Abstract

We report selective excitations of higher-order Hermite-Gaussian and Ince-Gaussian (IG) modes in a laser-diode-pumped microchip solid-state laser and controlled generation of corresponding higher-order and multiple optical vortex beams of different shapes using an astigmatic mode converter (AMC). Simply changing the pump-beam diameter, shape, and lateral off-axis position of the tight pump beam focus on the laser crystal within a microchip semispherical cavity can produce the desired optical vortex beams in a well controlled manner. Pattern changes featuring different IG and HG modes obtained by rotating the AMC are also demonstrated. Numerical simulation shows that the vortex structure is changed by controlled off-axis laser diode pumping, which could lead toward precise optical manipulation of small particles.

© 2008 Optical Society of America

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2007 (4)

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
[CrossRef] [PubMed]

S.-C. Chu and K. Otsuka, ""Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers," Opt. Express 15, 16506 (2007).
[CrossRef] [PubMed]

S. -C. Chu and K. Otsuka, "Stable donut-like vortex beam generation from lasers with controlled Ince-Gaussian modes," Appl. Opt. 46, (2007). http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-31-7709.
[CrossRef] [PubMed]

2006 (2)

2004 (3)

2002 (2)

2001 (3)

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

2000 (1)

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

1999 (2)

1998 (2)

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

1993 (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17, 221 (1992).
[CrossRef] [PubMed]

1983 (1)

1975 (1)

Allen, L.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Bandres, M. A.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Bentley, J. B.

Bhowmik, A.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Chu, S. -C.

S. -C. Chu and K. Otsuka, "Stable donut-like vortex beam generation from lasers with controlled Ince-Gaussian modes," Appl. Opt. 46, (2007). http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-31-7709.
[CrossRef] [PubMed]

Chu, S.-C.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

Courtial, J.

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

Davis, J. A.

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Elias, L. R.

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

Endo, M.

Fujioka, T.

Gahagan, K. T.

Gutiérrez-Vega, J. C.

Heckenberg, N. R.

Hill, W. T.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Jhe, W.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

Kamikariya, K.

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
[CrossRef] [PubMed]

Kawakami, M.

Kim, K.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

Kimel, I.

I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quantum Electron. 29, 2562 (1993).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Kwon, N.

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, "Revolving interference pattern for the rotation of optically trapped particles," Opt. Commun. 201, 21 (2002).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

McDuff, R.

Milam, D.

Nanri, K.

Nemoto, K.

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

Ohtomo, T.

Otsuka, K.

T. Ohtomo, K. Kamikariya, K. Otsuka, and S.-C. Chu, "Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers," Opt. Express 15, 10705 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-17-10705.
[CrossRef] [PubMed]

S.-C. Chu and K. Otsuka, ""Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers," Opt. Express 15, 16506 (2007).
[CrossRef] [PubMed]

K. Otsuka, K. Nemoto, K. Kamikariya, and S.-C. Chu, "Linearly-polarized, single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince-Gaussian mode operations," Jpn. J. Appl. Phys. 46, 5865 (2007).
[CrossRef]

S. -C. Chu and K. Otsuka, "Stable donut-like vortex beam generation from lasers with controlled Ince-Gaussian modes," Appl. Opt. 46, (2007). http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-31-7709.
[CrossRef] [PubMed]

Padgett, M. J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Piccirillo, B.

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum," Phys. Rev. Lett. 80, 013601 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

Santamato, E.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Sasso, A.

Schwarz, U. T.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912 (2001).
[CrossRef] [PubMed]

Siegman, A. E.

Simpson, N. B.

Smith, C. P.

Song, Y.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Swartzlander, G. A.

Sziklas, E. A.

Takeda, S.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Vella, A.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

White, A. G.

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

J. Opt. Soc. Am. A (1)

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[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313 (2001).
[CrossRef] [PubMed]

Opt. Commun. (3)

M. P. MacDonald, "Revolving interference pattern for the rotation of optically trapped particles," Opt. Commun. 201, 21 (2002).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

A. T. O’Neil and J. Courtial, "Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter," Opt. Commun. 181, 35 (2000).
[CrossRef]

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Opt. Photonics News (1)

G. A. Swartzlander Jr., "The Optical vortex lens," Opt. Photonics News 17, 39 (2006).
[CrossRef]

Optics Commun. (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phase plate," Optics Commun. 112, 321 (1994).
[CrossRef]

Phy. Rev. A (1)

X. Xu, K. Kim, W. Jhe, and N. Kwon, "Efficient optical guiding of trapped cold atoms by a hollow laser beam," Phy. Rev. A 63, 3401 (2001).
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

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T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, "Novel optical trap of atoms with a doughnut beam," Phys. Rev. Lett. 78, 4713 (1997).
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Supplementary Material (5)

» Media 1: MOV (1883 KB)     
» Media 2: MOV (1838 KB)     
» Media 3: MOV (2032 KB)     
» Media 4: MOV (2058 KB)     
» Media 5: MOV (1877 KB)     

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Figures (15)

Fig. 1.
Fig. 1.

Experimental setup.

Fig. 2.
Fig. 2.

Decomposition of the diagonal HG2,0 and LG2,0 modes.

Fig. 3.
Fig. 3.

Successive generation of LGl,0 modes with increasing d. Pump power P=182 mW. Threshold pump power: 36 mW.

Fig. 4.
Fig. 4.

(1884 KB) Structural change of the HG4,0 mode with a rotation of the AMC. P=182 mW. [Media 1]

Fig. 5.
Fig. 5.

Some analytic patterns of the Ince-Gaussian modes.

Fig. 6.
Fig. 6.

Approach from Ince-Gaussian to Hermite Gaussian modes with increasing ellipticity. (a) IGe p,p modes and (b) IGe p,0 and IGo p,1 . (Here, ellipticity ε : 4→103).

Fig. 7.
Fig. 7.

(1839 KB) Structural pattern change with AMC rotation for the IGo 3,1 mode. P=182 mW. [Media 2]

Fig. 8.
Fig. 8.

(2032 KB) Structural pattern change with AMC rotation for IGe 3,3 mode. P=182 mW. [Media 3]

Fig. 9.
Fig. 9.

(2000 KB) Structural pattern change with AMC rotation for HG3,1 mode. P=234 mW. [Media 4]

Fig. 10.
Fig. 10.

(1878 KB) Structural pattern change with AMC rotation for IGe 4,4 mode. P=234 mW. [Media 5]

Fig. 11.
Fig. 11.

Numerical result indicating a structural change in the HG4,0 mode intensity pattern with a rotation of AMC and the corresponding phase portrait of the donut-like multi-vortex at θ=45°.

Fig. 12.
Fig. 12.

Numerical result indicating a structural change in the IG° 3,1 mode intensity pattern with a rotation of AMC.

Fig. 13.
Fig. 13.

Numerical result indicating a structural change of the IGe 3,3 mode intensity pattern with a rotation of AMC.

Fig. 14.
Fig. 14.

Computed changes in phase portraits and interference fringes across θ=45°. (a) IGo 3,1 mode and (b) IGe 3,3 (right) mode. Points of phase singularity are indicated by red dots.

Fig. 15.
Fig. 15.

Computed structural change in vortex beam created from IG0 3,1 modes with different values of ellipticity parameter ε and θ=45°. (a) Intensity patterns of IG0 3,1 modes, (b) intensity profiles of vortices, (c) phase portraits, and (d) interference fringe patterns. Results for HG3,0 -originating vortices are shown in the bottom row for reference.

Tables (1)

Tables Icon

Table 1. Design of optical elements and their optical distances.

Equations (12)

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u l , p LG ( r , ϕ ) = k = 0 N B k iku N k , k HG ( x , y )
IG e p , m ( r , ε ) = C [ w 0 w ( z ) ] C p m ( i ξ , ε ) C p m ( η , ε ) exp [ r 2 w 2 ( z ) ]
× exp i [ kz + { kr 2 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
IG o p , m ( r , ε ) = S [ w 0 w ( z ) ] S p m ( i ξ , ε ) S p m ( η , ε ) exp [ r 2 w 2 ( z ) ]
× exp i [ kz + { kr 2 2 R ( z ) } ( p + 1 ) ψ z ( z ) ] ,
u σ l , p LG ( r , ϕ ) = k = 0 N D k u σ 2 l + p , k IG ( ξ , ε , η )
u σ p , m IG ( ξ , η , ε ) = l , p = 0 N D l , p u σ l , p LG ( r , ϕ )
g i ( x , y ) = g i 0 ( x , y ) ( 1 + I ~ i + ( x , y ) + I ~ i ( x , y ) I s ( x , y ) ) ,
I ~ i + ( x , y ) = ( 1 α ) i = 0 q α i I i + ( q i ) ,
I ~ i ( x , y ) = ( 1 α ) i = 0 q α i I i ( q i ) .
E i out ( x , y ) = E i in ( x , y ) exp [ 1 2 g i ( x , y ) d ] ,
E q + 1 ( x , y ) ~ E q ( x , y ) .

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