Abstract

We derive an explicit analytical relationship to describe the axial light intensity when a Gaussian beam is diffracted by the logarithmic axicon (LA). An evaluation formula for the effective radius of the diffraction pattern that we deduce shows the said radius to be in inverse proportion to the LA “force” parameter. The finite-difference time-domain-based simulation has shown that using the LA makes it possible to go beyond the diffraction limit: in the LA vicinity, the FWHM of the light beam can be as small as one fifth of the illumination wavelength.

© 2011 Optical Society of America

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  1. M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
    [CrossRef]
  2. J. Fu, H. Dong, and W. Fang, “Subwavelength focusing of light by a tapered microtube,” Appl. Phys. Lett. 97, 041114(2010).
    [CrossRef]
  3. V. V. Kotlyar and S. S. Stafeev, “Modeling the sharp focus of a radially polarized laser mode using a conical and a binary microaxicon,” J. Opt. Soc. Am. B 27, 1991–1997 (2010).
    [CrossRef]
  4. I. Golub, B. Chebi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett. 35, 2828–2830(2010).
    [CrossRef] [PubMed]
  5. M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).
  6. L. R. Staronski, J. Sochacki, Z. Jroszewicz, and A. Kolodziejcziwz, “Lateral distribution and flow of energy in uniform-intensity axicon,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
    [CrossRef]
  7. J. W. Lit and R. Tremblay, “Focal depth of a transmitting axicon,” J. Opt. Soc. Am. 63, 445–449 (1973).
    [CrossRef]
  8. W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference exited by a radially polarized beam,” Opt. Lett. 34, 722–724 (2009).
    [CrossRef] [PubMed]
  9. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math. Series (National Bureau of Standards, 1965).
  10. V. V. Kotlyar and A. A. Kovalev, “Family of hypergeometric laser beams,” J. Opt. Soc. Am. A 25, 262–270 (2008).
    [CrossRef]
  11. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28, 2440–2442 (2003).
    [CrossRef] [PubMed]
  12. V. V. Kotlyar and S. S. Stafeev, “Sharply focusing a radially polarized laser beam using a gradient Mikaelian’s microlens,” Opt. Commun. 282, 459–464 (2009).
    [CrossRef]
  13. RSoft Design Group, www.rsoftdesign.com.

2010

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

J. Fu, H. Dong, and W. Fang, “Subwavelength focusing of light by a tapered microtube,” Appl. Phys. Lett. 97, 041114(2010).
[CrossRef]

I. Golub, B. Chebi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett. 35, 2828–2830(2010).
[CrossRef] [PubMed]

V. V. Kotlyar and S. S. Stafeev, “Modeling the sharp focus of a radially polarized laser mode using a conical and a binary microaxicon,” J. Opt. Soc. Am. B 27, 1991–1997 (2010).
[CrossRef]

2009

W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference exited by a radially polarized beam,” Opt. Lett. 34, 722–724 (2009).
[CrossRef] [PubMed]

V. V. Kotlyar and S. S. Stafeev, “Sharply focusing a radially polarized laser beam using a gradient Mikaelian’s microlens,” Opt. Commun. 282, 459–464 (2009).
[CrossRef]

2008

2003

1992

1981

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

1973

1965

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math. Series (National Bureau of Standards, 1965).

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math. Series (National Bureau of Standards, 1965).

Bhuyan, M. K.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Chebi, B.

Chen, W.

Courvoisier, F.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Dong, H.

J. Fu, H. Dong, and W. Fang, “Subwavelength focusing of light by a tapered microtube,” Appl. Phys. Lett. 97, 041114(2010).
[CrossRef]

Duan, K.

Dudley, J. M.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Fang, W.

J. Fu, H. Dong, and W. Fang, “Subwavelength focusing of light by a tapered microtube,” Appl. Phys. Lett. 97, 041114(2010).
[CrossRef]

Fu, J.

J. Fu, H. Dong, and W. Fang, “Subwavelength focusing of light by a tapered microtube,” Appl. Phys. Lett. 97, 041114(2010).
[CrossRef]

Furfaro, L.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Golub, I.

Golub, M. A.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Jacquot, M.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Jroszewicz, Z.

Karpeev, S. V.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Kolodziejcziwz, A.

Kotlyar, V. V.

Kovalev, A. A.

Lacourt, P. A.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Lit, J. W.

Lü, B.

Nowacki, D.

Prokhorov, A. M.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Salut, R.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

Shaw, D.

Sisakyan, I. N.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Sochacki, J.

Soifer, V. A.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Stafeev, S. S.

V. V. Kotlyar and S. S. Stafeev, “Modeling the sharp focus of a radially polarized laser mode using a conical and a binary microaxicon,” J. Opt. Soc. Am. B 27, 1991–1997 (2010).
[CrossRef]

V. V. Kotlyar and S. S. Stafeev, “Sharply focusing a radially polarized laser beam using a gradient Mikaelian’s microlens,” Opt. Commun. 282, 459–464 (2009).
[CrossRef]

Staronski, L. R.

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math. Series (National Bureau of Standards, 1965).

Tremblay, R.

Zhan, Q.

Appl. Phys. Lett.

M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97, 081102 (2010).
[CrossRef]

J. Fu, H. Dong, and W. Fang, “Subwavelength focusing of light by a tapered microtube,” Appl. Phys. Lett. 97, 041114(2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

V. V. Kotlyar and S. S. Stafeev, “Sharply focusing a radially polarized laser beam using a gradient Mikaelian’s microlens,” Opt. Commun. 282, 459–464 (2009).
[CrossRef]

Opt. Lett.

Sov. Tech. Phys. Lett.

M. A. Golub, S. V. Karpeev, A. M. Prokhorov, I. N. Sisakyan, and V. A. Soifer, “Focusing light into a specified volume by computer-synthesized hologram,” Sov. Tech. Phys. Lett. 7, 264–266 (1981).

Other

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Applied Math. Series (National Bureau of Standards, 1965).

RSoft Design Group, www.rsoftdesign.com.

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Figures (10)

Fig. 1
Fig. 1

Axial intensity ( ρ = 0 ) produced by the focusing LA ( γ < 0 ), ( γ = 1 , w = λ , λ = 532 nm ) derived from Eq. (6) (solid curve) and the Rayleigh–Sommerfeld integral (dashed curve).

Fig. 2
Fig. 2

Intensity in the transverse plane z = 2 λ ( γ = 1 , w = λ , λ = 532 nm ) derived from the Rayleigh–Sommerfeld integral (curve 1) and the Fresnel integral (curve 2).

Fig. 3
Fig. 3

z dependence of the intensity in the vicinity of z = 0 and at near-zero values of ρ ( γ = 1 , w = λ , λ = 532 nm ).

Fig. 4
Fig. 4

Axial intensity of light field Eq. (10).

Fig. 5
Fig. 5

(a) Logarithmic microaxicon in the calculating domain and (b) magnified near-axis fragment thereof.

Fig. 6
Fig. 6

Radially polarized laser mode R - TEM 01 .

Fig. 7
Fig. 7

Intensity distribution along the z axis (the vertical dotted line is where the LA relief tops are found).

Fig. 8
Fig. 8

Intensity distribution in the focal spot.

Fig. 9
Fig. 9

Appearance of the 2D LA with radius w = 4 λ , microrelief height λ / ( n 0 1 ) = 2 λ , and γ = 20 .

Fig. 10
Fig. 10

Averaged intensity distribution directly behind the axicon.

Tables (1)

Tables Icon

Table 1 Radii of Rings in the Transverse Intensity Distribution

Equations (14)

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T ( r , ϕ ) = exp [ i γ ln ( r σ ) + i n ϕ ] ,
E 0 ( r , ϕ ) = exp [ ( r w ) 2 + i γ ln ( r σ ) + i n ϕ ] ,
E ( ρ , θ , z ) = ( i ) n + 1 n ! ( z 0 z ) ( w σ ) i γ ( k w ρ 2 z ) n × Γ ( n + 2 + i γ 2 ) ( 1 i z 0 z ) n + 2 + i γ 2 exp ( i n θ + i k ρ 2 2 z ) × F 1 1 [ n + 2 + i γ 2 ; n + 1 ; ( k w ρ 2 z ) 2 ( 1 i z 0 z ) 1 ] ,
I ( ρ , z ) = 1 ( n ! ) 2 ( z 0 z ) 2 ( k w ρ 2 z ) 2 n | Γ ( n + 2 + i γ 2 ) | 2 × ( 1 + z 0 2 z 2 ) n + 2 2 exp [ γ arctan ( z 0 z ) ] × | F 1 1 [ n + 2 + i γ 2 ; n + 1 ; ( k w ρ 2 z ) 2 ( 1 i z 0 z ) 1 ] | 2 .
I 0 ( ρ , z ) = z 0 2 z 2 + z 0 2 | Γ ( 1 + i γ 2 ) | 2 exp [ γ arctan ( z 0 z ) ] × | F 1 1 [ 1 + i γ 2 ; 1 ; ( k w ρ 2 z ) 2 ( 1 i z 0 z ) 1 ] | 2 .
I 0 ( z ) = ( π γ 2 ) sh 1 ( π γ 2 ) z 0 2 z 0 2 + z 2 exp [ π γ 2 + γ arctan ( z z 0 ) ] .
I 0 ( 0 ) = π γ [ exp ( π γ ) 1 ] 1 1 ( at   γ < 0 ) .
I 0 ( ρ , z ) = z 0 2 z 0 2 + z 2 exp [ 2 ρ 2 w 2 + 2 γ arctan ( z z 0 ) ] .
F 1 1 ( a ; c ; x ) = F ( c ) exp ( x ) Γ ( a ) x c a .
E 0 ( r , ϕ ) = { exp [ ( r w ) 2 + i γ ln ( r σ ) + i n ϕ ] , r r 0 , 1 , r < r 0 ,
x 1 = γ c 1 , 1 2 2 ( c 2 a ) ,
| x 1 | = γ c 1 , 1 2 2 | c 2 a | .
ρ 1 = 2 , 4 w [ 1 + z 2 z 0 2 2 ( 1 + γ 2 ) ] 1 / 4 .
ρ 1 2 w | γ | 1 / 2 .

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