Abstract

We derive an analytic expression for the instrument profile of a slit spectrograph, also known as the line spread function. While this problem is not new, our treatment relies on the operatorial approach to the description of diffractive optical systems, which provides a general framework for the analysis of the performance of slit spectrographs under different illumination conditions. Based on our results, we propose an approximation to the spectral resolution of slit spectrographs, taking into account diffraction effects and sampling by the detector, which improves upon the often adopted approximation based on the root-sum-square of the individual contributions from the slit, the grating, and the detector pixel.

© 2014 Optical Society of America

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References

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  1. J. M. Lerner and A. Thevenon, The Optics of Spectroscopy: A Tutorial (Jobin Yvon/Horiba Optical Systems, 1998).
  2. D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).
  3. F. L. O. Wadsworth, “On the resolving power of telescopes and spectroscopes for lines of finite width,” Philos. Mag. 43, 317–343 (1897).
  4. P. H. van Cittert, “Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien,” Z. Phys. 65, 547–563 (1930).
    [CrossRef]
  5. K. D. Mielenz, “Spectroscope slit images in partially coherent light,” J. Opt. Soc. Am. 57, 66–71 (1967).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  7. M. Nazarathy and J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [CrossRef]
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).
  9. D. F. Gray, The Observation and Analysis of Stellar Atmospheres (Wiley, 1976).
  10. D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000).

2008 (1)

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

1980 (1)

1967 (1)

1930 (1)

P. H. van Cittert, “Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien,” Z. Phys. 65, 547–563 (1930).
[CrossRef]

1897 (1)

F. L. O. Wadsworth, “On the resolving power of telescopes and spectroscopes for lines of finite width,” Philos. Mag. 43, 317–343 (1897).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Card, G. L.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Casini, R.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Davis, M.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Elmore, D. F.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gray, D. F.

D. F. Gray, The Observation and Analysis of Stellar Atmospheres (Wiley, 1976).

Lecinski, A.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Lerner, J. M.

J. M. Lerner and A. Thevenon, The Optics of Spectroscopy: A Tutorial (Jobin Yvon/Horiba Optical Systems, 1998).

Lull, R.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Mielenz, K. D.

Nazarathy, M.

Nelson, P. G.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000).

Shamir, J.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Thevenon, A.

J. M. Lerner and A. Thevenon, The Optics of Spectroscopy: A Tutorial (Jobin Yvon/Horiba Optical Systems, 1998).

Tomczyk, S.

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

van Cittert, P. H.

P. H. van Cittert, “Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien,” Z. Phys. 65, 547–563 (1930).
[CrossRef]

Wadsworth, F. L. O.

F. L. O. Wadsworth, “On the resolving power of telescopes and spectroscopes for lines of finite width,” Philos. Mag. 43, 317–343 (1897).

J. Opt. Soc. Am. (2)

Philos. Mag. (1)

F. L. O. Wadsworth, “On the resolving power of telescopes and spectroscopes for lines of finite width,” Philos. Mag. 43, 317–343 (1897).

Proc. SPIE (1)

D. F. Elmore, R. Casini, G. L. Card, M. Davis, A. Lecinski, R. Lull, P. G. Nelson, and S. Tomczyk, “A new spectro-polarimeter for solar prominence and filament magnetic field measurements,” Proc. SPIE 7014, 39–48 (2008).

Z. Phys. (1)

P. H. van Cittert, “Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien,” Z. Phys. 65, 547–563 (1930).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

D. F. Gray, The Observation and Analysis of Stellar Atmospheres (Wiley, 1976).

D. J. Schroeder, Astronomical Optics, 2nd ed. (Academic, 2000).

J. M. Lerner and A. Thevenon, The Optics of Spectroscopy: A Tutorial (Jobin Yvon/Horiba Optical Systems, 1998).

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Figures (5)

Fig. 1.
Fig. 1.

Layout of the imaging system. See text for details.

Fig. 2.
Fig. 2.

Ratio ρFWHM between the RSS estimate of the FWHM of the spectrograph LSF and the exact value as calculated with the model presented in this work, plotted as a function of the two contributions γS=wS/fC and γG=λ/wG: (left) case of coherent illumination and (right) case of incoherent illumination.

Fig. 3.
Fig. 3.

Cross section of the energy distribution at the detector (normalized to unit peak), from a fully illuminated slit, for a typical spectrograph configuration (λ=1μm, wS=50μm, fC=2.5m, wG=0.1m, fL=1m). The black (gray) curve is for the case of coherent (incoherent) illumination. The figure also gives the projected geometric slit width and the RSS estimate of the FWHM.

Fig. 4.
Fig. 4.

Plots of Eq. (30) with α=ln(4/3), for various values of the exponent q. See text for details.

Fig. 5.
Fig. 5.

Numerical tests of the resolution formula in Eq. (30), using the same spectrograph configuration of Fig. 3, for three different pixel sizes: (top) 6.25 μm, (center) 12.5 μm, and (bottom) 25 μm, corresponding to ζ=2.0, 1.0, 0.5, respectively. See text for details. The vertical dashed lines in the panels on the right define the geometric projection of the slit (cf. Fig. 3).

Equations (46)

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S=R(fL)Q(1fL)1LR(dL)1GR(dC)1CQ(1fC)R(fC)1S.
S=k{Q(1fL)V(1λfL)FQ(1fL)}Q(1fL)1L{F1Q(λ2dL)F}1G{F1Q(λ2dC)F}1CQ(1fC){Q(1fC)V(1λfC)FQ(1fC)}1S=kQ(1fL)V(1λfL)F1L{F1Q(λ2dL)F}1G×{F1Q(λ2dC)F}1CV(1λfC)FQ(1fC)1S,
S=kQ(1fL)V(1λfL)Q(λ2dL)F1G×{F1Q(λ2dC)F}1CV(1λfC)FQ(1fC)1S.
SC=kQ(1fL)V(1λfL)Q(λ2[dL+dC])F1CV(1λfC)FQ(1fC)1S=kQ(1fL)Q(dL+dCfL2)V(1λfL)F1CV(1λfC)FQ(1fC)1S=kQ(1fL2[fLdLdC])V(1λfL)F1CV(1λfC)FQ(1fC)1S,
F1=V(1)F=FV(1),
FV(1λfC)F=F2V(λfC)V(1)V(λfC)=V(λfC),
SG=kQ(1fL)V(1λfL)Q(λ2dL)F1GFV(1)Q(λ2dC)V(λfC)Q(1fC)1S=kQ(1fL)Q(dLfL2)V(1λfL)F1GFV(1)V(λfC)Q(dCfC2)Q(1fC)1S=kQ(1fL[1dLfL])V(1λfL)F1GFV(λfC)Q(1fC[1dCfC])1S=kQ(1fL[1dLfL])V(1λfL)F1GV(1λfC)FQ(1fC[1dCfC])1S.
SCU=kV(1λfL)FtCV(1λfC)FQ(1fC)tSU=kV(1λfL)FtCFV(λfC)Q(1fC)tSUkV(1λfL)FtCp=kV(1λfL)(t˜C*p˜),
p˜=F2V(λfC)Q(1/fC)tSUV(λfC)Q(1/fC)tSU,
SCU=kV(1λfL)t˜C*V(1λfL)p˜=k(t˜C[1λfL])*V(fCfL)Q(1fC)tSU=k(t˜C[1λfL])*Q(fCfL2)V(fCfL)tSU=k(t˜C[1λfL])*Q(fCfL2)(tS[fCfL])(U[fCfL]).
SGU=kV(1λfL)FtGV(1λfC)FtSU==k(t˜G[1λfL])*(tS[fCfL])(U[fCfL]).
UC(x)=kdχt˜C(1λfL(xχ))exp(iπλfCfL2|χ|2)tS(fCfLχ)U(fCfLχ),
UG(x)=kdχt˜G(1λfL(xχ))tS(fCfLχ)U(fCfLχ).
UG(x,y)=1π2[Si(π2wS/fC2x/fLλ/wG)+Si(π2wS/fC+2x/fLλ/wG)]×[Si(π2hS/fC2y/fLλ/hG)+Si(π2hS/fC+2y/fLλ/hG)],
Si(θ)=π2f(θ)cosθg(θ)sinθ,
Si(θ)θ(1θ26(13θ220(15θ2294))),
UG(x,y)UG(x)=1π[Si(π2wS/fC2x/fLλ/wG)+Si(π2wS/fC+2x/fLλ/wG)].
h(x,x)=t˜G(1λfL(xx)),
IG(x)=A1dx|h(x,x)|2tS(x)I(x),
IG(x)=kwGhGdχ|t˜G(1λfL(xχ))|2tS(fCfLχ)I(fCfLχ).
θx±=π2wS/fC±2x/fLλ/wG,θy±=π2hS/fC±2y/fLλ/hG.
IG(x,y)=1π2[Si(2θx)+Si(2θx+)sin2θxθxsin2θx+θx+]×[Si(2θy)+Si(2θy+)sin2θyθysin2θy+θy+].
UG(x,y)=1π2[Si(θx)+Si(θx+)][Si(θy)+Si(θy+)],
IG(x,y)IG(x)=1π[Si(2θx)+Si(2θx+)sin2θxθxsin2θx+θx+].
ζ=ΔLSFδcam.
R(λ;ζ)=λΔλ(ζ).
δλ=dλdβδβ=λcosβsinα+sinβδβ,
limζ0Δλ(ζ)2δcam,
limζΔλ(ζ)=ΔLSF.
Δλ(ζ)=δcamζ2+4exp(αζq),
α=ln(4/3).
R=sinα+sinβcosβwGrwSFD,
Rs.l.=sinα+sinβcosβfCrwS=sinα+sinβcosβ1rγS.
Rd.l.=sinα+sinβcosβwGrλ=sinα+sinβcosβ1rγG.
R=sinα+sinβcosβ1ΔLSF,
|UG(x)|2ϵ2sinc2(x/fLrγG),
Q(c){f(u)}=exp(iπλc|u|2)f(u),
V(c){f(u)}=f(cu),
V(c){f(u)g(u)}=V(c){f(u)}V(c){g(u)}.
F±1{f(u)}=+duexp(i2πu·u)f(u),
R(d){U(x)}=exp(i2πd/λ)iλd+dxexp(iπλd|xx|2)U(x),
R(d)=exp(i2πd/λ)iF1Q(λ2d)F,
R(d)=exp(i2πd/λ)iλdQ(1d)V(1λd)FQ(1d).
V(t)F=FV(1/t),
V(t)Q(c)=Q(t2c)V(t),
F2=V(1).

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