Abstract

In recent years, there has been a mounting interest in better methods of measuring nanoscale objects, especially in fields such as nanotechnology, biomedicine, cleantech, and microelectronics. Conventional methods have proved insufficient, due to the classical diffraction limit or slow and complicated measuring procedures. The purpose of this paper is to explore the special characteristics of singular beams with respect to the investigation of subwavelength objects. Singular beams are light beams that contain one or more singularities in their physical parameters, such as phase or polarization. We focus on the three-dimensional interaction between electromagnetic waves and subwavelength objects to extract information about the object from the scattered light patterns.

© 2010 Optical Society of America

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References

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  1. B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
    [CrossRef] [PubMed]
  2. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  3. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  4. G. Toker, A. Brunfeld, and J. Shamir, “High resolution inspection for in-line surface testing,” in 1995 Display Manufacturing Technology Conference (Society for Information Display, 1995), pp. 119–120.
  5. G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
    [CrossRef]
  6. A. Normatov, B. Spektor, and J. Shamir, “Singular beam scanning microscopy: preliminary experimental results,” Opt. Eng. 49, 048001 (2010).
    [CrossRef]
  7. G. Videen, “Light-scattering problem of a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
    [CrossRef]
  8. K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
    [CrossRef]
  9. J. Shamir and N. Karasikov, “A method for particle size and concentration measurement,” U.S. patent 7,746,469(29 June 2010).
  10. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
  11. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
  12. A. S. van de Nes and P. Torok, “Rigorous analysis of spheres in Gauss–Laguerre beams,” Opt. Express 15, 13360–13374(2007).
    [CrossRef] [PubMed]
  13. G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443(1988).
    [CrossRef]
  14. C. W. Qiu and L. W. Li, “Electromagnetic scattering by 3-D general anisotropic objects: a Hertz–Debye potential formulation,” in Proceedings of the 2005 IEEE Antennas and Propagation Society International Symposium and USNC/URSI National Radio Science Meeting (IEEE, 2005), pp. 422–425.
    [PubMed]
  15. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A 17, 294–303 (2000).
    [CrossRef]
  16. A. E. Siegman, Lasers (University Science, 1986).
  17. http://refractiveindex.info/—refractive index database.
  18. A. Normatov, B. Spektor, Y. Leviatan, and J. Shamir, “Plasmonic resonance scattering from a silver nanowire illuminated by a tightly focused singular beam,” Opt. Lett. 35, 2729–2731 (2010)
    [CrossRef] [PubMed]

2010 (2)

2008 (1)

2007 (1)

2006 (1)

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

2002 (1)

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
[CrossRef]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

2000 (1)

1995 (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

1991 (1)

1988 (1)

Aptowicz, K. B.

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Brunfeld, A.

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
[CrossRef]

G. Toker, A. Brunfeld, and J. Shamir, “High resolution inspection for in-line surface testing,” in 1995 Display Manufacturing Technology Conference (Society for Information Display, 1995), pp. 119–120.

Chang, R. K.

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

Gouesbet, G.

Grehan, G.

Hill, S. C.

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

Karasikov, N.

J. Shamir and N. Karasikov, “A method for particle size and concentration measurement,” U.S. patent 7,746,469(29 June 2010).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

Leviatan, Y.

Li, L. W.

C. W. Qiu and L. W. Li, “Electromagnetic scattering by 3-D general anisotropic objects: a Hertz–Debye potential formulation,” in Proceedings of the 2005 IEEE Antennas and Propagation Society International Symposium and USNC/URSI National Radio Science Meeting (IEEE, 2005), pp. 422–425.
[PubMed]

Maheu, B.

Normatov, A.

Pan, Y. L.

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

Piestun, R.

Pinnick, R. G.

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

Qiu, C. W.

C. W. Qiu and L. W. Li, “Electromagnetic scattering by 3-D general anisotropic objects: a Hertz–Debye potential formulation,” in Proceedings of the 2005 IEEE Antennas and Propagation Society International Symposium and USNC/URSI National Radio Science Meeting (IEEE, 2005), pp. 422–425.
[PubMed]

Schechner, Y. Y.

Shamir, J.

A. Normatov, B. Spektor, and J. Shamir, “Singular beam scanning microscopy: preliminary experimental results,” Opt. Eng. 49, 048001 (2010).
[CrossRef]

A. Normatov, B. Spektor, Y. Leviatan, and J. Shamir, “Plasmonic resonance scattering from a silver nanowire illuminated by a tightly focused singular beam,” Opt. Lett. 35, 2729–2731 (2010)
[CrossRef] [PubMed]

B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
[CrossRef] [PubMed]

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
[CrossRef]

R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A 17, 294–303 (2000).
[CrossRef]

J. Shamir and N. Karasikov, “A method for particle size and concentration measurement,” U.S. patent 7,746,469(29 June 2010).

G. Toker, A. Brunfeld, and J. Shamir, “High resolution inspection for in-line surface testing,” in 1995 Display Manufacturing Technology Conference (Society for Information Display, 1995), pp. 119–120.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Spektor, B.

A. Normatov, B. Spektor, and J. Shamir, “Singular beam scanning microscopy: preliminary experimental results,” Opt. Eng. 49, 048001 (2010).
[CrossRef]

A. Normatov, B. Spektor, Y. Leviatan, and J. Shamir, “Plasmonic resonance scattering from a silver nanowire illuminated by a tightly focused singular beam,” Opt. Lett. 35, 2729–2731 (2010)
[CrossRef] [PubMed]

B. Spektor, A. Normatov, and J. Shamir, “Singular beam microscopy,” Appl. Opt. 47, A78–A87 (2008).
[CrossRef] [PubMed]

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
[CrossRef]

Toker, G.

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
[CrossRef]

G. Toker, A. Brunfeld, and J. Shamir, “High resolution inspection for in-line surface testing,” in 1995 Display Manufacturing Technology Conference (Society for Information Display, 1995), pp. 119–120.

Torok, P.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

van de Nes, A. S.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Videen, G.

Appl. Opt. (1)

J. Geophys. Res. (1)

K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: a classification relating to particle morphologies,” J. Geophys. Res. 111, D12212(2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Opt. Eng. (1)

A. Normatov, B. Spektor, and J. Shamir, “Singular beam scanning microscopy: preliminary experimental results,” Opt. Eng. 49, 048001 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

G. Toker, A. Brunfeld, J. Shamir, and B. Spektor, “In-line optical surface roughness determination by laser scanning,” Proc. SPIE 4777, 323–329 (2002).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Other (7)

G. Toker, A. Brunfeld, and J. Shamir, “High resolution inspection for in-line surface testing,” in 1995 Display Manufacturing Technology Conference (Society for Information Display, 1995), pp. 119–120.

J. Shamir and N. Karasikov, “A method for particle size and concentration measurement,” U.S. patent 7,746,469(29 June 2010).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).

A. E. Siegman, Lasers (University Science, 1986).

http://refractiveindex.info/—refractive index database.

C. W. Qiu and L. W. Li, “Electromagnetic scattering by 3-D general anisotropic objects: a Hertz–Debye potential formulation,” in Proceedings of the 2005 IEEE Antennas and Propagation Society International Symposium and USNC/URSI National Radio Science Meeting (IEEE, 2005), pp. 422–425.
[PubMed]

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Figures (16)

Fig. 1
Fig. 1

Definition of the coordinate system.

Fig. 2
Fig. 2

Field component distribution, E x of an x- polarized incident beam: (a) amplitude and (b) phase for three GL beams—amplitude scale is arbitrary and phase scale is in radians (white or red represents high intensity and black or blue represents low intensity).

Fig. 3
Fig. 3

Field amplitude distribution of a DB; the amplitude scale is arbitrary. The two sides possess opposite phase.

Fig. 4
Fig. 4

Center sensor output for moving sphere, a = 100 nm , NA = 0.125 .

Fig. 5
Fig. 5

Differential detection with DB illumination, a = 100 nm , NA = 0.125 .

Fig. 6
Fig. 6

Total forward intensity-integrating detector output, a = 100 nm , NA = 0.125 .

Fig. 7
Fig. 7

Comparison between the signal outputs from the central sensor for DB illumination with NA = 0.125 and NA = 0.25 ( a = 100 nm , n = 1.5 ).

Fig. 8
Fig. 8

Same as Fig. 7 but for the differential detection configuration.

Fig. 9
Fig. 9

Comparison of center sensor output for DB illumination with NA = 0.125 for three sphere sizes ( a = 50 ,100, 200 nm , n = 1.5 ).

Fig. 10
Fig. 10

Same as Fig. 9 but for differential detection.

Fig. 11
Fig. 11

Same as Fig. 9 but for total power detection.

Fig. 12
Fig. 12

Comparison of the differential sensor for illumination with various beam structures, a = 100 nm , NA = 0.125 , n = 1.5 .

Fig. 13
Fig. 13

Total integrated intensity for the three beams, a = 100 nm , NA = 0.125 , n = 1.5 .

Fig. 14
Fig. 14

Signal comparison for the three illuminating beams in the presence of static noise, a = 100 nm , NA = 0.125 , n = 1.5 .

Fig. 15
Fig. 15

Same as Fig. 14 but with semidynamic noise.

Fig. 16
Fig. 16

Same as Fig. 14 but with semidynamic noise for a 50 nm sphere.

Equations (33)

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2 U r 2 + k 2 U + 1 r 2 sin ( θ ) θ ( sin ( θ ) U θ ) + 1 r 2 sin 2 ( θ ) 2 U φ 2 = 0 ,
U TM = E 0 k n = 1 m = n + n c n g n , TM m exp ( im φ ) { Ψ n ( k r ) ζ n ( k r ) } P n | m | ( cos θ ) ,
U TE = H 0 k n = 1 m = n + n c n g n , TE m exp ( im φ ) { Ψ n ( k r ) ζ n ( k r ) } P n | m | ( cos θ ) ,
c n = 1 ik ( i ) n 2 n + 1 n ( n + 1 ) .
Ψ n ( k r ) = k r · j n ( k r ) = ( π k r 2 ) 1 2 J n + 1 2 ( k r ) ,
ζ n ( k r ) = k r · h n ( 2 ) ( k r ) = ( π k r 2 ) 1 2 H n + 1 2 ( 2 ) ( k r ) ,
E r , TE = 0 , E θ , TE = i ω μ r sin θ U TE φ , E φ , TE = i ω μ r U TE θ ,
E r , TM = 2 U TM r 2 + k 2 U TM , E θ , TM = 1 r 2 U TM r θ , E φ , TM = 1 r sin θ 2 U TM r φ ,
H r , TM = 0 , H θ , TM = i ω ϵ r sin θ U TM φ , H φ , TM = i ω ϵ r U TM θ ,
H r , TE = 2 U TE r 2 + k 2 U TE , H θ , TE = 1 r 2 U TE r θ , H φ , TE = 1 r sin θ 2 U TE r φ .
E r = k E 0 n = 1 m = n + n c n g n , TM m a n [ ζ n ( k r ) + ζ n ( k r ) ] P n | m | ( cos θ ) exp ( im φ ) ,
E θ = E 0 r n = 1 m = n + n c n [ g n , TM m a n ζ n ( k r ) τ n | m | ( cos θ ) + m g n , TE m b n ζ n ( k r ) Π n | m | ( cos θ ) ] exp ( i m φ ) ,
E φ = ı E 0 r n = 1 m = n + n c n [ m g n , TM m a n ζ n ( k r ) Π n | m | ( cos θ ) + g n , TE m b n ζ n ( k r ) τ n | m | ( cos θ ) ] exp ( i m φ ) ,
τ n m ( cos θ ) = d d θ P n m ( cos θ ) ,
Π n m ( cos θ ) = P n m ( cos θ ) sin θ .
a n = Ψ n ( x ) Ψ n ( y ) M Ψ n ( x ) Ψ n ( y ) ζ n ( x ) Ψ n ( y ) M ζ n ( x ) Ψ n ( y ) ,
b n = M Ψ n ( x ) Ψ n ( y ) Ψ n ( x ) Ψ n ( y ) M ζ n ( x ) Ψ n ( y ) ζ n ( x ) Ψ n ( y ) ,
E r incident = E 0 k r 2 n = 1 m = n + n c n g n , TM m n ( n + 1 ) Ψ n ( k r ) P n | m | ( cos θ ) exp ( i m φ ) ,
H r incident = H 0 k r 2 n = 1 m = n + n c n g n , TE m n ( n + 1 ) Ψ n ( k r ) P n | m | ( cos θ ) exp ( i m φ ) ,
0 2 π exp [ i ( m m ) φ ] d φ = 2 π δ m m ,
0 π P n m ( cos θ ) P n m ( cos θ ) sin θ d θ = 2 2 n + 1 ( n + m ) ! ( n m ) ! δ n n ,
0 π P n | m | ( cos θ ) sin θ d θ 0 2 π exp ( i m φ ) E r incident E 0 d φ = 1 i ( k r 2 ) ( i ) n ( 2 n + 1 ) Ψ n ( k r ) g n , TM m 4 π 2 n + 1 ( n + m ) ! ( n m ) ! .
g n , TM m = i n + 1 4 π k r j n ( k r ) ( n m ) ! ( n + m ) ! 0 π P n | m | ( cos θ ) sin θ d θ 0 2 π exp ( i m φ ) E r incident E 0 d φ .
g n , TM m = i n + 1 4 π k r j n ( k r ) ( n m ) ! ( n + m ) ! 0 π P n | m | ( cos θ ) sin θ d θ 0 2 π exp ( i m φ ) H r incident H 0 d φ .
u n m ( r ) = G ( ρ ˜ , z ˜ ) R n m ( ρ ˜ ) Φ m ( ϕ ) Z n ( z ˜ ) ,
G ( ρ ˜ , z ˜ ) = ω 0 ω ( z ˜ ) exp ( ρ ˜ 2 ) exp ( i ρ ˜ 2 z ˜ ) exp [ i ψ ( z ˜ ) ] ,
R n m ( ρ ˜ ) = ( 2 ρ ˜ ) | m | L ( n | m | ) / 2 | m | ( 2 ρ ˜ 2 ) ,
Φ m ( ϕ ) = exp ( i m ϕ ) ,
Z n ( z ˜ ) = exp [ i n ψ ( z ˜ ) ] ,
n = | m | , | m | + 2 , | m | + 4 , | m | + 6
D = G L 0 , 1 G L 0 , 1 2 .
n m , n k = ( 1 2 ) m k .
σ i = 2 e I Δ F ,

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