Abstract

To our knowledge, we demonstrate a new scheme for passband engineering of a grating spectrometer. Through spatial masking of the input beam and translation of the optical components we present shaping, shifting, and scaling of the passband in optical frequency. Specifically, we demonstrate the relationship between the applied spatial masking function and the spectrometer passband may be tuned from an exact Fourier transform to a direct scaling through longitudinal displacement of the spectrometer lens and sampling slit and that this operation is independent of the choice of spectrometer center frequency.

© 2004 Optical Society of America

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References

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Appl. Opt.

IEEE J. Quantum Electron.

D. E. Leaird and A. M. Weiner, �??Femtosecond direct space-to-time pulse shaping,�?? IEEE J. Quantum Electron. 37, 494�??504 (2001).
[CrossRef]

J. D. McKinney and A. M.Weiner, �??Direct Space-to-Time Pulse Shaping at 1.5 μm,�?? IEEE J. Quantum Electron. 39, 1635�??1644 (2003).
[CrossRef]

S. Xiao, A. M. Weiner, and C. Lin, �??A Dispersion Law for Virtually Imaged Phased-Array Spectral Dispersers Based on Paraxial Wave Theory,�?? IEEE J. Quantum Electron. 40, 420�??426 (2004).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Rev. Sci. Instrum.

A. M. Weiner, �??Femtosecond pulse shaping using spatial light modulators,�?? Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Sensors and Actuators B

H. O. Edwards and J. P. Daikin, �??Gas Sensors using Correlation Spectroscopy Compatible with Fibre-optic Operation,�?? Sensors and Actuators B 11, 9�??19 (1993).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (The McGraw-Hill Companies, Inc., New York, 1996).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, 1987).

A. E. Siegman, Lasers (University Science Books, Sausalito, 1986).

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Figures (8)

Fig. 1.
Fig. 1.

Experimental setup. Our apparatus consists of a 600 l/mm diffraction grating, 150 mm achromat lens, and a singlemode fiber which functions as the output slit. In this work, the input spectrum Ein (λ) is amplified spontaneous emission from an erbium doped fiber amplifier. The applied spatial masking function m(x′) is defined immediately before the diffraction grating.

Fig. 2.
Fig. 2.

A ray-tracing picture of the generalized spectrometer. When the apparatus is aligned in an f - f configuration, the center wavelength of the output is chosen via the transverse position of the output slit and the shape of the optical spectrum is determined by the Fourier transform of the input spatial masking function. The colored lines represent the angular dispersion due to the diffraction grating.

Fig. 3.
Fig. 3.

As the longitudinal position of the output slit is varied, the slit selects different wavelength portions from each input spatial feature. In the spectral domain, each input spatial feature is mapped to a different portion of the optical spectrum. The sense of the mapping from space to time and optical frequency is determined by whether d 2 is greater or less than the focal length f of the spectrometer lens as illustrated in (a) where d 2 > f and (b) where d 2 < f.

Fig. 4.
Fig. 4.

For a perfectly aligned spectrometer (d 1 = d 2 = f in Fig. 1), the shape of the pass-band function is related to the Fourier transform of the input masking function. As the transverse position of the output fiber is changed, the passband function center wavelength changes while the overall shape is maintained. Here, the input masking function consists of 10 periodically-spaced spots and the center wavelength is varied from 1545 nm in (a) to 1555 nm in (c). The spatial period of ~3.3 mm corresponds to a modulation of ~0.75 nm (~94 GHz at λo = 1550) as shown by the sidebands in (a) – (c).

Fig. 5.
Fig. 5.

Space-to-wavelength conversion constant as a function of normalized lens position as calculated from Eq. (30) (solid line) and measured (circles) for our system. The mapping from space to wavelength varies over ~60% for a total change in lens position of 1.5 cm. The dashed lines show the valid range for the direct scaling assumption (d 1/ f <~ 0.83 or d 1/f >~ 1.17) implied by Eq. (25) and the spot spacing of ∆xmin ≈3.3 mm of our input patterns.

Fig. 6.
Fig. 6.

Spectral scaling properties of the generalized spectrometer. As the spectrometer lens position is varied, the imposed quadratic spatial phase causes the spectrometer passband to be scaled about the center wavelength. In (a)–(d) the measured space-to-wavelength conversion constant and lens distance from the grating (d 1 in Fig. 1) are (a) d 1 = 12.5cm, /dx′ =~ 0.74 nm/mm, (b) d 1 = 12.0cm, /dx′ =~ 0.91 nm/mm, (c) d 1 = 11.5cm, /dx′ =~ 1.08 nm/mm, and (d) d 1 = 11.0cm, /dx′ =~ 1.24 nm/mm. The solid curves are numerical fits to the spectral aperture determined by the mode of the output fiber.

Fig. 7.
Fig. 7.

Scaling of asymmetric spectral passbands about center wavelengths of ~1552 nm and ~1550 nm. The lens positions and space-to-wavelength conversion constants correspond to those of Fig. 6 (a) – (d).

Fig. 8.
Fig. 8.

The passband width and center wavelength may be tuned independently by the spectrometer lens position and output slit position. The space-to-wavelength conversion constant and center wavelength are: (a) /dx′ =~ 0.91 nm/mm, λs =1550.0 nm, (b) /dx′ =~ 1.08 nm/mm, λs =1548.0 nm, and (c) /dx′ =~ 1.24 nm/mm, λs =1544.0 nm.

Equations (32)

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E out ( x ) = P ( k o x f ) d x p ( x ) exp ( j k o x f x ) ,
k o = 2 π λ o ,
E out ( x , λ ) E in ( λ ) d x p ( x ) exp ( j k o f x x ) .
p ( x ) = m ( α x ) exp [ j k o ( λ λ o ) d θ D d λ x ] .
d θ D d λ = 1 d cos θ D o ,
α = cos θ i cos θ D o .
E out ( x , λ ) E in ( λ ) d x m ( α x ) exp { j k o f [ x f ( λ λ o ) d θ D d λ ] x } .
E out ( x , λ ) E in ( λ ) M { k o α f [ x f ( λ λ o ) d θ D d λ ] } ,
E out ( x s , λ ) E in ( λ ) M { k o α f [ x f ( λ λ o ) d θ D d λ ] } δ ( x x s ) .
x s = f [ λ s ( x s ) λ o ] d θ D d λ .
λ s ( x s ) = λ o + x s f ( d θ D d λ ) 1 .
E out ( x s , λ ) E in ( λ ) M { k o α [ λ s ( x s ) λ d cos θ D o ] } .
M = [ A B C D ] = [ 1 d 2 f f ( d 1 + d 2 ) d 1 d 2 f 1 f 1 d 1 f ] .
x s = A x + B θ i ,
θ i = θ D = ( λ λ o ) d θ D d λ .
x s = B [ λ s ( x s ) λ o ] d θ D d λ .
x = [ λ λ s ( x s ) ] B A ( d θ D d λ ) .
E ( x s , λ ) m { [ λ λ s ( x s ) ] α B A ( d θ D d λ ) } .
d λ ͂ d x = A α B ( d θ D d λ ) 1 ( nm mm ) ,
λ ͂ = λ λ s ( x s )
E ( λ ͂ ) m [ λ ͂ ( d λ ͂ d x ) 1 ] .
E out ( x , λ ) d x ' m ( α x ' ) exp ( j k o A 2 B x 2 ) exp ( j ξ x ) .
ξ = k o [ x B ( λ λ o ) d θ D d λ ] .
E out ( x , λ ) M ( ξ α ) * exp ( j B 2 k o A ξ 2 ) .
d 2 ϕ ( x ) d x 2 = k o A B α 2 π 2 2 Δ x min 2
E out ( x , λ ) d ξ M ( ξ α ) exp ( j B k o A ξ ξ ) .
E out ( x , λ ) m { α A [ x B ( λ λ o ) d θ D d λ ] } .
E out ( x s , λ ) m { α A [ x s B ( λ λ o ) d θ D d λ ] } .
x ' = [ λ λ s ( x s ) ] α B A ( d θ D d λ ) .
d λ ͂ d x = A α B ( d θ D d λ ) 1 ( nm mm ) .
d λ ͂ d x = f d 2 f ( d 1 + d 2 ) d 1 d 2 d cos θ D 2 cos θ i ( nm mm ) .
16.6 mm = 2 In 2 f λ π w o

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