Abstract

Bessel beams feature a very large depth-of-focus (DOF) compared to conventional focusing schemes, but their central lobe carries only a small fraction of the total beam power, leading to a strongly reduced peak irradiance. This is problematic for power-limited applications, such as optical coherence tomography (OCT) or optical coherence microscopy, as it can result in a prohibitive reduction of the signal-to-noise ratio (SNR). Using scalar diffraction theory, we show that the trade-off between DOF and peak irradiance of Bessel beams depends solely on the Fresnel number N. We demonstrate the existence of a low-Fresnel-number regime, N<10, in which axicons with Gaussian illumination can generate energy-efficient Bessel beams with a small number of sidelobes. In the context of OCT, this translates into DOF enhancements of up to 13× for a SNR penalty below 20 dB, which is confirmed by our experiments. We expect that these findings will enable improved performance of optical systems with extended DOF.

© 2014 Optical Society of America

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D. Lorenser, X. Yang, and D. D. Sampson, IEEE Photon. J. 5, 3900015 (2013).
[CrossRef]

2012

2011

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

2010

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2007

2006

2003

2002

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, Opt. Lett. 27, 243 (2002).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

1996

1987

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Amako, J.

Bachmann, A. H.

Bouma, B. E.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Chen, N.

Chen, Z.

Dholakia, K.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

Ding, Z.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, Nat. Photonics 4, 780 (2010).
[CrossRef]

Ferguson, R. A.

Friberg, A. T.

Fujii, E.

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

Gardecki, J. A.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Grimwood, A.

Hart, C.

Howe, W. C.

Lasser, T.

Lee, K. S.

Leitgeb, R. A.

Liu, C.

Liu, L.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

L. Liu, C. Liu, W. C. Howe, C. J. R. Sheppard, and N. Chen, Opt. Lett. 32, 2375 (2007).
[CrossRef]

Lorenser, D.

D. Lorenser, X. Yang, and D. D. Sampson, IEEE Photon. J. 5, 3900015 (2013).
[CrossRef]

D. Lorenser, X. Yang, and D. D. Sampson, Opt. Lett. 37, 1616 (2012).
[CrossRef]

McGloin, D.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

Melville, H.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Nadkarni, S. K.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Nelson, J. S.

Ren, H.

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, Nat. Photonics 4, 780 (2010).
[CrossRef]

Rolland, J. P.

Sampson, D. D.

D. Lorenser, X. Yang, and D. D. Sampson, IEEE Photon. J. 5, 3900015 (2013).
[CrossRef]

D. Lorenser, X. Yang, and D. D. Sampson, Opt. Lett. 37, 1616 (2012).
[CrossRef]

Sanchez-Losa, A.

Sawaki, D.

Seifert, A.

Sheppard, C. J. R.

Sibbett, W.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, Nat. Photonics 4, 780 (2010).
[CrossRef]

Spether, D.

Steinmann, L.

Tearney, G. J.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Tomlins, P. H.

Toussaint, J. D.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Villiger, M.

Weber, N.

Woolliams, P. D.

Yagi, Y.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Yang, X.

D. Lorenser, X. Yang, and D. D. Sampson, IEEE Photon. J. 5, 3900015 (2013).
[CrossRef]

D. Lorenser, X. Yang, and D. D. Sampson, Opt. Lett. 37, 1616 (2012).
[CrossRef]

Zapata-Rodriguez, C. J.

Zappe, H.

Zhao, Y.

Appl. Opt.

IEEE Photon. J.

D. Lorenser, X. Yang, and D. D. Sampson, IEEE Photon. J. 5, 3900015 (2013).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nat. Med.

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, Nat. Med. 17, 1010 (2011).
[CrossRef]

Nat. Photonics

F. O. Fahrbach, P. Simon, and A. Rohrbach, Nat. Photonics 4, 780 (2010).
[CrossRef]

Nature

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(See text for definition of symbols) (a) Normalized magnitude of the diffracted field amplitude from an axicon illuminated by a Gaussian beam, obtained via numerical solution of Eq. (1) for N=4. Red lines delineate the extent of the Bessel beam region predicted by geometrical optics. Dashed blue lines indicate the approximate extent of the focused Gaussian beam if the axicon were replaced by a lens with the same NA. (b) Transverse irradiance profile at the axial peak location of the field shown in (a). Comparison of the on-axis (c) and transverse (d) irradiance distributions of the numerical solution of Eq. (1) and the analytical approximation given by Eq. (2). The corresponding curves for the focused Gaussian beam of same NA are also shown. (e) Power in the central lobe of a Bessel beam as a function of Fresnel number. (f) SNR penalty and DOF gain factor of a Bessel beam compared to a Gaussian beam of same NA as a function of Fresnel number. The circles indicate values calculated via numerical solution of Eq. (1) and the lines represent the analytical expressions obtained via Eq (2).

Fig. 2.
Fig. 2.

Experimental setup for OCT imaging using low-Fresnel-number Bessel beams. DC, dispersion compensator; L, lens; M, mirror; PC, polarization controller; SF, spatial filter; other abbreviations defined in text.

Fig. 3.
Fig. 3.

Experimental results comparing the imaging performance of low-Fresnel-number Bessel beams and a Gaussian beam of same transverse resolution. (a) Measured beam profiles. (b) Normalized on-axis irradiance showing the DOF. (c) OCT images of the PSF phantom. Red bars indicate the DOF based on (b) and assuming a refractive index of n=1.49 for epoxy. (d) Scatter plots of the SNR values of a large number of point scatterers from the PSF phantom image data, indicating the SNR penalties of the three Bessel configurations relative to the Gaussian. The dashed red curves are fits of the assumed on-axis irradiance profiles to a subset of the point cloud comprising only the strongest signals (see text for details). (e) OCT images of a lemon sample. Red bars indicate the DOF based on (b) and assuming a refractive index of n=1.33 for water.

Tables (1)

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Table 1. Experimental Comparison Bessel versus Gaussian

Equations (4)

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t(ρ)=Uill(ρ)exp(ikβρ)=2π1wexp[(ρw)2]exp(ikβρ),
U(ρ˜,z˜)=i22πwNz˜exp[iπN(ρ˜2z˜z˜)]×0exp[(ρ˜0z˜)2]exp[iπNz˜(ρ˜01)2]J0(2πNρ˜ρ˜0)ρ˜0dρ˜0,
I(ρ˜,z˜)=8πw2Nz˜exp(2z˜2)J02(2πNρ˜).
SNRpenalty(dB)=20log(2eN)=20log(1.213N),

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