Abstract

We report a mathematically rigorous technique which facilitates the optimization of various optical properties of electromagnetic fields in free space and including scattering interactions. The technique exploits the linearity of electromagnetic fields along with the quadratic nature of the intensity to define specific Optical Eigenmodes (OEi) that are pertinent to the interaction considered. Key applications include the optimization of the size of a focused spot, the transmission through sub-wavelength apertures, and of the optical force acting on microparticles. We verify experimentally the OEi approach by minimising the size of a focused optical field using a superposition of Bessel beams.

© 2011 Optical Society of America

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  1. C. Cohen-Tannoudji, Quantum Mechanics (Wiley, New York, 1977).
  2. M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. 73, 1–23 (1966).
    [CrossRef]
  3. A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211 (1993).
    [CrossRef]
  4. P. Bienstman, and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
    [CrossRef]
  5. J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
    [CrossRef]
  6. H. Kogelnik, and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
    [CrossRef] [PubMed]
  7. J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989).
    [CrossRef]
  8. M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A: Pure Appl. Opt. 11, 094005 (2009).
    [CrossRef]
  9. F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
    [CrossRef] [PubMed]
  10. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
    [CrossRef] [PubMed]
  11. M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
    [CrossRef]
  12. L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008).
    [CrossRef]
  13. R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley-VCH, 2008).
  14. T. Sales, and G. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
    [CrossRef] [PubMed]
  15. M. Berry, “Faster than Fourier,” in Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov, J. S. Anandan and J. L. Safko, eds., (World Scientific, Singapore, 1994), pp. 55–65.
    [PubMed]
  16. M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008).
    [PubMed]
  17. J. Durnin, “Exact solutions for nondiffracting beams. I. The Scalar Theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  18. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
    [CrossRef]
  19. B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. Lond. A 253, 357–379 (1959).
  20. R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913 (2007).
  21. D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992), 2nd ed.
  22. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
    [CrossRef]
  23. I. Vellekoop, and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
    [CrossRef]
  24. T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
    [CrossRef]

2010 (2)

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

2009 (1)

M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A: Pure Appl. Opt. 11, 094005 (2009).
[CrossRef]

2008 (4)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

I. Vellekoop, and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
[CrossRef]

L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008).
[CrossRef]

M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008).
[PubMed]

2007 (1)

2005 (1)

F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

2002 (1)

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

2001 (1)

P. Bienstman, and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[CrossRef]

1998 (1)

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

1997 (2)

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

T. Sales, and G. Morris, “Fundamental limits of optical superresolution,” Opt. Lett. 22, 582–584 (1997).
[CrossRef] [PubMed]

1993 (1)

A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211 (1993).
[CrossRef]

1989 (1)

J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989).
[CrossRef]

1987 (1)

1966 (2)

H. Kogelnik, and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
[CrossRef] [PubMed]

M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. 73, 1–23 (1966).
[CrossRef]

1959 (1)

B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. Lond. A 253, 357–379 (1959).

Alexander, D.

J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989).
[CrossRef]

Arlt, J.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Assion, A.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Baets, R.

P. Bienstman, and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[CrossRef]

Barton, J.

J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989).
[CrossRef]

Baumert, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Bergt, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Bienstman, P.

P. Bienstman, and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[CrossRef]

Brixner, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Chavez-Cerda, S.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Cižmár, T.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Courtial, J.

Dennis, M. R.

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

M. R. Dennis, A. C. Hamilton, and J. Courtial, “Superoscillation in speckle patterns,” Opt. Lett. 33, 2976–2978 (2008).
[PubMed]

Dholakia, K.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Di Leonardo, R.

Durnin, J.

Forchel, A.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Garcés-Chávez, V.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

García-Vidal, F.

F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Gerber, G.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Hamilton, A. C.

Holleran, K. O.

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

Ianni, F.

Jack, B.

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

Kac, M.

M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. 73, 1–23 (1966).
[CrossRef]

Kiefer, B.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

King, R. P.

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

Knipp, P.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Kogelnik, H.

Li, T.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Martín-Moreno, L.

F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Mazilu, M.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A: Pure Appl. Opt. 11, 094005 (2009).
[CrossRef]

L. C. Thomson, G. Whyte, M. Mazilu, and J. Courtial, “Simulated holographic three-dimensional intensity shaping of evanescent-wave fields,” J. Opt. Soc. Am. B 25, 849–853 (2008).
[CrossRef]

Moreno, E.

F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Morris, G.

Mosk, A.

I. Vellekoop, and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
[CrossRef]

Padgett, M. J.

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

Porto, J.

F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Reinecke, T.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Reithmaier, J.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Richards, B.

B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. Lond. A 253, 357–379 (1959).

Röhner, M.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Ruocco, G.

Sales, T.

Schäfer, F.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Schaub, S.

J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989).
[CrossRef]

Seyfried, V.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Strehle, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Sudbo, A.

A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211 (1993).
[CrossRef]

Thomson, L. C.

Vellekoop, I.

I. Vellekoop, and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
[CrossRef]

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Whyte, G.

Wolf, E.

B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. Lond. A 253, 357–379 (1959).

Zull, H.

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

Am. Math. Mon. (1)

M. Kac, “Can one hear the shape of a drum?” Am. Math. Mon. 73, 1–23 (1966).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

J. Barton, D. Alexander, and S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4603 (1989).
[CrossRef]

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

M. Mazilu, “Spin and angular momentum operators and their conservation,” J. Opt. A: Pure Appl. Opt. 11, 094005 (2009).
[CrossRef]

J. Opt. B (1)

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4, S82–S89 (2002).
[CrossRef]

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

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

Nat. Photonics (2)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4, 388–394 (2010).
[CrossRef]

Nat. Phys. (1)

M. R. Dennis, R. P. King, B. Jack, K. O. Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6, 118 (2010).
[CrossRef]

Opt. Commun. (1)

I. Vellekoop, and A. Mosk, “Phase control algorithms for focusing light through turbid media,” Opt. Commun. 281, 3071–3080 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

P. Bienstman, and R. Baets, “Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers,” Opt. Quantum Electron. 33, 327–341 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

J. Reithmaier, M. Röhner, H. Zull, F. Schäfer, A. Forchel, P. Knipp, and T. Reinecke, “Size dependence of confined optical modes in photonic quantum dots,” Phys. Rev. Lett. 78, 378–381 (1997).
[CrossRef]

F. García-Vidal, E. Moreno, J. Porto, and L. Martín-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic Systems,” Proc. R. Soc. Lond. A 253, 357–379 (1959).

Pure Appl. Opt. (1)

A. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2, 211 (1993).
[CrossRef]

Science (1)

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919 (1998).
[CrossRef] [PubMed]

Other (4)

C. Cohen-Tannoudji, Quantum Mechanics (Wiley, New York, 1977).

R. Paschotta, Encyclopedia of Laser Physics and Technology (Wiley-VCH, 2008).

M. Berry, “Faster than Fourier,” in Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov, J. S. Anandan and J. L. Safko, eds., (World Scientific, Singapore, 1994), pp. 55–65.
[PubMed]

D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992), 2nd ed.

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Figures (10)

Fig. 1
Fig. 1

(a) Transversal and (b) longitudinal 2D intensity cross sections of the OEi superposition delivering the smallest focal spot in the ROI (R = λ) considering 25 LG modes. w/w0 is the relative spot size measured according to Eq. (3). The Strehl ratio in (a) is 4.5%.

Fig. 2
Fig. 2

(a) Spot size as a function of the radius of the ROI for different number of LG modes considered. The right hand scale and filled curve indicate the numbers of intensity eigenmodes N(0) fulfilling the intensity criteria for the N = 11 case. The arrows indicate the corresponding scales. (b) Ratio between the ROI intensity of the smallest spot size eigenmode and the largest intensity achievable in the ROI (Strehl ratio).

Fig. 3
Fig. 3

(a) Radial wavevector spectral density. Yellow highlights regions outside the spectral bandwidth. (b) Transversal cross section of the OEi spot size optimized field intensity with yellow showing super-oscillating regions.

Fig. 4
Fig. 4

Intensity cross sections: (a) Airy disk for the maximum numerical aperture considered NA = sin(θmax) = 0.1. The yellow dashed circle shows the position of the smallest zero-intensity circle taken as the ROI inside which the spot size is calculated. The spot size is normalized to the spot size of the reference Bessel beam. (b) Reference Bessel beam corresponding to the largest cone angle θmax. The spot size of the reference Bessel beam is denoted as wB. (c) OEi spot size optimized beam for a superposition of Bessel beams (θ ∈ [0, θmax]) for a large ROI highlighted by the dashed yellow circle. Strehl ratio: 2%. (d) OEi spot size optimized beam for a small ROI. Strehl ratio: 0.2%. The gray-scaled region shows the sidebands while the color range the ROI. Notice that the two scales are different.

Fig. 5
Fig. 5

(a) Relative spot size Δr/w B of the Bessel beam superposition as a function of the relative ROI radius R/RB. The spot size wB and the ROI radius RB are associated with the reference Bessel beam shown in Fig. 4(b), where the ROI is indicated as dashed circle. For comparison, the red dot indicates the location of the reference beam in the Δr /w B vs. R/RB plot. (b) Strehl ratio vs relative ROI radius R/RB.

Fig. 6
Fig. 6

Experimental setup. FP = focal plane, L = Lens. Focal widths: f1 = 50 mm, f2 = 500 mm, f3 = f4 = 400 mm, f5 = 1 m. Laser: JDS Uniphase HeNe laser, Pmax = 10 mW, λ = 633 nm, SLM: Holoeye HEO 1080 P dual display system, resolution = 1920 pixel × 1080 pixel, display size = 1 in × 0.7 in. CCD camera: Basler pilot piA640-210gm, resolution = 648 pixel × 488 pixel, pixel size = 7.4 μm × 7.4 μm.

Fig. 7
Fig. 7

SLM encoded field modulations and resulting beam profiles. (a) Ring mask RGB image as encoded onto the dual panel SLM. (b) Associated Bessel beam created in the CCD camera plane. (c) Aperture RGB image as encoded onto the dual panel SLM. (d) Associated Airy disk as detected by the CCD camera. The yellow bar in (b) represents 2 times the spot size wB of the Bessel beam’s central core. w in (d) is the spot size of the Airy disk.

Fig. 8
Fig. 8

Experimental OEi spot size minimization. Top row: INum-S(z2) for different ROI radii in pixel as indicated in the top left corner of all graphs shown. The ROI is exemplary indicated as a dashed ring in the left hand side intensity distribution. The number in the bottom left corner represents the spot size w in units of the reference spot size wB. Central row: Optimized experimental distribution as RGB encoded onto the SLM. Bottom row: Intensity distributions IExp-S(z2). The relative spot size w/wB is indicated in the lower left corner.

Fig. 9
Fig. 9

(a–c) Cross section plot of the electric field amplitude, |E|, for a sub-wavelength aperture (diameter=200nm) in a thin layer of silver (thickness=200nm, refractive index n = 0.12 – 3.7i at a wavelength λ = 600nm) illuminated from below. The yellow lines represent the boundary of the structure. (a) Intensity OEi ensuring the largest transmission (transmission enhancement factor 2.1 with respect to the tightest Bessel beam and 1.55 with respect to the Airy disk illumination). (b) Incident intensity OEi without the structure. (c) Tightest Bessel beam illumination. (d–f) Electric field amplitude, |E|, in a cross section for a high refractive index (n = 1.8) microparticle (diameter=800nm) illuminated from below with a wavelength (λ = 504nm). (d) Momentum OEi ensuring the largest momentum transfer (enhancement factor 49.3 with respect to the plane wave and 1.33 with respect the Airy disk illumination). (e) Incident momentum OEi without the structure. (f) Plane wave illumination.

Fig. 10
Fig. 10

(a) Comparison between the enhancement factor achieved using standard phase front correction techniques (in black [23, 24]), the phase front correction from the intensity eigenmode (in red) and from the smallest spot size eigenmode (in blue). (b) Normalised beam spot size for the standard phase front correction techniques (in black [23, 24]) and from the smallest spot size eigenmode (in blue). ω0 corresponds to the spot size of the Airy disk.

Tables (1)

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Table 1 Time averaged quadratic measures m of common light-matter interactions. The integration either over a volume V or a surface S which in general corresponds to the Range of interest = ROI of the measure. In the optical momentum case, it corresponds to a closed surface surrounding the scattering object with F · u representing the optical force in the direction defined by the unit vector u. For surface integrals, n is the normal unit vector to the surface considered. The definition of the electromagnetic energy density is ℰ = 1/2(ε0E · E* + μ0H · H*)

Equations (8)

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m ( A ) ( E , H ) = a M ( A ) a
4 M j k ( A ) = m ( A ) ( E j + E k , H k + H k ) i m ( A ) ( E j + i E k , H k + i H k ) m ( A ) ( E j E k , H k H k ) + i m ( A ) ( E j i E k , H k i H k ) .
w = 2 m ( 2 ) m ( 0 ) = 2 a M ( 2 ) a b M ( 0 ) b ,
{ E min , H min } = { j = 1 N k = 1 N v min , k ( 2 ) v k , j ( 0 ) λ k ( 0 ) . E j , j = 1 N k = 1 N v min , k ( 2 ) v k , j ( 0 ) λ k ( 0 ) . H j } .
M j k ( 0 ) = S E j * E k d σ
M j k ( 2 ) = S r 2 E j * E k d σ ,
E min = j = 1 N k = 1 N v min , k ( 2 ) v k , j ( 0 ) λ k ( 0 ) . E j .
E = E 0 exp ( i ϕ + i k t z ) ) ( ( α e x + β e y ) J ( k t r ) + i k t 2 k Z ( ( α + i β ) exp ( i ϕ ) J 1 ( k t r ) ( α i β ) exp ( i ϕ ) J + 1 ( k t r ) ) e z )

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