Abstract

The analytical solution is derived, within the Rayleigh–Sommerfeld formulation of diffraction, for the on-axis spectral irradiance of a broadband source after diffracting through a circular symmetric hard aperture. By using this solution, and within the paraxial approximation, we investigate several diffraction-induced effects originated by binary diffractive optical elements made up of a set of annular apertures with equal areas and periodic in the squared radial coordinate. In particular, the ability to focus femtosecond pulses is investigated. In addition, the analysis of the spectral modifier function associated with these elements allows us to simulate spectral shifts at focus positions. Finally, we introduce a relatively simple and low-cost technique to slice the spectrum of a broadband source in order to generate narrow bands or wavelength channels.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. H. C. Kandpal, "Experimental observation of the phenomenon of spectral switch," J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
    [CrossRef]
  5. J. T. Foley and E. Wolf, "Phenomenon of spectral switches as a new effect in singular optics with polychromatic light," J. Opt. Soc. Am. A 19, 2510-2516 (2002).
    [CrossRef]
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    [CrossRef]
  7. L. E. Helseth, "Spectral density of polychromatic electromagnetic waves," Phys. Rev. E 73, 026602 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. J. S. Lee and C. S. Shim, "Characteristic of spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photon. Technol. Lett. 7, 905-907 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2007 (2)

J. D. Taylor, L. R. Chen, and X. Gu, "Simple reconfigurable photonic microwave filter using an arrayed waveguide grating and fiber Bragg gratings," IEEE Photon. Technol. Lett. 19, 510-512 (2007).
[CrossRef]

O. Mendoza-Yero, G. Mínguez-Vega, J. Lancis, M. Fernández-Alonso, and V. Climent, "On-axis diffraction of an ultrashort light pulse by circularly symmetric hard apertures," Opt. Express 15, 4546-4556 (2007).
[CrossRef] [PubMed]

2006 (6)

C. J. Zapata-Rodríguez, "Temporal effects in ultrashort pulsed beams focused by planar diffracting elements," J. Opt. Soc. Am. A 23, 2335-2341 (2006).
[CrossRef]

H. Zhang, J. Li, D. W. Doerr, and D. R. Alexander, "Diffraction characteristics of a Fresnel zone plate illuminated by 10fs laser pulses," Appl. Opt. 45, 8541-8546 (2006).
[CrossRef] [PubMed]

G. Mínguez-Vega, J. Lancis, J. Caraquitena, and V. Torres-Company, "High spatiotemporal resolution in multifocal processing with femtosecond laser pulses," Opt. Lett. 31, 2631-2633 (2006).
[CrossRef] [PubMed]

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, "Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities," Appl. Phys. Lett. 89, 041119 (2006).
[CrossRef]

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Shimmel, and F. Vyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

L. E. Helseth, "Spectral density of polychromatic electromagnetic waves," Phys. Rev. E 73, 026602 (2006).
[CrossRef]

2004 (3)

J. Pu, C. Cai, and S. Nemoto, "Spectral anomalies in Young's double-slit interference experiment," Opt. Express 12, 5131-5139 (2004).
[CrossRef] [PubMed]

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture," Optik (Stuttgart) 115, 447-454 (2004).
[CrossRef]

S. Mansoori and A. Mitchell, "RF transversal filter using an AOTF," IEEE Photon. Technol. Lett. 16, 879-881 (2004).
[CrossRef]

2003 (3)

2002 (3)

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. Popescu and A. Dogariu, "Spectral anomalies at wave-front dislocations," Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

J. T. Foley and E. Wolf, "Phenomenon of spectral switches as a new effect in singular optics with polychromatic light," J. Opt. Soc. Am. A 19, 2510-2516 (2002).
[CrossRef]

2001 (4)

H. C. Kandpal, "Experimental observation of the phenomenon of spectral switch," J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
[CrossRef]

D. Pastor, J. Capmany, and B. Ortega, "Broad-band tunable microwave transversal notch filter based on tunable uniform fiber Bragg gratings as slicing filters," IEEE Photon. Technol. Lett. 13, 726-729 (2001).
[CrossRef]

T. Tomaru, "Two-element-cavity femtosecond Cr4+:YAG laser operating at a 2.6GHz repetition rate," Opt. Lett. 26, 1439-1441 (2001).
[CrossRef]

L. Boivin and B. C. Collings, "Spectrum slicing of coherent sources in optical communications," Opt. Fiber Technol. 7, 1-20 (2001).
[CrossRef]

2000 (1)

J. Pu and S. Nemoto, "Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture," IEEE J. Quantum Electron. 36, 1407-1411 (2000).
[CrossRef]

1999 (1)

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

1998 (1)

R. Mellish, S. V. Chernikov, P. M. W. French, and J. E. Taylor, "All-solid-state compact high repetition rate modelocked Cr4+:YAG laser," Electron. Lett. 34, 552-553 (1998).
[CrossRef]

1995 (1)

J. S. Lee and C. S. Shim, "Characteristic of spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photon. Technol. Lett. 7, 905-907 (1995).
[CrossRef]

1994 (1)

J. S. Lee, Y. C. Chung, and C. S. Shim, "Bandwidth optimization of a spectrum-slicing fiber amplifier source using an angle-tuned Fabry-Perot filter and a double-stage structure," IEEE Photon. Technol. Lett. 6, 1197-1199 (1994).
[CrossRef]

1988 (1)

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Hornung, "LED spectral slicing for single-mode local loop applications," Electron. Lett. 24, 389-390 (1988).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

D. Ganic, J. W. M. Chon, and M. Gu, "Effect of numerical aperture on the spectral splitting feature near phase singularities of focused waves," Appl. Phys. Lett. 82, 1527-1528 (2003).
[CrossRef]

S. P. Veetil, N. K. Viswanathan, C. Vijayan, and F. Wyrowski, "Spectral and temporal evolutions of ultrashort pulses diffracted through a slit near phase singularities," Appl. Phys. Lett. 89, 041119 (2006).
[CrossRef]

Electron. Lett. (2)

R. Mellish, S. V. Chernikov, P. M. W. French, and J. E. Taylor, "All-solid-state compact high repetition rate modelocked Cr4+:YAG laser," Electron. Lett. 34, 552-553 (1998).
[CrossRef]

M. H. Reeve, A. R. Hunwicks, W. Zhao, S. G. Methley, L. Bickers, and S. Hornung, "LED spectral slicing for single-mode local loop applications," Electron. Lett. 24, 389-390 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Pu and S. Nemoto, "Spectral shifts and spectral switches in diffraction of partially coherent light by a circular aperture," IEEE J. Quantum Electron. 36, 1407-1411 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

D. Pastor, J. Capmany, and B. Ortega, "Broad-band tunable microwave transversal notch filter based on tunable uniform fiber Bragg gratings as slicing filters," IEEE Photon. Technol. Lett. 13, 726-729 (2001).
[CrossRef]

J. D. Taylor, L. R. Chen, and X. Gu, "Simple reconfigurable photonic microwave filter using an arrayed waveguide grating and fiber Bragg gratings," IEEE Photon. Technol. Lett. 19, 510-512 (2007).
[CrossRef]

S. Mansoori and A. Mitchell, "RF transversal filter using an AOTF," IEEE Photon. Technol. Lett. 16, 879-881 (2004).
[CrossRef]

J. S. Lee, Y. C. Chung, and C. S. Shim, "Bandwidth optimization of a spectrum-slicing fiber amplifier source using an angle-tuned Fabry-Perot filter and a double-stage structure," IEEE Photon. Technol. Lett. 6, 1197-1199 (1994).
[CrossRef]

J. S. Lee and C. S. Shim, "Characteristic of spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photon. Technol. Lett. 7, 905-907 (1995).
[CrossRef]

J. Mod. Opt. (1)

S. P. Veetil, C. Vijayan, D. K. Sharma, H. Shimmel, and F. Vyrowski, "Diffraction induced space-time splitting effects in ultra-short pulse propagation," J. Mod. Opt. 53, 1819-1828 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

H. C. Kandpal, "Experimental observation of the phenomenon of spectral switch," J. Opt. A, Pure Appl. Opt. 3, 296-299 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. Pu, H. Zhang, and S. Nemoto, "Spectral shifts and spectral switches of partially coherent light passing through an aperture," Opt. Commun. 162, 57-63 (1999).
[CrossRef]

Opt. Express (2)

Opt. Fiber Technol. (1)

L. Boivin and B. C. Collings, "Spectrum slicing of coherent sources in optical communications," Opt. Fiber Technol. 7, 1-20 (2001).
[CrossRef]

Opt. Lett. (3)

Optik (Stuttgart) (1)

Z. Liu and B. Lü, "Spectral shifts and spectral switches in diffraction of ultrashort pulsed beams passing through a circular aperture," Optik (Stuttgart) 115, 447-454 (2004).
[CrossRef]

Phys. Rev. E (1)

L. E. Helseth, "Spectral density of polychromatic electromagnetic waves," Phys. Rev. E 73, 026602 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

G. Gbur, T. D. Visser, and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. Popescu and A. Dogariu, "Spectral anomalies at wave-front dislocations," Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

PDOEs with (a) ε = 2 , (b) ε = 3 , and (c) ε = 4 .

Fig. 2
Fig. 2

Irradiance function versus the axial coordinate z. Solid curves, (a) ε = 3 and (b) ε = 4 ; diamonds, case of the Fresnel zone plate.

Fig. 3
Fig. 3

Normalized irradiance versus the axial coordinate z. Solid curves, monochromatic illumination using PDOEs with (a) ε = 2 , (b) ε = 3 , and (c) ε = 4 ; diamonds, case of femtosecond pulse illumination.

Fig. 4
Fig. 4

Spectra at different axial positions (indicated by the insets) for the third focus of PDOEs with ε = 3 and ε = 4 .

Fig. 5
Fig. 5

Spectra in the focus located at z = 12.67 cm of PDOEs with (a) ε = 2 and (b) ε = 4 .

Fig. 6
Fig. 6

Differences between axial irradiances derived within Fresnel (solid curves) and Rayleigh–Sommerfeld (diamonds) regimes, for (a) ε = 3 and (b) ε = 4 when ω = ω 0 .

Equations (8)

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A ( z , ω ) = U ( z , t ) exp ( i ω t ) d t.
U ( z , t ) = m = 1 N z z 2 + r i m 2 u 0 ( t z 2 + r i m 2 c ) m = 1 N z z 2 + r o m 2 u 0 ( t z 2 + r o m 2 c ) .
A ( z , ω ) = A 0 ( ω ) m = 1 N z z 2 + r i m 2 exp ( i ω z 2 + r i m 2 c ) A 0 ( ω ) m = 1 N z z 2 + r o m 2 exp ( i ω z 2 + r o m 2 c ) .
S ( z , ω ) = S 0 ( ω ) { z 2 m = 1 N n = 1 N cos [ ω c ( d i m d i n ) ] d i m d i n + cos [ ω c ( d o m d o n ) ] d o m d o n 2 cos [ ω c ( d i m d o n ) ] d i m d o n } ,
M ( z , ω ) = 1 + z 2 z 2 + a 2 2 z z 2 + a 2 cos [ ω c ( z z 2 + a 2 ) ] .
M ( z , ω ) = 4 m = 1 N n = 1 N sin [ ω ( r o n 2 r i n 2 ) 4 c z ] sin [ ω ( r o m 2 r i m 2 ) 4 c z ] cos [ ω ( r o n 2 r o m 2 ) 4 c z ω ( r i m 2 r i n 2 ) 4 c z ] .
M ( z , ω ) = 4 sin 2 ( ω Σ 0 4 π c z ) sin 2 ( N ω ε Σ 0 4 π c z ) sin 2 ( ω ε Σ 0 4 π c z ) .
I ( z , ω 0 ) = 0 M ( z , ω ) S 0 ( ω ) d ω .

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