Abstract

Current methods for generating Bessel beams are limited to fixed beam sizes or, in the case of conventional adaptive optics, relatively long switching times between beam shapes. We analyze the multiscale Bessel beams created using an alternative rapidly switchable device: a tunable acoustic gradient index (TAG) lens. The shape of the beams and their nondiffracting, self-healing characteristics are studied experimentally and explained theoretically using both geometric and Fourier optics. By adjusting the electrical driving signal, we can tune the ring spacings, the size of the central spot, and the working distance of the lens. The results presented here will enable researchers to employ dynamic Bessel beams generated by TAG lenses.

© 2008 Optical Society of America

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References

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    [CrossRef]
  5. V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
    [CrossRef]
  6. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  26. C. B. Arnold and E. McLeod, “A new approach to adaptive optics for materials processing,” Photonics Spectra 41, 78-84(2007).
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    [CrossRef]
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    [CrossRef]
  29. T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
    [CrossRef]
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  31. The thin lens approximation is valid for our system because the deflection that a ray experiences when traveling through the lens is <50 μm, which is much smaller than the acoustic wavelength within the lens, which is on the order of a few millimeters.
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  33. A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High speed varifocal lens using a tunable acoustic gradient index of refraction lens,” Opt. Lett. (to be published).
    [PubMed]
  34. E. McLeod and C. B. Arnold, “Complex beam sculpting with tunable acoustic gradient index lenses,” Proc. SPIE 6483, 64830I (2007).
    [CrossRef]

2007

C. B. Arnold and E. McLeod, “A new approach to adaptive optics for materials processing,” Photonics Spectra 41, 78-84(2007).

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102, 033104 (2007).
[CrossRef]

E. McLeod and C. B. Arnold, “Complex beam sculpting with tunable acoustic gradient index lenses,” Proc. SPIE 6483, 64830I (2007).
[CrossRef]

I. Grulkowski, D. Jankowski, and P. Kwiek, “Acousto-optic interaction of a Gaussian laser beam with an ultrasonic wave of cylindrical symmetry,” Appl. Opt. 46, 5870-5876 (2007).
[CrossRef] [PubMed]

2006

E. McLeod, A. B. Hopkins, and C. B. Arnold, “Multiscale Bessel beams generated by a tunable acoustic gradient index of refraction lens,” Opt. Lett. 31, 3155-3157 (2006).
[CrossRef] [PubMed]

P. Dufour, M. Pich, Y. de Koninck, and N. McCarthy, “Two-photon excitation fluorescence microscopy with a high depth of field using an axicon,” Appl. Opt. 45, 9246-9252 (2006).
[CrossRef] [PubMed]

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys. A 84, 423-430 (2006).
[CrossRef]

2005

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

2004

K. A. Higginson, M. A. Costolo, and E. A. Rietman, “Tunable optics derived from nonlinear acoustic effects,” J. Appl. Phys. 95, 5896-5904 (2004).
[CrossRef]

2003

K. Wang, L. Zeng, and C. Yin, “Influence of the incident wavefront on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216, 99-103(2003).
[CrossRef]

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537-624 (2003).
[CrossRef]

J. Amako, D. Sawaki, and E. Fujii, “Microstructuring transparent materials by use of nondiffracting ultrashort pulse beams generated by diffractive optics,” J. Opt. Soc. Am. B 20, 2562-2568 (2003).
[CrossRef]

2002

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27, 243-245(2002).
[CrossRef]

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

2001

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239-245 (2001).
[CrossRef]

2000

V. Magni, “Optimum beam for second harmonic generation,” Opt. Commun. 176, 245-251 (2000).
[CrossRef]

1999

1998

1996

1993

1992

1989

1988

1987

Amako, J.

Arimoto, R.

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Arnold, C. B.

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102, 033104 (2007).
[CrossRef]

E. McLeod and C. B. Arnold, “Complex beam sculpting with tunable acoustic gradient index lenses,” Proc. SPIE 6483, 64830I (2007).
[CrossRef]

C. B. Arnold and E. McLeod, “A new approach to adaptive optics for materials processing,” Photonics Spectra 41, 78-84(2007).

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

E. McLeod, A. B. Hopkins, and C. B. Arnold, “Multiscale Bessel beams generated by a tunable acoustic gradient index of refraction lens,” Opt. Lett. 31, 3155-3157 (2006).
[CrossRef] [PubMed]

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “Dynamic pulsed-beam shaping using a tag lens in the near UV,” Appl. Phys. A (in press).

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High speed varifocal lens using a tunable acoustic gradient index of refraction lens,” Opt. Lett. (to be published).
[PubMed]

Aruga, T.

Bouchal, Z.

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537-624 (2003).
[CrossRef]

Brown, D. L.

Carcole, E.

Cerda, S. C.

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Chen, N.-X.

Chen, Z.

Cizmar, T.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Cong, W.-X.

Costolo, M. A.

K. A. Higginson, M. A. Costolo, and E. A. Rietman, “Tunable optics derived from nonlinear acoustic effects,” J. Appl. Phys. 95, 5896-5904 (2004).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

de Koninck, Y.

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Ding, Z.

Dufour, P.

Durnin, J.

Eberly, J. H.

Friberg, A. T.

Fujii, E.

Garces-Chavez, V.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005.

Grulkowski, I.

Gu, B.-Y.

Guertin, J.

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401-1405 (1993).
[CrossRef] [PubMed]

Higginson, K. A.

K. A. Higginson, M. A. Costolo, and E. A. Rietman, “Tunable optics derived from nonlinear acoustic effects,” J. Appl. Phys. 95, 5896-5904 (2004).
[CrossRef]

Hopkins, A. B.

Huang, H.

Indebetouw, G.

Inoue, T.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys. A 84, 423-430 (2006).
[CrossRef]

Jankowski, D.

Kawata, S.

Kizuka, Y.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys. A 84, 423-430 (2006).
[CrossRef]

Kwiek, P.

Li, R.

Li, S. W.

Lin, Y.

Magni, V.

V. Magni, “Optimum beam for second harmonic generation,” Opt. Commun. 176, 245-251 (2000).
[CrossRef]

Matsuoka, Y.

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys. A 84, 423-430 (2006).
[CrossRef]

McCarthy, N.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

McLeod, E.

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102, 033104 (2007).
[CrossRef]

C. B. Arnold and E. McLeod, “A new approach to adaptive optics for materials processing,” Photonics Spectra 41, 78-84(2007).

E. McLeod and C. B. Arnold, “Complex beam sculpting with tunable acoustic gradient index lenses,” Proc. SPIE 6483, 64830I (2007).
[CrossRef]

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

E. McLeod, A. B. Hopkins, and C. B. Arnold, “Multiscale Bessel beams generated by a tunable acoustic gradient index of refraction lens,” Opt. Lett. 31, 3155-3157 (2006).
[CrossRef] [PubMed]

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “Dynamic pulsed-beam shaping using a tag lens in the near UV,” Appl. Phys. A (in press).

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High speed varifocal lens using a tunable acoustic gradient index of refraction lens,” Opt. Lett. (to be published).
[PubMed]

Melville, H.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Mermillod-Blondin, A.

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “High speed varifocal lens using a tunable acoustic gradient index of refraction lens,” Opt. Lett. (to be published).
[PubMed]

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “Dynamic pulsed-beam shaping using a tag lens in the near UV,” Appl. Phys. A (in press).

Nelson, J. S.

Pich, M.

Ren, H.

Rietman, E. A.

K. A. Higginson, M. A. Costolo, and E. A. Rietman, “Tunable optics derived from nonlinear acoustic effects,” J. Appl. Phys. 95, 5896-5904 (2004).
[CrossRef]

Saloma, C.

Sawaki, D.

Seka, W.

Sepulveda, K. V.

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Sibbett, W.

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Takabe, M.

Tanaka, T.

Tsai, T.

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

Turunen, J.

Vasara, A.

Wang, K.

K. Wang, L. Zeng, and C. Yin, “Influence of the incident wavefront on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216, 99-103(2003).
[CrossRef]

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401-1405 (1993).
[CrossRef] [PubMed]

Yin, C.

K. Wang, L. Zeng, and C. Yin, “Influence of the incident wavefront on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216, 99-103(2003).
[CrossRef]

Yoshikado, S.

Zemnek, P.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Zeng, L.

K. Wang, L. Zeng, and C. Yin, “Influence of the incident wavefront on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216, 99-103(2003).
[CrossRef]

Zhao, Y.

Appl. Opt.

Appl. Phys. A

Y. Matsuoka, Y. Kizuka, and T. Inoue, “The characteristics of laser micro drilling using a Bessel beam,” Appl. Phys. A 84, 423-430 (2006).
[CrossRef]

A. Mermillod-Blondin, E. McLeod, and C. B. Arnold, “Dynamic pulsed-beam shaping using a tag lens in the near UV,” Appl. Phys. A (in press).

Appl. Phys. Lett.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemnek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86, 174101 (2005).
[CrossRef]

Contemp. Phys.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Czech. J. Phys.

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537-624 (2003).
[CrossRef]

J. Appl. Phys.

K. A. Higginson, M. A. Costolo, and E. A. Rietman, “Tunable optics derived from nonlinear acoustic effects,” J. Appl. Phys. 95, 5896-5904 (2004).
[CrossRef]

E. McLeod and C. B. Arnold, “Mechanics and refractive power optimization of tunable acoustic gradient lenses,” J. Appl. Phys. 102, 033104 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Opt. Commun.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239-245 (2001).
[CrossRef]

V. Magni, “Optimum beam for second harmonic generation,” Opt. Commun. 176, 245-251 (2000).
[CrossRef]

K. Wang, L. Zeng, and C. Yin, “Influence of the incident wavefront on intensity distribution of the nondiffracting beam used in large-scale measurement,” Opt. Commun. 216, 99-103(2003).
[CrossRef]

Opt. Lett.

Photonics Spectra

C. B. Arnold and E. McLeod, “A new approach to adaptive optics for materials processing,” Photonics Spectra 41, 78-84(2007).

Phys. Rev. A

V. Garces-Chavez, K. V. Sepulveda, S. C. Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66, 063402 (2002).
[CrossRef]

Phys. Rev. Lett.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401-1405 (1993).
[CrossRef] [PubMed]

Proc. SPIE

E. McLeod and C. B. Arnold, “Complex beam sculpting with tunable acoustic gradient index lenses,” Proc. SPIE 6483, 64830I (2007).
[CrossRef]

T. Tsai, E. McLeod, and C. B. Arnold, “Generating Bessel beams with a tunable acoustic gradient index of refraction lens,” Proc. SPIE 6326, 63261F (2006).
[CrossRef]

Other

The thin lens approximation is valid for our system because the deflection that a ray experiences when traveling through the lens is <50 μm, which is much smaller than the acoustic wavelength within the lens, which is on the order of a few millimeters.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005.

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Figures (12)

Fig. 1
Fig. 1

(Color online) Components of the TAG lens.

Fig. 2
Fig. 2

(Color online) Characteristic TAG-generated multiscale Bessel beams. (a) and (b) each show two major rings plus the central major spot. (a) The pattern at a low driving amplitude ( 30 V ) without minor rings. (b) The pattern at a higher driving amplitude ( 65 V ) with many minor rings. These images are both taken 50 cm behind the lens with driving frequency 257 kHz .

Fig. 3
Fig. 3

(a) Experimental setup used to study the TAG beam characteristics. (b) The coordinate system used.

Fig. 4
Fig. 4

(Color online) (a) Predicted index profile at one instant in time with a linear approximation to the central peak (red dashed line). (b) The predicted index profile one half-period later in time. A linear approximation is made to the central annular peak (red dashed line). The scale of the spatial axis is set by the driving frequency, in this case, 497.5 kHz . (c) and (d) The theoretical predictions for the instantaneous intensity patterns corresponding to (a) and (b) observed with 355 nm laser light 50 cm behind the TAG lens with n A = 1.5 × 10 5 . Scale bars are 2 mm long. (e) and (f) The stroboscopic experimental images obtained in conditions identical to (c) and (d) with the laser repetition rate synchronized to the TAG driving frequency. The TAG lens driving amplitude is 5 V .

Fig. 5
Fig. 5

Simulated Fraunhofer far field diffraction pattern corresponding to the state of the TAG lens in Fig. 4a. The TAG lens is illuminated with a Gaussian plane wave with a 1 / e 2 width, w, given by the location of the maximum absolute gradient in the refractive index. The intensity is normalized so the peak value is 1.

Fig. 6
Fig. 6

Illustrates the experimentally determined time-average intensity enhancement and the propagation of the TAG central spot and the first major ring. The x and z axes have significantly different scales. Note the characteristic fringe patterns emanating from each peak in the index profile (see Fig. 4). This image was acquired by driving the lens at 257 kHz with an amplitude of 37.2 V . Images are taken every 10 cm , azimuthally averaged, then interpolated along the z axis to yield the above continuous plot.

Fig. 7
Fig. 7

(Color online) Experimental and theoretical intensity profile of the TAG beam imaged 70 cm behind the lens. The TAG lens is driven at 257 kHz with an amplitude of 37.2 V . For the theory the value of n A is 4 × 10 5 . Note that the way the fringe patterns extend is similar to what one would expect from an axicon.

Fig. 8
Fig. 8

(Color online) Beam divergence of the theoretical TAG, experimental TAG, Gaussian, and exact Bessel beams. The TAG and Gaussian beams achieve their maximum intensity approximately 58 cm behind the lens, and all beams have the same beam width at this location. The TAG lens is driven at 257 kHz with an amplitude of 37.2 V . For the theory the value of n A is 4 × 10 5 .

Fig. 9
Fig. 9

Propagation similar to Fig. 6 with a 1.25 mm diameter circular obstruction placed 27 cm behind the lens. The TAG lens is driven at 332.1 kHz with an amplitude of 5 V .

Fig. 10
Fig. 10

(Color online) Experimental and theoretical locations of the first major ring as a function of driving frequency. The solid curve represents the theory given by Eq. (13). The red squares represent this theory but also account for the deflection in optical propagation due to the asymmetry of the refractive index on either side of the major ring. The remaining black symbols represent experimental results from various trials.

Fig. 11
Fig. 11

(a) Experimental variation in the intensity enhancement 50 cm behind the lens as a function of driving amplitude. The TAG lens is driven at 257 kHz . Images are taken at 26 different voltages, azimuthally averaged, then interpolated along the voltage axis to yield the above continuous plot. (b) Theoretical variation in the intensity enhancement 50 cm behind the lens as a function of driving amplitude when driving the lens at 257 kHz . Note the agreement with (a). The colormap has been scaled down for clarity. The actual peak intensity is 51.

Fig. 12
Fig. 12

(Color online) Experimental and theoretical central spot size as a function of driving amplitude when the lens is driven at 257 kHz and the beam is imaged 50 cm behind the lens. Error bars represent the size of a camera pixel.

Equations (13)

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n ( ρ , t ) = n 0 + n A J 0 ( ω ρ c s ) sin ( ω t ) ,
t l ( ξ , η ) = exp ( i k 0 ( n L + L 0 L ) ) ,
n ( ρ ) ( L 0 + z ˜ ( ρ ) ) = C z ˜ ( ρ ) = C n ( ρ ) L 0 ,
tan ( θ ˜ ( ρ ) ) = z ˜ ( ρ ) ρ = C n 2 ( ρ ) n ( ρ ) ρ = L 0 + z ˜ ( ρ ) n ( ρ ) n ( ρ ) ρ L 0 n ( ρ ) n ( ρ ) ρ .
sin ( θ ( ρ ) ) = n ( ρ ) sin ( θ ˜ ( ρ ) ) ,
θ ( ρ ) = L 0 n ( ρ ) ρ .
sin ( ϕ + θ a x ) = n 0 sin ϕ .
α = π 2 ϕ = π + 2 L 0 n 0 1 n ρ ,
U TAG ( ξ , η ) = t l ( ξ , η ) U 0 ( ξ , η ) ,
U img ( x , y , z ) = z i λ U TAG ( ξ , η ) exp ( i k 0 s ( x , y , ξ , η ) ) s 2 ( x , y , ξ , η ) d ξ d η ,
s = z 2 + ( x ξ ) 2 + ( y η ) 2 .
I img ( x , y , z ) = 1 2 ϵ 0 μ 0 | U img ( x , y , z ) | 2 .
ρ * = 3.832 c s ω ,

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