Abstract

Optical singularities are localized regions in a light field where one or more of the field parameters, such as phase or polarization, become singular with associated zero intensity. Singular beam microscopy exploits the fact that the strong variations of the optical field around the singularities are highly sensitive to changes in their neighborhood. As a consequence, analysis of the light field scattered from the object during a scanning process can yield useful information about the object features. We present a theoretical background, numerical simulations, and experimental results. Preliminary experiments have demonstrated a sensitivity of 20nm in the position and size of simple objects, with theoretically estimated 1nm capability under the assumption of a reasonable and conservative 30dB signal to noise ratio.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. E. N. Leith, "Space bandwidth requirements for three-dimensional imagery," Appl. Opt. 10, 2419-2422 (1971).
    [CrossRef] [PubMed]
  2. M.S.Soskin, ed., International Conference on Singular Optics, Proc. SPIE 3487 (1998).
  3. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
    [CrossRef]
  4. J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. London Ser. A 414, 433-446 (1987).
    [CrossRef]
  5. J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. II. Observations of the electric field," Proc. R. Soc. London Ser. A 414, 447-468 (1987).
    [CrossRef]
  6. G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
    [CrossRef]
  7. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
    [CrossRef]
  8. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064-4075 (1997).
    [CrossRef]
  9. I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997).
    [CrossRef]
  10. I. Freund, "Vortex derivatives," Opt. Commun. 137, 118-126 (1997).
    [CrossRef]
  11. Y. J. Schechner and J. Shamir, "Parameterization and orbital angular momentum of anisotropic dislocations," J. Opt. Soc. Am. A 13, 967-973 (1996).
    [CrossRef]
  12. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 294-372 (1999).
  13. W. M. Lee, X.-C. Yuan, and K. Dholakia, "Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step," Opt. Commun. 239, 129-135 (2004).
    [CrossRef]
  14. M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptical stationary points," Opt. Lett. 28, 1475-1477 (2003).
    [CrossRef] [PubMed]
  15. B. Spektor, R. Piestun, and J. Shamir, "Dark beams with a constant notch," Opt. Lett. 21, 456-458 (1996).
    [CrossRef] [PubMed]
  16. A. Tavrov, N. Kerwien, R. Berger, H. Tiziani, M. Totzek, B. Spektor, J. Shamir, G. Toker, and A. Brunfeld, "Vector simulations of dark beam interaction with nano-scale surface features," Proc. SPIE 5144, 26-36 (2003).
    [CrossRef]
  17. B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, "High resolution surface feature evaluation using multi-wavelength optical transforms," Proc. SPIE 4777, 345-351 (2002).
    [CrossRef]
  18. G. Toker, A. Brunfeld, J. Shamir, B. Spektor, E. Cromwell, and J. Adam, "In-line optical surface roughness determination by laser scanning," Proc. SPIE 4777, 323-329 (2002).
    [CrossRef]
  19. Y. Parkhomenko, B. Spektor, and J. Shamir, "Laser mode selection by combining a bi-prism-like reflectors with a narrow amplitude masks," Appl. Opt. 45, 2761-2765 (2006).
    [CrossRef] [PubMed]
  20. J. Shamir, Optical Systems and Processes (SPIE, 1999).
    [CrossRef]
  21. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an optical system," Proc. R. Soc. London Ser. A 253, 358-379 (1959).
    [CrossRef]
  22. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2005).

2006 (1)

2004 (1)

W. M. Lee, X.-C. Yuan, and K. Dholakia, "Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step," Opt. Commun. 239, 129-135 (2004).
[CrossRef]

2003 (2)

A. Tavrov, N. Kerwien, R. Berger, H. Tiziani, M. Totzek, B. Spektor, J. Shamir, G. Toker, and A. Brunfeld, "Vector simulations of dark beam interaction with nano-scale surface features," Proc. SPIE 5144, 26-36 (2003).
[CrossRef]

M. S. Soskin, V. Denisenko, and I. Freund, "Optical polarization singularities and elliptical stationary points," Opt. Lett. 28, 1475-1477 (2003).
[CrossRef] [PubMed]

2002 (2)

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, "High resolution surface feature evaluation using multi-wavelength optical transforms," Proc. SPIE 4777, 345-351 (2002).
[CrossRef]

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

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 294-372 (1999).

1997 (3)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997).
[CrossRef]

I. Freund, "Vortex derivatives," Opt. Commun. 137, 118-126 (1997).
[CrossRef]

1996 (2)

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]

1993 (1)

G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

1987 (2)

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. London Ser. A 414, 433-446 (1987).
[CrossRef]

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. II. Observations of the electric field," Proc. R. Soc. London Ser. A 414, 447-468 (1987).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

1971 (1)

1959 (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an optical system," Proc. R. Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

G. Indebetouw, "Optical vortices and their propagation," J. Mod. Opt. 40, 73-87 (1993).
[CrossRef]

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

Opt. Commun. (4)

I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997).
[CrossRef]

I. Freund, "Vortex derivatives," Opt. Commun. 137, 118-126 (1997).
[CrossRef]

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

W. M. Lee, X.-C. Yuan, and K. Dholakia, "Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step," Opt. Commun. 239, 129-135 (2004).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, "Topological charge and angular momentum of light beams carrying optical vortices," Phys. Rev. A 56, 4064-4075 (1997).
[CrossRef]

Proc. R. Soc. London Ser. A (4)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. I. Theory," Proc. R. Soc. London Ser. A 414, 433-446 (1987).
[CrossRef]

J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves. II. Observations of the electric field," Proc. R. Soc. London Ser. A 414, 447-468 (1987).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an optical system," Proc. R. Soc. London Ser. A 253, 358-379 (1959).
[CrossRef]

Proc. SPIE (3)

A. Tavrov, N. Kerwien, R. Berger, H. Tiziani, M. Totzek, B. Spektor, J. Shamir, G. Toker, and A. Brunfeld, "Vector simulations of dark beam interaction with nano-scale surface features," Proc. SPIE 5144, 26-36 (2003).
[CrossRef]

B. Spektor, G. Toker, J. Shamir, M. Friedman, and A. Brunfeld, "High resolution surface feature evaluation using multi-wavelength optical transforms," Proc. SPIE 4777, 345-351 (2002).
[CrossRef]

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

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 294-372 (1999).

Other (3)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2005).

J. Shamir, Optical Systems and Processes (SPIE, 1999).
[CrossRef]

M.S.Soskin, ed., International Conference on Singular Optics, Proc. SPIE 3487 (1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1
Fig. 1

(Color online) Schematic diagram of an experimental setup of a SB microscope. A Gaussian laser beam is converted to a SB by a phase mask and a spatial filtering stage. The SB is focused onto the investigated object, and light scattered off the object is recorded with the digital camera.

Fig. 2
Fig. 2

Cross section of (a) the normalized intensity distribution and (b) the complex amplitude at the focal plane of the microscope objective with N A = 0.4 . Solid curve represents an idealized SB, which is compared to a Gaussian beam (dotted curve).

Fig. 3
Fig. 3

Simulated output intensity distributions, at the recording plane for 4 × 4 = 16 parameter values: PS positions, S = x 0 (in microns) relative to the SB center and heights, H = h , of the phase step (in nanometers).

Fig. 4
Fig. 4

Simulated output intensity differences (absolute value in percents of the maximum intensity from Fig. 3), at the recording plane. The cases are defined as in Fig. 3. The black curve corresponds to output intensity difference resulting from a shift of the PS by 10 n m relative to the singular beam focal intensity distribution. The gray curve corresponds to the output intensity difference resulting from a height variation of the phase step by 5 n m .

Fig. 5
Fig. 5

Object parameter dependence of sensitivity. (a) Maximum intensity difference at the recording plane versus PS position when the PS height is π. The gray curve corresponds to a 5 n m phase step height variation (from 488 n m to 493 n m ), while the black curve corresponds to output intensity difference following a shift of 10 n m from a given position of the PS. (b) The same as (a) but as a function of PS height for PS positioned in 0.

Fig. 6
Fig. 6

Absolute difference (in percents of peak intensity) of | E | 2 between the electric field calculated using paraxial, scalar operators and the electric field obtained from rigorous calculations using the Richards–Wolf approach. The ellipticity, corresponding to the simulated N A = 0.4 , of the rigorously calculated field is represented by the two orthogonal cross sections along x and y .

Fig. 7
Fig. 7

Electric field intensity at the recording plane. TE, TM correspond to rigorous vector simulations, and Scalar corresponds to paraxial, scalar simulation. The phase step is situated in the vicinity of the focal SB intensity distribution center. Screen coordinates (the horizontal axis) represent the relative recording plane coordinates.

Fig. 8
Fig. 8

Absolute value of the difference, in percents of maximum intensity at the recording plane, between normalized results of the scalar and vector simulations, as shown in Fig. 7. The solid curve corresponds to the absolute value of the difference between scalar and TE results, while the dashed curve corresponds to the TM result. Screen coordinates (the horizontal axis) represent the relative recording plane coordinates.

Fig. 9
Fig. 9

Output intensity difference for distributions of Fig. 7 following a 5 n m shift in the position of the PS. Screen coordinates (the horizontal axis) represent the relative recording plane coordinates.

Fig. 10
Fig. 10

Experimental intensity distributions of (a) the Gaussian beam and (b) SB, taken at the recording plane. (c) Cross-section profiles corresponding to the output intensity distributions of (a) and (b). Gaussian beam (dotted curve) and a SB (solid curve). Screen coordinates (the horizontal axis) represent the relative recording plane coordinates.

Fig. 11
Fig. 11

Profiles of the intensity distribution in the recording plane. The semi-dotted curve denotes simulation; other curves denote experiments. The phase step is situated at the center of the SB focal intensity distribution. Screen coordinates (the horizontal axis) represent the relative recording plane coordinates.

Fig. 12
Fig. 12

Same as Fig. 11 but with the phase step situated outside the singular beam focal intensity distribution.

Fig. 13
Fig. 13

Output intensity distribution (a)–(e) and corresponding profiles for the PS positions (shown from left to right): 4 μ m , 2 μ m , 0 μ m , 1.8 μ m , and 4 μ m . The vertical axis for profiles is relative intensity.

Fig. 14
Fig. 14

Integration window application to output intensity distributions shown in Fig. 13. The integration (summation of intensity) is performed only on the pixels inside the window.

Fig. 15
Fig. 15

Integration window power as a function of PS position. The vertical axis is relative power; the horizontal axis is the PS position in microns. The semi-dotted curve is simulation; other curves denote experiments.

Fig. 16
Fig. 16

Integration window power derivative as a function of PS position. The results are normalized to 20 nm PS shifts that were used in experiment C. The vertical axis is in percents of maximum integration window power; the horizontal axis is PS position in microns. The semi-dotted curve is simulation;the other curves denote experiments.

Fig. 17
Fig. 17

Simulated output intensity distribution in the form of a profile map. The result represents a simulation of an experiment with the phase step scanning across the focal intensity distribution of the singular beam in shifts of 20 nm. For each position of the phase step the intensity profile on the recording plane is coded in gray scale. Screen coordinates (the horizontal axis) represent the relative recording plane coordinates.

Fig. 18
Fig. 18

Same as Fig. 17 but for experimental data.

Fig. 19
Fig. 19

Simulated results for the derivative of the profile map of Fig. 17. The derivative value corresponds to a simulation of an experiment with the PS scanning across the focal intensity distribution of the singular beam in shifts of 20 n m . For each position of the PS the derivative profile is coded in gray scale. Screen coordinates (the horizontal axis) represent the relative recording plane coordinates. Color bar values represent percents of the peak intensity of Fig. 17.

Fig. 20
Fig. 20

Same as Fig. 19 but for the experimental results for the derivative of the profile map of Fig. 18.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

U i n = R [ d 2 ] T m a s k A q 0 exp ( j k d 1 ) q ( d 1 ) Q [ 1 q ( d 1 ) ] ,
R [ d ] = exp ( j k d ) F Q [ λ 2 d ] F 1 ,
R [ d ] = exp ( j k d ) j λ d Q [ 1 d ] V [ 1 λ d ] F Q [ 1 d ] ,
Q [ 1 a ] = exp [ j k 2 a ( x 2 + y 2 ) ] ,
q 0 = k w 0 2 2 j = π w 0 2 j λ ,
U o u t = R [ d 3 ] T o b j R [ f ] T a p e r t u r e Q [ 1 f ] U i n .
Δ I n = | U o u t ( p n + Δ p n ) | 2 | U o u t ( p n ) | 2 max { | U o u t ( p n ) | 2 } .
S N R = max ( I o u t ) max ( N o u t ) .
α Δ I n > 1 / S N R ,
T o b j [ h , x 0 ] = exp [ j u ( x x 0 ) π h λ ] ,
u ( x ) = { 0 x < 0 1 x 0 .
Δ I n = 0 + [ | U o u t ( p n + Δ p n ) | 2 | U o u t ( p n ) | 2 ] max { | U o u t ( p n ) | 2 } [ Δ p n ] | Δ p n = 0 Δ p n 1 ! + O ( [ Δ p n ] 2 ) .
Δ I n C ( p n ) Δ p n ,

Metrics