## Limits for reduction of effective focal volume in multiple-beam light microscopy

Optics Express, Vol. 17, Issue 4, pp. 2861-2870 (2009)

http://dx.doi.org/10.1364/OE.17.002861

Acrobat PDF (557 KB)

### Abstract

Employing interference patterns for illumination has been shown to reduce the focal volume in fluorescence microscopy. For example, the 4Pi technique employs two interfering laser beams and significantly decreases the focal volume, as compared to conventional microscopy. We study theoretically the effect of using multiple interfering laser beams on the focal volume. In realistic setups with three or four beams, the focal volume is about half of that from the 4Pi case. This improvement reaches a limit quickly as more beams are added, and for the idealized case of an infinite number of beams the focal volume is rather close to the three- or four-beam cases. Thus, our study suggests a limit for the possible reduction of the focal volume in a purely optical far-field setup.

© 2009 Optical Society of America

## 1. Introduction

*λ*/(2

*n*sin

*α*) (or 200–300 nm in practice), where

*λ*is the wavelength of the light used in the microscope,

*n*is the refraction index, and

*α*is the half-aperture angle of the light focused by the lens. This barrier, known as the Abbe diffraction limit, has been overcome with the emergence of approaches that employ structured illumination, or, in other words, interference of the illumination light [1

1. S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. **93**, 277–282 (1992). [CrossRef]

2. B. Bailey, D. L. Farkas, D. L. Taylor, and F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature **366**, 44–48 (1993). [CrossRef] [PubMed]

3. S. Lindek, R. Pick, and E. H. K. Stelzer, “Confocal Theta Microscope with Three Objective Lenses,” Rev. Sci. Instrum. **65**, 3367–3372 (1994). [CrossRef]

4. M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE **2412**, 147–156 (1995). [CrossRef]

5. V. Krishnamurthi, B. Bailey, and F. Lanni, “Image processing in 3D standing-wave fluorescence microscopy,” Proc. SPIE **2655**, 18–25 (1996). [CrossRef]

6. M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. **22**, 1905–1907 (1997). [CrossRef]

7. R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE **3568**, 185–196 (1999). [CrossRef]

8. M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. **195**, 10–16 (1999). [CrossRef] [PubMed]

9. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. **198**, 82–87 (2000). [CrossRef] [PubMed]

10. J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA **97**, 7232–7235 (2000). [CrossRef] [PubMed]

11. G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. **25**, 46–48 (2000). [CrossRef]

12. O. Haeberle, C. Xu, A. Dieterlen, and S. Jacquey, “Multiple-objective microscopy with three-dimensional resolution near 100 nm and a long working distance,” Opt. Lett. **26**, 1684–1686 (2001). [CrossRef]

13. P. T. C. So, H.-S. Kwon, and C. Y. Dong, “Resolution enhancement in standing-wave total internal reflection microscopy: a point-spread-function engineering approach,” J. Opt. Soc. Am.Ã **18**, 2833–2845 (2001). [CrossRef]

14. J. Ryu, S. S. Hong, B. K. P. Horn, D. M. Freeman, and M. S. Mermelstein, “Multibeam interferometric illumination as the primary source of resolution in optical microscopy,” Appl. Phys. Lett. **88**, 171,112 (2006). [CrossRef]

1. S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. **93**, 277–282 (1992). [CrossRef]

3. S. Lindek, R. Pick, and E. H. K. Stelzer, “Confocal Theta Microscope with Three Objective Lenses,” Rev. Sci. Instrum. **65**, 3367–3372 (1994). [CrossRef]

12. O. Haeberle, C. Xu, A. Dieterlen, and S. Jacquey, “Multiple-objective microscopy with three-dimensional resolution near 100 nm and a long working distance,” Opt. Lett. **26**, 1684–1686 (2001). [CrossRef]

15. J. Swoger, J. Huisken, and E. H. K. Stelzer, “Multiple imaging axis microscopy improves resolution for thick-sample applications,” Opt. Lett. **28**, 1654–1656 (2003). [CrossRef] [PubMed]

16. S. W. Hell, “Far-Field Optical Nanoscopy,” Science **316**, 1153–1158 (2007). [CrossRef] [PubMed]

17. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission,” Opt. Lett. **19**, 780–782 (1994). [CrossRef] [PubMed]

18. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA **102**, 13,081–13,086 (2005). [CrossRef]

19. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-High Resolution Imaging by Fluorescence Photoactivation Localization Microscopy,” Biophys. J. **91**, 4258–4272 (2006). [CrossRef] [PubMed]

20. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science **313**, 1642–1645 (2006). [CrossRef] [PubMed]

21. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**, 793–796 (2006). [CrossRef] [PubMed]

22. L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence Fluctuation Spectroscopy in Subdiffraction Focal Volumes,” Phys. Rev. Lett. **94**, 178104 (2005). [CrossRef] [PubMed]

23. A. Arkhipov, J. Hüve, M. Kahms, R. Peters, and K. Schulten, “Continuous fluorescence microphotolysis and correlation spectroscopy using 4Pi microscopy,” Biophys. J. **93**, 4006–4017 (2007). [CrossRef] [PubMed]

1. S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. **93**, 277–282 (1992). [CrossRef]

3. S. Lindek, R. Pick, and E. H. K. Stelzer, “Confocal Theta Microscope with Three Objective Lenses,” Rev. Sci. Instrum. **65**, 3367–3372 (1994). [CrossRef]

12. O. Haeberle, C. Xu, A. Dieterlen, and S. Jacquey, “Multiple-objective microscopy with three-dimensional resolution near 100 nm and a long working distance,” Opt. Lett. **26**, 1684–1686 (2001). [CrossRef]

23. A. Arkhipov, J. Hüve, M. Kahms, R. Peters, and K. Schulten, “Continuous fluorescence microphotolysis and correlation spectroscopy using 4Pi microscopy,” Biophys. J. **93**, 4006–4017 (2007). [CrossRef] [PubMed]

## 2. Results

**65**, 3367–3372 (1994). [CrossRef]

*h*(

*r*→), are computed. The PSF resolution can be defined as the full-width-half-maximum (FWHM) of the PSF, and the effective volume is defined [24

24. P. Schwille and E. Haustein, “Fluorescence Correlation Spectroscopy: A Tutorial for the Biophysics Textbook Online (BTOL)“ (2002), http://www.biophysics.org/education/techniques.htm.

*α*for each beam in the same setup (larger

*α*leads to a narrower illumination spot). Theoretically,

*α*∈ [0,90°], but technically

*α*is usually limited to < 70°, although higher values, e.g.,

*α*= 74°, have been reported recently [25

25. M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. **10**, 043041 (2008). [CrossRef]

*α*has to be limited to even smaller values due to the spatial restrictions, e.g., to 60° for 3 beams in one plane. For the limiting cases “Inf. beams, 2D” and “Inf. beams, 3D”, the value of a has to be zero, which would lead to a divergence of the illumination spot. Below, we formally set

*α*≤ 90° for these arrangements, as an idealized and abstract case of the most narrow illumination idealistically possible. Thus, the 3-, 4-, or 6-beam setups are realistic, as they can be in principle realized with technically available lens parameters (e.g., using

*α*= 60° in the 3-beam case or

*α*= 45° for 4- and 6-beam cases). In practice, the values of a for these setups may need to be somewhat smaller, so that the objective edges for each beam are not overlapping. The values

*α*= 60° and

*α*= 45° are thus the maximal possible values for the considered setups. The cases with an infinite number of beams are unrealistic in the sense that a is set to values incompatible with the setup geometry, but the consideration of such idealized cases gives us an estimate for the absolute limit of the light focusing in a multi-beam setup.

*z*direction and focused by a lens, resulting in the spherical wavefront converging around

*r*→ = 0. The electric field [26

26. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A. (Math. Phys. Sci.) **253**, 349–357 (1959). [CrossRef]

27. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A. (Math. Phys. Sci.) **253**, 358–379 (1959). [CrossRef]

*E*→(

*r*) = -

*i*[

*I*

_{0}(

*r*→) +

*I*

_{2}(

*r*→)cos

^{2}ϕ,

*I*

_{2}(

*r*→)sin

^{2}ϕ, -2

*i*

*I*

_{1}(

*r*→)cosϕ], where

*f*

_{0}(0) = 1 + cos

*θ*,

*f*

_{1}(

*θ*) = sin

*θ*,

*f*

_{2}(

*θ*) = 1 - cos

*θ*. Here, ϕ is the angle between the plane of oscillation of

*E*→(

*r*→) and the plane of the objective lens (we assume the most common case of ϕ = 90°);

*J*

_{0, 1,2}are Bessel functions of the first kind. For a beam with an arbitrary direction

*k*→, we can define a coordinate frame (

*x*′,

*y*′,

*z*′), such that the beam is traveling in the -

*z*′ direction, and the electric field is described by the above formula with

*r*→ replaced by

*r*→′. Then, it is simple to obtain

*E*→(

*r*→) for this beam in the original coordinate frame (

*x*,

*y*,

*z*) through rotation of all vectors by the angle defined by the direction and polarization of the beam. For arbitrary beams

*i*,

*i*= 1,2, …,

*N*, we obtain the total electric field

*E*→

_{tot}(

*r*→) =ℑ

^{N}

_{i-1}

*E*→

*i*→(

*r*→) and PSF

*h*(

*r*→) = ∣

*E*→

_{tot}(

*r*→)

^{2nexc}(

*n*-photon excitation is used; we set

_{exc}*n*= 2 as commonly used in 4Pi microscopy). We assume that all beams are of the same intensity and are coherent. Phases of all beams are matched in such a way that, at the common focus of all objectives, electric field vectors of all beams are always of the same magnitude (meaning that collinear beams have identical phases).

_{exc}*fmtool*[23

23. A. Arkhipov, J. Hüve, M. Kahms, R. Peters, and K. Schulten, “Continuous fluorescence microphotolysis and correlation spectroscopy using 4Pi microscopy,” Biophys. J. **93**, 4006–4017 (2007). [CrossRef] [PubMed]

*E*→(

*r*→) from single beams; visualization is done with VMD [28

28. W. Humphrey, A. Dalke, and K. Schulten, “VMD - Visual Molecular Dynamics,” J. Mol. Graphics **14**, 33–38 (1996). [CrossRef]

**93**, 4006–4017 (2007). [CrossRef] [PubMed]

*λ*/

*n*. Typical values of

*λ*and

*n*are, e.g.,

*λ*= 910 nm (the high value is due to the two-photon excitation) and

*n*= 1.46. For the 1-beam case, we assume one-photon excitation and a wavelength of

*λ*/2, where

*λ*is the wavelength used for all other cases. Using two-photon excitation for the 1-beam case decreases the PSF size by the factor ~ √2, but, due to the two-fold increase of the wavelength, the size actually increases by ~ √2. For other cases, two-photon excitation leads to a smaller PSF volume than that for one-photon excitation, due to the constructive interference of multiple beams.

*z*-direction than that of the 1-beam setup, and its volume is further decreased by using 3, 4, or more beams (Fig. 2). However, as the number of beams grows, one has to employ smaller a. For example, if all beams are in the same plane, the maximal a is 60° for 3 beams, 45° for 4, 30° for 6, etc., and the PSF elongates in the

*x*-direction as more beams are used (can be noticed already for 3 vs. 4 beams). A similar effect is observed if out-of-plane beams are added (such as 6 beams in Fig. 2). We found that the PSF volume for setups with >4 beams in one plane or >6 in 3D is larger than that for 4- or 6-beam cases. Even if an idealized value

*α*= 70° is used, the PSF size is not decreased much beyond the 3-or 4-beam case. For example, 60 beams in Fig. 2 correspond to an overlap of three sets of 20 beams uniformly distributed in the

*x*,

*y*-,

*y*,

*z*-, and

*x*,

*z*-planes. The resulting PSF does not have side lobes or other outstanding features, but its size is not significantly reduced in comparison with the case of 3 beams. Likewise, the PSFs for an infinite number of beams with

*α*= 70° are similar in size to the 3-beam PSF with

*α*= 60°.

*V*

_{eff}are plotted in Fig. 3. The 4Pi Veff is about 1/2 of that for the 1-beam microscope. Another decrease by a factor of 2 results from employing 3, 4, or 6 beams. Even with

*α*= 60° for 3 beams or

*α*= 45° for 4 beams in one plane, or 6 beams in 3D,

*V*

_{eff}in these cases is about 1/2 of Veff of the 4Pi PSF with

*α*= 70°. For 3 beams at

*α*= 60°,

*V*

_{eff}≈ 0.24 (

*λ*/

*n*)

^{3}vs.

*V*

_{eff}≈ 0.4 (

*λ*/

*n*)

^{3}for the 4Pi at

*α*= 70°. For an infinite number of beams in 2D or 3D,

*V*

_{eff}is not reduced much further than in the case of 3 beams, even if an idealized and unrealistic value of

*α*= 90° is employed, resulting in

*V*

_{eff}≈ 0.14 (

*λ*/

*n*)

^{3}. Thus, the realistic setups with 3, 4, or 6 beams allow for values of

*V*

_{eff}which are about twice smaller than the best 4Pi values, and that are already very close to the minimal theoretically possible, but probably practically inaccessible, limit of light focusing.

*V*

_{eff}is decreased by using more than 2 beams, the imaging resolution cannot be improved beyond that of the 4Pi microscope, as suggested by the FWHM values in Table 1 (see also Fig. 4). The resolution along

*x*- and

*y*-directions can be somewhat improved by using 4 or an infinite number of beams, but in the

*z*-direction, the 4Pi FWHM is the smallest. Interestingly, the FWHM of the 3-beam PSF is greater than that of the 4Pi PSF in each dimension, while

*V*

_{eff}is smaller for the 3-beam case (the same is true for “Inf. beams, 3D” vs. “Inf. beams, C;2D”). This is because the side lobes or exterior rings present in the 4Pi and “Inf. beams, 2D” PSFs contribute significantly to the value of

*V*

_{eff}, even though the central part of the PSF is relatively narrow.

*π*. The resulting electric field is therefore obtained through double integration of the electric field for one beam: one integration is over the polar angle

*φ*(from 0 to 2

*π*) corresponding to the in-plane rotation for “Inf. beams, 2D”, and the other is over the azimuthal angle

*χ*(from 0 to

*π*). The integration can be performed independently for each of the functions

*I*

_{0}, 1,2 in Eq. (2), since the expression for the electric field is linear over these functions. Let us use spherical coordinates:

*x*=

*r*sin

*θ*cos

_{r}*ϕ*,

_{r}*y*=

*r*sin

*θ*sin

_{r}*θ*,

_{r}*z*=

*r*cos

*θ*. Then, contributions to the total electric field of “Inf. beams, 3D” at point

_{r}*r*→ = (

*x*,

*y*,

*z*) from functions

*I*

_{0,1,2}are proportional to

*g*(

*φ*) = cos

*φ*for the

*x*-component of the electric field, and

*g*(

*φ*) = sin

*φ*for the

*y*-component; the

*z*-component is zero. Note that

*θ*and

*θ*, as well as

_{r}*ϕ*and

_{r}*φ*, are different variables. The integration over

*φ*, originally from 0 to 2

*π*, can be replaced by an integration from -

*ϕ*to

_{r}*π*-

*ϕ*due to the symmetry of the “Inf. beams, 2D” case, where, for each beam, there is always another beam traveling in the opposite direction in the same plane (see Fig. 1).

_{r}*A*

_{0,1,2}(

*r*→) above is not, in general, spherically symmetric. However, we saw for the numerically constructed PSFs for the “Inf. beams, 3D” case that the asymmetry in those PSFs is not exceptionally high. Also, we would like to enforce the condition of spherical symmetry on

*A*

_{0,1,2}(

*r*→) to obtain a simple analytical approximation for the PSF in the case of an infinite number of beams in 3D. If spherical symmetry applies,

*A*

_{0,1,2}(

*r*→) has to assume the same value for any values of

*ϕ*and

_{r}*θ*, i.e., we can choose any fixed value to substitute for these variables. Since we are looking for the PSF with the smallest effective volume, we should choose

_{r}*ϕ*and

_{r}*θ*that correspond to the narrowest PSF. From the computationally obtained PSFs for the “Inf. beams, 3D” case, these are

_{r}*ϕ*= 0 and

_{r}*θ*= 0 (or

_{r}*x*=

*y*= 0, see Figs. 2 and 4, and Table 1). The expression for

*A*

_{0,1,2}(

*r*→) in this case becomes 0 for the

*x*-component; the

*y*-component is

*y*-component of

*E*→(

*r*→),

*E*(

_{y}*r*→) ∝

*A*

_{0}(

*r*→) -

*A*

_{2}(

*r*→), is not zero. Using numerical integration, we found

*A*

_{0}>>

*A*

_{2}for all values of

*α*in the relevant interval. We further notice that the narrowest

*A*

_{0}(

*r*) is obtained if one sets

*θ*=

*α*and

*χ*= 0 for the functions in the integral. Therefore, we neglect

*A*

_{2}and approximate

*E*→(

*r*→) as

*A*

_{0}(

*r*) is almost the same (within a few percent deviation) for all values of

*α*, and that an accurate representation of

*A*

_{0}(

*r*) (with any

*α*) is given by Eq. (5) if a is set to ≈ 60° in this equation (a reasonable agreement is found if

*α*is between ≈ 60° and 90°). Then, the PSF (

*n*= 2) is

_{exc}*λ*/

*n*(cf. Table 1). This approximation is valid only for the central peak of the PSF, because the function in Eq. (5) features side lobes that are missing in most of the space around the central peak of the real “Inf. beams, 3D” PSF (Fig. 2). If the side lobes are ignored, the estimate for Veff based on Eq. (5) is

*V*

_{1}≈ 0.069 (

*λ*/

*n*)

^{3}. However, the real PSF is about 1.5 times wider in the

*y*- than in the

*z*- or

*x*-directions, and the estimate for

*V*

_{eff}with

*r*replaced by 1.5

*r*is

*V*

_{2}≈ 0.231 (

*λ*/

*n*)

^{3}. Using these values, we estimate

*V*

_{eff}= (

*V*

^{2}

_{1}

*V*

_{2})

^{1/3}, which gives

*V*

_{eff}≈ 0.1 (

*λ*/

*n*)

^{3}, close to the computed value (Fig. 3).

## 3. Conclusion

*V*

_{eff}can be decreased two-fold in comparison with the 4Pi

*V*

_{eff}, which itself is ≈ 1/2 of the

*V*

_{eff}for the conventional 1-beam microscope. With more than 3 or 4 beams, further decrease in

*V*

_{eff}is not significant. In the limiting case of an infinite number of beams,

*V*

_{eff}is decreased by another 30 to 50 %, but this is achieved only if one employs an abstract and idealized assumption of arbitrary

*α*. The smallest possible

*V*

_{eff}achieved under this assumption is ≈ 1/2 of that in the 3-beam case. Thus, the 3-beam setup provides the smallest value of

*V*

_{eff}for realistic

*α*, and this value is close to the theoretical limit.

*α*= 60°) may be difficult to realize in practice, and the actual focal volume of such setup may be somewhat larger than in theory. The setups with three, four, or six symmetrically focused illumination beams have been constructed [3

**65**, 3367–3372 (1994). [CrossRef]

**26**, 1684–1686 (2001). [CrossRef]

15. J. Swoger, J. Huisken, and E. H. K. Stelzer, “Multiple imaging axis microscopy improves resolution for thick-sample applications,” Opt. Lett. **28**, 1654–1656 (2003). [CrossRef] [PubMed]

*z*) dimension is the highest one among the systems studied. Techniques such as STED [17

17. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission,” Opt. Lett. **19**, 780–782 (1994). [CrossRef] [PubMed]

18. M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA **102**, 13,081–13,086 (2005). [CrossRef]

21. M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods **3**, 793–796 (2006). [CrossRef] [PubMed]

19. S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-High Resolution Imaging by Fluorescence Photoactivation Localization Microscopy,” Biophys. J. **91**, 4258–4272 (2006). [CrossRef] [PubMed]

20. E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science **313**, 1642–1645 (2006). [CrossRef] [PubMed]

*V*

_{eff}furnished by, e.g., 3-beam setup, might be useful.

## Acknowledgments

## References and links

1. | S. Hell and E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation,” Opt. Commun. |

2. | B. Bailey, D. L. Farkas, D. L. Taylor, and F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature |

3. | S. Lindek, R. Pick, and E. H. K. Stelzer, “Confocal Theta Microscope with Three Objective Lenses,” Rev. Sci. Instrum. |

4. | M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Sevenfold improvement of axial resolution in 3D wide-field microscopy using two objective lenses,” Proc. SPIE |

5. | V. Krishnamurthi, B. Bailey, and F. Lanni, “Image processing in 3D standing-wave fluorescence microscopy,” Proc. SPIE |

6. | M. A. A. Neil, R. Juskaitis, and T. Wilson, “Method of obtaining optical sectioning by using structured light in a conventional microscope,” Opt. Lett. |

7. | R. Heintzmann and C. G. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE |

8. | M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “I5M: 3D widefield light microscopy with better than 100 nm axial resolution,” J. Microsc. |

9. | M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. |

10. | J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA |

11. | G. E. Cragg and P. T. C. So, “Lateral resolution enhancement with standing evanescent waves,” Opt. Lett. |

12. | O. Haeberle, C. Xu, A. Dieterlen, and S. Jacquey, “Multiple-objective microscopy with three-dimensional resolution near 100 nm and a long working distance,” Opt. Lett. |

13. | P. T. C. So, H.-S. Kwon, and C. Y. Dong, “Resolution enhancement in standing-wave total internal reflection microscopy: a point-spread-function engineering approach,” J. Opt. Soc. Am.Ã |

14. | J. Ryu, S. S. Hong, B. K. P. Horn, D. M. Freeman, and M. S. Mermelstein, “Multibeam interferometric illumination as the primary source of resolution in optical microscopy,” Appl. Phys. Lett. |

15. | J. Swoger, J. Huisken, and E. H. K. Stelzer, “Multiple imaging axis microscopy improves resolution for thick-sample applications,” Opt. Lett. |

16. | S. W. Hell, “Far-Field Optical Nanoscopy,” Science |

17. | S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission,” Opt. Lett. |

18. | M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA |

19. | S. T. Hess, T. P. K. Girirajan, and M. D. Mason, “Ultra-High Resolution Imaging by Fluorescence Photoactivation Localization Microscopy,” Biophys. J. |

20. | E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging Intracellular Fluorescent Proteins at Nanometer Resolution,” Science |

21. | M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods |

22. | L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence Fluctuation Spectroscopy in Subdiffraction Focal Volumes,” Phys. Rev. Lett. |

23. | A. Arkhipov, J. Hüve, M. Kahms, R. Peters, and K. Schulten, “Continuous fluorescence microphotolysis and correlation spectroscopy using 4Pi microscopy,” Biophys. J. |

24. | P. Schwille and E. Haustein, “Fluorescence Correlation Spectroscopy: A Tutorial for the Biophysics Textbook Online (BTOL)“ (2002), http://www.biophysics.org/education/techniques.htm. |

25. | M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, “4Pi microscopy with negligible sidelobes,” New J. Phys. |

26. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. Lond. A. (Math. Phys. Sci.) |

27. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A. (Math. Phys. Sci.) |

28. | W. Humphrey, A. Dalke, and K. Schulten, “VMD - Visual Molecular Dynamics,” J. Mol. Graphics |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(180.1790) Microscopy : Confocal microscopy

(180.2520) Microscopy : Fluorescence microscopy

(180.6900) Microscopy : Three-dimensional microscopy

(350.5730) Other areas of optics : Resolution

**ToC Category:**

Microscopy

**History**

Original Manuscript: December 22, 2008

Revised Manuscript: February 4, 2009

Manuscript Accepted: February 7, 2009

Published: February 11, 2009

**Virtual Issues**

Vol. 4, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Anton Arkhipov and Klaus Schulten, "Limits for reduction of effective focal volume in multiple-beam light microscopy," Opt. Express **17**, 2861-2870 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2861

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### References

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