Abstract

A fundamental formalism featuring the common working principle of different spectroscopic devices is introduced. General formulas for angular dispersion, free spectral range, and spectral resolution are deduced from both the impulse response function and the spatial transmission function of the device, based on the assumption that these functions can be written up as the product of a finite width, real-aperture function, and a complex periodic function. The method will also be shown to work in specific cases.

© 2006 Optical Society of America

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References

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  1. W. Demtröder, Laser Spectroscopy (Springer-Verlag, 1982).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).
  3. M. V. Klein, Optics (Wiley, 1970).
  4. F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1957).
  5. J. Hebling, "Spectral resolving power," J. Opt. Soc. Am. A 11, 2900-2904 (1994).
    [CrossRef]
  6. D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), pp. 91-99.
  7. D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), pp. 9-39.
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48-56.
  9. S. Szatmári, G. Kuhnle, and P. Simon, "Pulse compression and traveling wave excitation scheme using a single dispersive element," Appl. Opt. 29, 5372-5379 (1990).
    [CrossRef] [PubMed]
  10. J. Hebling, "Derivation of the pulse front tilt caused by angular dispersion," Opt. Quantum Electron. 28, 1759-1763 (1996).
    [CrossRef]
  11. Zs. Bor and B. Rácz, "Group velocity dispersion in prisms and its application to pulse compression and travelling wave excitation," Opt. Commun. 54, 165-170 (1985).
    [CrossRef]

1996 (1)

J. Hebling, "Derivation of the pulse front tilt caused by angular dispersion," Opt. Quantum Electron. 28, 1759-1763 (1996).
[CrossRef]

1994 (1)

1990 (1)

1985 (1)

Zs. Bor and B. Rácz, "Group velocity dispersion in prisms and its application to pulse compression and travelling wave excitation," Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Bor, Zs.

Zs. Bor and B. Rácz, "Group velocity dispersion in prisms and its application to pulse compression and travelling wave excitation," Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Champeney, D. C.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), pp. 9-39.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), pp. 91-99.

Demtröder, W.

W. Demtröder, Laser Spectroscopy (Springer-Verlag, 1982).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48-56.

Hebling, J.

J. Hebling, "Derivation of the pulse front tilt caused by angular dispersion," Opt. Quantum Electron. 28, 1759-1763 (1996).
[CrossRef]

J. Hebling, "Spectral resolving power," J. Opt. Soc. Am. A 11, 2900-2904 (1994).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1957).

Klein, M. V.

M. V. Klein, Optics (Wiley, 1970).

Kuhnle, G.

Rácz, B.

Zs. Bor and B. Rácz, "Group velocity dispersion in prisms and its application to pulse compression and travelling wave excitation," Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Simon, P.

Szatmári, S.

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1957).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Zs. Bor and B. Rácz, "Group velocity dispersion in prisms and its application to pulse compression and travelling wave excitation," Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Opt. Quantum Electron. (1)

J. Hebling, "Derivation of the pulse front tilt caused by angular dispersion," Opt. Quantum Electron. 28, 1759-1763 (1996).
[CrossRef]

Other (7)

W. Demtröder, Laser Spectroscopy (Springer-Verlag, 1982).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

M. V. Klein, Optics (Wiley, 1970).

F. A. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1957).

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), pp. 91-99.

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, 1973), pp. 9-39.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48-56.

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Figures (5)

Fig. 1
Fig. 1

Schematic construction of a spectroscopic device.

Fig. 2
Fig. 2

Wavefronts of a plane wave diffracted by an angle θ. The reciprocal spatial frequency in the x direction is indicated.

Fig. 3
Fig. 3

Scheme for the discussion of the special formulas of angular dispersion and spectral resolution in the simple case when n = 1 and Λ and τ are independent of λ.

Fig. 4
Fig. 4

(a) Diffraction grating used in normal incidence in a medium with refractive index n, (b) spatial transmission function, (c) impulse response function of a grating.

Fig. 5
Fig. 5

Plane wave passing through a prism.

Tables (2)

Tables Icon

Table 1 General Formulas for Angular Dispersion, Free Spectral Range, and Spectral Resolving Power a

Tables Icon

Table 2 Angular Dispersion, Free Spectral Range, and Spectral Resolving Power, in the Case When n = 1 and Λ and τ Are Independent of λ

Equations (70)

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d θ d λ = m d cos θ ,
d ϵ d λ = d ϵ d n d n d λ ,
d s d λ = f D ,
R = λ 0 Δ λ ,
E out ( x ) = t ( x ) E in ( x ) ,
t ( x ) = a ( x ) p ( x ) ,
H ( ω ) = FT [ h ( t ) ] ,
E out ( t ) = h ( t ) E in ( t ) ,
h ( t ) = a ( t ) p ( t ) .
H ( ω ) = FT [ h ( t ) ] = FT [ a ( t ) ] FT [ p ( t ) ] = FT [ a ( t ) ] p m δ ( ω 2 π m τ ) ,
τ = m λ c ,
d τ d λ = τ λ + τ θ d θ d λ = m c = τ λ .
d θ d λ = 1 λ [ τ λ ( τ λ ) ] τ θ = τ λ θ τ ,
d θ d λ = m c τ τ θ τ ,
d θ d λ = m c θ τ .
A ( f x ) = FT [ E out ( x ) ] ,
sin θ = λ f x .
A ( f x ) = E in FT [ t ( x ) ] = E in FT [ a ( x ) ] FT [ p ( x ) ] = E in FT [ a ( x ) ] p m δ ( f x m Λ ) ,
sin θ = m λ Λ .
d θ d λ = m Λ cos θ .
sin θ = m λ n Λ
d θ d λ = m { n Λ λ [ ( n Λ ) λ ] } cos θ ( n Λ ) 2 = m ( n Λ ) cos θ ( n Λ ) 2 ,
d θ d λ = m [ Λ λ ( Λ λ ) ] cos θ Λ 2 = m Λ cos θ Λ 2 ,
p ( x ) = p 1 ( x ) p 2 ( x ) ,
A ( f x ) = E in FT [ t ( x ) ] = E in FT [ a ( x ) ] FT [ p 1 ( x ) ] FT [ p 2 ( x ) ]
= E in FT [ a ( x ) ] p m 1 δ ( f x m 1 Λ ) p m 2 δ ( f x m 2 Λ ) .
sin θ = m 1 λ n Λ 1 + m 2 λ n Λ 2 .
d θ d λ = m 1 ( n Λ 1 ) cos θ ( n Λ 1 ) 2 + m 2 ( n Λ 2 ) cos θ ( n Λ 2 ) 2 ,
θ ( λ , m + 1 ) = θ ( λ + Δ λ FSR , m ) .
τ ( λ , m + 1 ) = τ ( λ + Δ λ FSR , m ) Δ τ .
Δ τ τ λ Δ λ FSR .
λ ( m + 1 ) c = ( λ + Δ λ FSR ) m c τ λ Δ λ FSR ,
Δ λ FSR = λ m c ( τ λ ) = λ m τ T = λ m L λ ,
Δ λ FSR = λ m τ τ λ ( τ λ ) = λ m τ τ .
sin θ = ( m + 1 ) λ n Λ = m ( λ + Δ λ FSR ) n Λ .
Δ λ FSR = λ m .
sin θ = ( m + 1 ) λ ( n Λ ) λ = m ( λ + Δ λ ) ( n Λ ) λ + Δ λ ,
m λ [ 1 ( n Λ ) λ 1 ( n Λ ) λ + Δ λ ] + λ ( n Λ ) λ = m Δ λ ( n Λ ) λ + Δ λ .
m λ [ ( 1 n Λ ) λ ] λ Δ λ + λ ( n Λ ) λ = m ( n λ ) λ + Δ λ Δ λ .
Δ λ FSR = λ m { 1 ( n Λ ) λ + Δ λ + λ ( ( 1 n Λ ) λ ) λ } ( n Λ ) λ .
Δ λ FSR = λ m n Λ { n Λ λ [ ( n Λ ) λ ] } = λ m ( n Λ ) ( n Λ ) .
Δ θ Λ cos θ = m Δ λ ,
Δ θ Δ λ = m Λ cos θ ,
d θ d λ = m n g n 2 d cos θ .
d θ d λ = 1 λ n g n tan θ
n λ d θ d λ = n g tan θ .
n λ d θ d λ = n g tan γ .
E out ( x ) = E 0 exp i 2 π x Λ ,
sin θ = n sin β 2 = sin α 2 ;
d θ d λ = ( Λ ) cos θ ( Λ ) 2 = tan α 2 n d n d λ .
sin θ = 1 n sin α 1 = sin β 1 ,
d θ d λ = d β 1 d λ = tan β 1 n d n d λ .
d α 2 d λ = α 2 λ + α 2 β 2 d β 2 d λ = α 2 λ α 2 β 2 d β 1 d λ .
α 2 β 2 = n cos β 2 cos α 2 .
d α 2 d λ = ( tan α 2 n + cos β 2 cos α 2 sin β 1 cos β 1 ) d n d λ .
d α d λ = 2 tan α n d n d λ .
τ = Λ sin α 2 c = λ c sin α 2 n sin β 2 ,
d α 2 d λ = tan α 2 n d n d λ ,
d θ d λ = m τ τ n Λ cos θ
d θ d λ = m ( n Λ ) ( n Λ ) 2 cos θ
Δ λ F S R = λ m τ τ
Δ λ F S R = λ m n Λ ( n Λ )
* R = b i Δ t T τ τ
R = b t a sin θ λ ( n Λ ) n Λ
d θ d λ = m Λ cos θ
d θ d λ = m Λ cos θ
Δ λ F S R = λ m
Δ λ F S R = λ m
R = b i Δ t T
R = b t a sin θ λ

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