).
, the new method is the only one that meets the requirement for the application at hand of 0.1 nm spectral resolution and more than 50% throughput over the bandwidth. The computed spectral resolutions are shown in Fig. 4(a). For the new design method, the spectral resolution ranges from 0.07 to 0.09 nm for the FWHM and 0.08 to 0.1 nm for the 80% dip over the full bandwidth as shown in Fig. 4(a). Results demonstrate an improvement of at least three times compared to the other solutions. The power percentages collected by the pixel array with 10 μm width were measured and the results are shown in Fig. 4(b). The throughput collected over the full bandwidth is more than 50% as compared to the other solutions that yield collecting powers of less than 20% over most of the spectrum.

5. Conclusion

We have designed a Czerny-Turner type spectrometer using an off-the-shelf cylindrical lens to provide a broadband astigmatism-corrected spectrometer with tilted spherical mirrors. The traditional Czerny-Turner spectrometer, while configured to cancel coma, suffers from astigmatism due to off-axis use of the spherical mirrors. There have been a number of attempts to overcome this limitation including the use of toroidal mirrors. This design enables low cost, practical construction using off-the-shelf optics rather than a custom toroidal mirror or cylindrical grating. This solution is effective over a broad range of parameters according to the specification of the spectrometer designed and does not require restrictions in 1st order parameter selection for astigmatism compensation.

Acknowledgments

This research was funded by the NYSTAR Foundation. We thank Optical Research Associate for providing an educational license for CODE V®.

References and links

1.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995). [CrossRef]

2.

J. P. Rolland, P. Meemon, S. Murali, K. P. Thompson, and K. S. Lee, “Gabor-based fusion technique for Optical Coherence Microscopy,” Opt. Express 18(4), 3632–3642 (2010). [CrossRef] [PubMed]

3.

S. Murali, P. Meemon, K. S. Lee, W. P. Kuhn, K. P. Thompson, and J. P. Rolland, “Assessment of a liquid lens enabled in vivo optical coherence microscope,” Appl. Opt. 49(16), D145–D156 (2010). [CrossRef] [PubMed]

4.

A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31(12), 1914–1916 (2006). [CrossRef] [PubMed]

5.

M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11-12), 792–797 (1930). [CrossRef]

6.

A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny-Turner spectrometer,” J. Opt. Soc. Am. 54(7), 879–887 (1964). [CrossRef]

7.

Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009). [CrossRef]

8.

G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. III. Some applications,” J. Opt. Soc. Am. 52(4), 412–415 (1962). [CrossRef]

9.

M. Goto and S. Morita, “Spatial distribution measurement of atomic radiation with an astigmatism-corrected Czerny–Turner-type spectrometer in the Large Helical Device,” Rev. Sci. Instrum. 77, 10F124 (2006). [CrossRef]

10.

A. B. Shafer, “Correcting for astigmatism in the czerny-turner spectrometer and spectrograph,” Appl. Opt. 6(1), 159–160 (1967). [CrossRef] [PubMed]

11.

M. L. Dalton Jr., “Astigmatism compensation in the Czerny-turner spectrometer,” Appl. Opt. 5(7), 1121–1123 (1966). [CrossRef] [PubMed]

12.

B. Bates, M. McDowell, and A. C. Newton, “Correction of astigmatism in a Czerny–Turner spectrograph using a plane grating in divergent illumination,” J. Phys. E 3(3), 206–210 (1970). [CrossRef]

13.

M. McDowell, “Design of Czerny–Turner spectrographs using divergent grating illumination,” Opt. Acta (Lond.) 22, 473–475 (1975).

14.

D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef] [PubMed]

15.

M. Born, and E. Wolf, Principles of Optics, (Cambridge University Press, 2002), Chap. 8.6.

OCIS Codes
(110.4500) Imaging systems : Optical coherence tomography
(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments
(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: July 29, 2010
Revised Manuscript: October 3, 2010
Manuscript Accepted: October 4, 2010
Published: October 21, 2010

Citation
Kye-Sung Lee, Kevin P. Thompson, and Jannick P. Rolland, "Broadband astigmatism-corrected Czerny–Turner spectrometer," Opt. Express 18, 23378-23384 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23378


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References

  1. A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995). [CrossRef]
  2. J. P. Rolland, P. Meemon, S. Murali, K. P. Thompson, and K. S. Lee, “Gabor-based fusion technique for Optical Coherence Microscopy,” Opt. Express 18(4), 3632–3642 (2010). [CrossRef] [PubMed]
  3. S. Murali, P. Meemon, K. S. Lee, W. P. Kuhn, K. P. Thompson, and J. P. Rolland, “Assessment of a liquid lens enabled in vivo optical coherence microscope,” Appl. Opt. 49(16), D145–D156 (2010). [CrossRef] [PubMed]
  4. A. S. Wyatt, I. A. Walmsley, G. Stibenz, and G. Steinmeyer, “Sub-10 fs pulse characterization using spatially encoded arrangement for spectral phase interferometry for direct electric field reconstruction,” Opt. Lett. 31(12), 1914–1916 (2006). [CrossRef] [PubMed]
  5. M. Czerny and A. F. Turner, “On the Astigmatism of Mirror Spectrometers,” Z. Phys. 61(11-12), 792–797 (1930). [CrossRef]
  6. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny-Turner spectrometer,” J. Opt. Soc. Am. 54(7), 879–887 (1964). [CrossRef]
  7. Q. Xue, S. Wang, and F. Lu, “Aberration-corrected Czerny-Turner imaging spectrometer with a wide spectral region,” Appl. Opt. 48(1), 11–16 (2009). [CrossRef]
  8. G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. III. Some applications,” J. Opt. Soc. Am. 52(4), 412–415 (1962). [CrossRef]
  9. M. Goto and S. Morita, “Spatial distribution measurement of atomic radiation with an astigmatism-corrected Czerny–Turner-type spectrometer in the Large Helical Device,” Rev. Sci. Instrum. 77, 10F124 (2006). [CrossRef]
  10. A. B. Shafer, “Correcting for astigmatism in the czerny-turner spectrometer and spectrograph,” Appl. Opt. 6(1), 159–160 (1967). [CrossRef] [PubMed]
  11. M. L. Dalton., “Astigmatism compensation in the Czerny-turner spectrometer,” Appl. Opt. 5(7), 1121–1123 (1966). [CrossRef] [PubMed]
  12. B. Bates, M. McDowell, and A. C. Newton, “Correction of astigmatism in a Czerny–Turner spectrograph using a plane grating in divergent illumination,” J. Phys. E 3(3), 206–210 (1970). [CrossRef]
  13. M. McDowell, “Design of Czerny–Turner spectrographs using divergent grating illumination,” Opt. Acta (Lond.) 22, 473–475 (1975).
  14. D. R. Austin, T. Witting, and I. A. Walmsley, “Broadband astigmatism-free Czerny-Turner imaging spectrometer using spherical mirrors,” Appl. Opt. 48(19), 3846–3853 (2009). [CrossRef] [PubMed]
  15. M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 2002), Chap. 8.6.

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