Abstract

This paper examines the use of Gauss-Laguerre beams in STED microscopy. These types of beams are shown to have beneficial properties that can be utilised to generate stable, high quality STED beams resulting in an aberration-resilient generation volume. In this paper we obtain general expressions for Gauss-Laguerre beams being focused through a stratified medium and describe their optimization for STED microscopy purposes. Our results show that the circularly polarised, lowest order “dark” beam is the most beneficial for STED purposes.

© 2004 Optical Society of America

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References

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Appl. Opt.

Appl. Phys. Lett.

M. Dyba, T. Klar, S. Jakobs, and S. Hell, �??Ultrafast dynamics microscopy,�?? Appl. Phys. Lett. 77, 597�??599 (2000).
[CrossRef]

J. Mod. Opt.

S. N. Khonina, V. Kotlyar, V. Soifer, J. Honkanen, M. Lautanen, and T. J., �??Generation of rotating Gauss-Laguerre modes with binary-phase diffractive optics,�?? J. Mod. Opt. 46, 227�??238 (1999).

J. Opt. Soc. Am. A

Nature Biotech.

S. Hell, �??Toward fluorescence nanoscopy,�?? Nature Biotech. 21, 1347�??1355 (2003).
[CrossRef]

Opt. Commun.

P. Torok, P. Higdon, and T. Wilson, �??On the general properties of polarising conventional and confocal microscopes,�?? Opt. Commun. 148, 300�??315 (1998).
[CrossRef]

C. Paterson and R. Smith, �??Helicon waves: propagation-invariant waves in a rotating coordinate system,�?? Opt. Commun. 124, 131�??140 (1996).
[CrossRef]

M. Padgett, L. Allen, �??The Poynting vector in Laguerre-Gaussian laser modes,�?? Opt. Commun. 121, 36�??40 (1995).
[CrossRef]

Opt. Comunm.

P. Torok and F.-J. Kao, �??Point-Spread Function Reconstruction in High Aperture Lenses Focusing Ultra-Short Laser Pulses,�?? Opt. Comunm. 213, 97�??102 (2002).
[CrossRef]

Opt. Lett.

Optik

S. Lindek, N. Salmon, C. Cremer, and E. H. K. Stelzer, �??Theta microscopy allows phase regulation in 4Pi(A)-confocal two-photon fluorescence microscopy,�?? Optik 98, 15�??20 (1994).

Phys. Rev. E

T. A. Klar, E. Engel, and S. Hell, �??Breaking Abbe�??s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,�?? Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Phys. Rev. Lett.

M. Dyba, S.W. Hell, �??Focal spots of size ë/23 open up far-field fluorescence microscopy at 33 nm axial resolution,�?? Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Proc. Roy. Soc. A

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

Other

R. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, 1966).

Supplementary Material (4)

» Media 1: GIF (561 KB)     
» Media 2: MPG (100 KB)     
» Media 3: MPG (93 KB)     
» Media 4: MPG (98 KB)     

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Figures (12)

Fig. 1.
Fig. 1.

Schematic diagram of the illumination.

Fig. 2.
Fig. 2.

Distribution of the modulus square of the Cartesian components of the electric field vector for a linearly polarized illumination for a water immersion lens of NA=1.2 and λ=558nm, in the focal plane.

Fig. 3.
Fig. 3.

Distribution of the modulus square of the Cartesian components of the electric field vector for an x-polarized Gauss-Laguerre beam (m=1, n=1) illumination for a water immersion lens of NA=1.2, F=1.0 and λ=775nm, in the focal plane. Note that while individual sub-plots are comparable to each other, the absolute scaling is different from that in Fig. 2. Also note that here a larger section of the focal plane is shown.

Fig. 4.
Fig. 4.

Distribution of the modulus square of the Cartesian components of the electric field vector for a circularly polarized Gauss-Laguerre beam (m=1, n=1) illumination for a water immersion lens of NA=1.2, F=1.0 and λ=775nm, in the focal plane.

Fig. 5.
Fig. 5.

The time evolution of the transverse component of the electric field vector for a circularly polarized Gauss-Laguerre beam (m=1, n=1) illumination for a water immersion lens of NA=1.2, F=1.0 and λ=775nm, in the focal plane. The length of the arrows is proportional to their relative strength. (gif, 206kB)

Fig. 6.
Fig. 6.

The FWHM and the peak intensity of the transverse component of the electric field vector for a circularly polarized Gauss-Laguerre beam (m=1, n=1) as a function of numerical aperture of the water immersion lens and the fill factor F for λ=775nm, in the focal plane.

Fig. 7.
Fig. 7.

The x-z distribution of the transverse component of the time-averaged electric field of a circularly polarized Gauss-Laguerre beam (m=1, n=1) when the fill factor is varied from 0.1 to 2.0. The numerical aperture of the water immersion lens is NA=1.2 and λ=775nm. The dimensions in the figure are shown in µm. The horizontal axis is the x direction. (gif, 560kB)

Fig. 8.
Fig. 8.

The encircled energy as a function of normalized radius for various orders of Gauss-Laguerre beam (left). The total energy carried in a beam has been normalized to unity. a/w 0=1 corresponds to fill factor F=1. The figure on the right shows the FWHM of the transverse component of the focused Gauss-Laguerre beam as a function on the fill factor, F for a water immersion lens of NA=1.2. The inset shows the radial distribution of U 2,4 for F=1.7 along the x-direction.

Fig. 9.
Fig. 9.

Animation showing the evolution of STED generation volume as depth of water is increased to 15 µm using a conventional STED beam. The excitation beam had wavelength λ exc=558nm while the wavelength of the STED beam was λSTED=775nm. The objective lens had NA 1.4 (oil immersion).(mpg, 100kB)

Fig. 10.
Fig. 10.

Animation showing the evolution of STED generation volume as depth of water is increased to 15 µm for a circularly polarised m=1, n=1 Gauss-Laguerre STED beam. The excitation beam had wavelength λexc=558nm while the wavelength of the STED beam was λSTED=775nm. The objective lens had NA 1.4 (oil immersion). (mpg, 93kB)

Fig. 11.
Fig. 11.

Animation showing the evolution of STED generation volume as depth of water is increased to 15 µm for two orthogonally directed circularly polarised m=1, n=1 Gauss-Laguerre STED beams. The excitation beam had wavelength λ exc=558nm while the wavelength of the STED beam was λ STED=775nm. The numerical aperture of the conventional and rotated objective lenses were 1.4 (oil immersion) and 0.45 (oil immersion), respectively. (mpg, 93kB)

Fig. 12.
Fig. 12.

Figure (a) showing the intensity distributions in the x-z meridional plane for the conventional and Gauss-Laguerre STED beams focused from glass to 2µm deep water but otherwise for the same conditions than applied for Fig. 10. Figure (b) are line scans taken along the lines indicated in figure (a). For reference the intensity distribution corresponding to the unaberrated conventional STED beam is shown.

Equations (38)

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U m , n ( ρ , ϕ , z = 0 ) = exp [ ( ρ w 0 ) 2 ] ( 2 ρ w 0 ) m L n m 2 m [ 2 ( ρ w 0 ) 2 ] exp ( im ϕ )
E = Ω E ( s x , s y ) s z exp ( ik r p · s ) d s x d s y
E = R 1 · L · R · E in
E = R 1 · L · R · BS λ 4 · E in
E m , n ; x x ( r p ) = 2 π ( i ) m exp ( im ϕ p ) I m n π ( i ) m + 2 exp [ i ( m + 2 ) ϕ p ] I m + 2 n π ( i ) m 2 exp [ i ( m 2 ) ϕ p ] I m 2 n
E m , n ; y x ( r p ) = i π ( i ) m 2 exp [ i ( m 2 ) ϕ p ] I m 2 n i π ( i ) m + 2 exp [ i ( m + 2 ) ϕ p ] I m + 2 n
E m , n ; z x ( r p ) = 2 π ( i ) m + 1 exp [ i ( m + 1 ) ϕ p ] I m + 1 n 2 π ( i ) m 1 exp [ i ( m 1 ) ϕ p ] I m 1 n
E m , n ; x LH ( r p ) = 2 π ( i ) m exp ( im ϕ p ) I m n 2 π ( i ) m + 2 exp [ i ( m + 2 ) ϕ p ] I m + 2 n
E m , n ; y LH ( r p ) = i 2 π ( i ) m exp ( im ϕ p ) I m n + i 2 π ( i ) m + 2 exp [ i ( m + 2 ) ϕ p ] I m + 2 n
E m , n ; z LH ( r p ) = 4 π ( i ) m + 1 exp [ i ( m + 1 ) ϕ p ] I m + 1 n
0 2 π cos ( n ξ ) cos ( m ξ ) exp [ i a cos ( ξ ζ ) ] d ξ =
= π ( i ) n + m J n + m ( a ) cos [ ( n + m ) ζ ] + π ( i ) n m J n m ( a ) cos [ ( n m ) ζ ]
0 2 π sin ( n ξ ) sin ( m ξ ) exp [ i a cos ( ξ ζ ) ] d ξ =
= π ( i ) n + m J n + m ( a ) cos [ ( n + m ) ζ ] + π ( i ) n m J n m ( a ) cos [ ( n m ) ζ ]
0 2 π cos ( n ξ ) sin ( m ξ ) exp [ i a cos ( ξ ζ ) ] d ξ =
= π ( i ) n + m J n + m ( a ) sin [ ( n + m ) ζ ] π ( i ) n m J n m ( a ) sin [ ( n m ) ζ ]
I m n ( r p , z p ) = 0 α Ψ m , n ( θ ) cos θ ( 1 + cos θ ) J m ( k r p sin θ ) exp ( i k z p cos θ ) sin θ d θ
I m ± 1 n ( r p , z p ) = 0 α Ψ m ± 1 , n ( θ ) cos θ J m ± 1 ( k r p sin θ ) exp ( i k z p cos θ ) sin 2 θ d θ
I m ± 2 n ( r p , z p ) = 0 α Ψ m ± 2 , n ( θ ) cos θ ( 1 cos θ ) J m ± 2 ( k r p sin θ ) exp ( i k z p cos θ ) sin θ d θ
K l p ( r , φ , z ) = 2 π ( i ) l exp ( i l φ ) I l p ( r , z )
E m , n ; x x ( r p ) = K m n 1 2 K m + 2 n 1 2 K m 2 n
E m , n ; y x ( r p ) = i 2 K m + 2 n i 2 K m 2 n
E m , n ; z x ( r p ) = K m + 1 n K m 1 n
E m , n ; x LH ( r p ) = K m n K m + 2 n
E m , n ; y LH ( r p ) = iK m n + iK m + 2 n
E m , n ; z LH ( r p ) = 2 K m + 1 n
E m , n ; x RH ( r p ) = K m n K m 2 n
E m , n ; y RH ( r p ) = iK m n iK m 2 n
E m , n ; z RH ( r p ) = 2 K m 1 n
I m n ( r p , z p ) = 0 α 1 Ψ m , n ( θ 1 ) cos θ 1 sin θ 1 ( T s ( N 1 ) + T p ( N 1 ) cos θ N ) J m ( k 1 r p sin θ 1 ) ×
× exp ( i k 0 Φ i ) exp ( i k N z p cos θ N ) d θ 1
I m ± 1 n ( r p , z p ) = 0 α 1 Ψ m + 1 , n ( θ 1 ) cos θ 1 sin θ 1 T p ( N 1 ) sin θ N J m ± 1 ( k 1 r p sin θ 1 ) ×
× exp ( i k 0 Φ i ) exp ( i k N z p cos θ N ) d θ 1
I m ± 2 n ( r p , z p ) = 0 α 1 Ψ m + 2 , n ( θ 1 ) cos θ 1 sin θ 1 ( T s ( N 1 ) T p ( N 1 ) cos θ N ) J m ± 2 ( k 1 r p sin θ 1 ) ×
× exp ( i k 0 Φ i ) exp ( i k N z p cos θ N ) d θ 1
Φ i = h N 1 n N s N z n 1 h 1 s 1 z
W m , n ( a ) = 0 a 0 2 π U m , n ( ρ , ϕ , z = 0 ) 2 ρ d ϕ d ρ = 2 π 0 a U m , n ( ρ , z = 0 ) 2 ρ d ρ
I SV = I exc exp ( σ max α 1 I 1 STED ) exp ( σ max α 2 I 2 STED )

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