## Excitation strategies for optical lattice microscopy

Optics Express, Vol. 13, Issue 8, pp. 3021-3036 (2005)

http://dx.doi.org/10.1364/OPEX.13.003021

Acrobat PDF (2576 KB)

### Abstract

Recently, new classes of optical lattices were identified, permitting the creation of arbitrarily large two- and three-dimensional arrays of tightly confined excitation maxima of controllable periodicity and polarization from the superposition of a finite set of plane waves. Here, experimental methods for the generation of such lattices are considered theoretically in light of their potential applications, including high resolution dynamic live cell imaging, photonic crystal fabrication, and quantum simulation and quantum computation using ultracold atoms.

© 2005 Optical Society of America

## 1. Introduction

6. LZ. Cai, X.L. Wang, and Y.R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. **27**, 900–902 (2002). [CrossRef]

7. L. Yuan, G.P. Wang, and X. Huang, “Arrangements of four beams for any Bravais lattice,” Opt. Lett.1769–1771 (2003). [CrossRef] [PubMed]

8. 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]

10. 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]

11. 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]

12. 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**, 7232–7236 (2000). [CrossRef] [PubMed]

13. V. Berger, O. Gauthler-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. **82**, 60–64 (1997). [CrossRef]

14. M. Campbell, D.N. Sharp, M.T. Harrison, R.G. Denning, and A.J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature **404**, 53–56 (2000). [CrossRef] [PubMed]

15. D.C. Meisel, M. Wegener, and K. Busch, “Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: Symmetries and complete photonic band gaps,” Phys. Rev. B **70**, 165104 (2004). [CrossRef]

16. P.S. Jessen, *et al*., “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. **69**, 49–52 (1992). [CrossRef] [PubMed]

18. G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. **70**, 2249–2252 (1993). [CrossRef] [PubMed]

19. M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature **415**, 39 (2002). [CrossRef] [PubMed]

20. J.I. Cirac and P. Zoller, “How to manipulate cold atoms,” Science **301**, 176–177 (2003). [CrossRef] [PubMed]

21. R. Dumke, *et al*, “Micro-optical realization of arrays of selectively addressable dipole traps: A scalable configuration for quantum computation with atomic qubits,” Phys. Rev. Lett. **89**, 097903 (2002). [CrossRef] [PubMed]

*λ*, and that the intensity has varied on the same order as the periodicity. Thus, they have been too close packed to serve as three-dimensional arrays of individually resolvable, diffraction-limited excitation foci or trapping potentials. Expanding upon an existing crystallographic formalism [23

23. K.I. Petsas, A.B. Coates, and G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A **50**, 5173–5189 (1994). [CrossRef] [PubMed]

**e**

_{n}and wavevector

**k**

_{n}properties of their constituent plane waves necessary to create, at many periodicities both comparable to and large compared to

*λ*, lattices of isolated excitation maxima of controllable polarization, each tightly confined in all directions, [24]. An example is shown in Fig. 1.

## 2. Confined beams and bound lattices

**k**

_{n}. The first issue is therefore to determine the effect of this substitution on the properties of the lattice.

*bound lattices*intentionally confined to

*excitation zones*defined by the intersection of multiple beams of finite width. First, for nonlinear applications, absorption or scattering cross-sections are often sufficiently small to require excitation intensities accessible only with highly focused illumination. Second, confined excitation limits photobleaching and potential photodamage outside the field of view within biological specimens. Third, phase errors introduced by dynamically varying refractive index inhomogeneities within such specimens can be greatly reduced at a single lattice point with adaptive optical techniques [26

26. M.J. Booth, M.A.A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA **99**, 5788–5792 (2002). [CrossRef] [PubMed]

*a/f*→0 limit of a lens of radius

*a*and focal length

*f*, so we initially concentrate on the latter, using the geometry shown in Fig. 2(a). Thus, an ideal plane wave

*x,t*)=e

_{n}exp[

*i*(

*kz-ωt*)],

**e**

_{n}·

**ê**

_{z}=0 normally incident from

*z*<0 on a circular lens centered at (

*x,y*)=(0,0) in an opaque screen at

*z*=0 creates a diffraction field

**e**

_{n}(

**x**,

*t*) in the

*z*>0 half space that, for

*ka*>>1, can be estimated using the generalized Kirchhoff integral [27]:

*W*(yellow in Fig. 2(a)) of radius

*f*and solid angle Ω subtended by the lens,

**ê**

_{W}is the unit vector normal to

*W*, and [

**e**

_{W}(

**x**′,

*t*)]

_{n}is given by the projection of the incident field

**x**,

*t*) onto

*W*.

*na*) limit and use the paraxial approximation

*a/f*≡sin

*α*≈

*α*≪1, since only a fairly narrow distribution of wavevectors is thereby introduced. Then, [

**e**

_{W}(

**x**′,

*t*)]

_{n}≈

*x*′,

*y*′,0,

*t*). Further restricting our attention to field points

**x**satisfying

*ka*(

*a/z*)

^{3}≪1, Eq. (1) can then be expressed in cylindrical coordinates

**x**≡(

*ρcosθ,ρ*sin

*θ, z*),

**x**′≡(

*ρ*′cos

*θ*′,

*ρ*′sin

*θ′,z*′) as [28]:

*z*

_{f}≡

*z-f*from the focal point. For

*zf*satisfying

*ka*(

*a/f*)(

*z*

_{f}

*/f*)

^{2}<<1, we find:

**ê**

_{z}, modified by a complex multiplier that determines the nature of the field confinement. As described elsewhere [28], the numerical evaluation of the integrals

*C*(

*u,v*) and

*S*(

*u,v*) in these multipliers is expedited by relating each to Lommel functions, a group of rapidly converging infinite series of Bessel functions.

*a*=30

*λ*to those of a lens with

*a/f*=0.012, expected to yield a beam waist at the focal point of similar size. Although either method provides the desired amplitude confinement, the lens approach is preferred, because it introduces negligible phase variation is over the entire central amplitude peak in the neighborhood of the focal point. Thus, throughout the excitation zone, the phase relationship between lens-confined beams of a bound lattice will remain nearly identical to the phase relationship between the corresponding plane waves of the associated ideal lattice, and lattice distortion will be minimized. Furthermore, whereas the diffraction field from the aperture diverges with increasing distance from its defining screen, the lens field converges to yield a higher intensity

*I*(0,

*v*)/

*I*

_{o}=2

*J*

_{1}(

*v*)/

*v*)

^{2}and minimum width:

*λ*and the ratio

*a/f*. Thus, extremely small excitation zones of width 2

*ρ*<<

*a*can be created at arbitrarily large distances through appropriate choice of

*a*and

*f*. This simplifies the task of constructing an optical system to produce the large number of beams from diverse angles necessary for many lattices (e.g., Fig. 1(a)).

*na*circular lens, uniformly illuminated at normal incidence, with the lens axis aligned to the plane wave propagation vector

**k**

_{n}, and having input polarization

**e**

_{n}and phase set to achieve the desired basis field (i.e., the field that identically exists in every primitive cell of the ideal lattice) as described in Ref. [24]. Initially we consider only identical lenses sharing a common focal point, deferring more sophisticated geometries until later.

**e**(

**x**,

*t*) resulting when this arrangement is applied with lenses of

*na*=0.08 to the 24 plane waves of an ideal simple cubic lattice of period

**e**(

**x**)|

^{2}. Maximally symmetric composite lattices of the cubic crystal group such as this, created with all plane waves consistent with the symmetry operations of the group, are of particular interest because they can produce isolated excitation maxima confined to near the diffraction limit in all directions when the basis is chosen as described in Ref. [24]. As expected for the wide range of ideal wavevector directions involved, the excitation zone is nearly spherically symmetric and, as indicated in the central slice in Fig. 4(c), exhibits an envelope full width at half maximum 2

*ρ*≈9.6

*λ*in rough agreement with Eq. (5). Furthermore, the widths of the

*individual*excitation maxima and the background strength elsewhere within the excitation zone are nearly identical those of the ideal lattice, which is consistent with expectations based on the uniform phase observed in Fig 3(b).

*na*of the lenses is confirmed in the animation accompanying Fig. 4. Although beyond the limit of

*a/f*<<1 assumed above, the

*na*can be increased until the envelope width as given by Eq. (5) becomes smaller than the lattice spacing, at which point the single central lattice point dominates. In this limit, the total solid angle subtended by all

*N*lenses,

*π*, or 2

*π*if only the beams within one half space are included. Thus, excitation in single focus imaging systems such as the confocal [29

29. M. Gu, *Principles of Three-Dimensional Imaging in Confocal Microscopes* (World Scientific, 1996). [CrossRef]

*π*microscope [30

30. S. Hell and E.H.K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A **9**, 2159–2166 (1992). [CrossRef]

## 3. Lattice excitation through high numerical aperture objectives

**k**

_{n}outside the solid angle occluded by the objective, confinement of the excitation maxima along the objective axis

**ê**

_{z}would be compromised, as in the confocal microscope, since the maximum spatial frequency along

**ê**

_{z}would be reduced from 2|

**k**

_{n}·

**ê**

_{z}|

_{max}to approximately |

**k**

_{n}·

**ê**

_{z}|

_{max}-|

**k**

_{n}·

**ê**

_{z}|

_{min}.

*NA*objectives both to deliver multiple beams to the excitation zone and to collect the resulting signal. This is analogous to the 4

*π*microscope, except that illumination at specific, discrete points within the rear pupil of each objective creates a bound lattice with a plurality of excitation maxima, as opposed to the single, multi-lobed focus of the 4

*π*microscope under uniform illumination.

*na*≡

*a/f*<<1 propagating in the direction

**k**

_{n}defined by a corresponding plane wave of a given ideal lattice, ray tracing indicates that the rear pupil (radius

*A*) of an infinity corrected objective of numerical aperture

*NA*, focal length

*F*

_{o}, and object space refractive index

*n*should be illuminated at position:

*b=na*·

*F*

_{o}. The bound lattice resulting from the superposition of all such beams is then derived by determining, in the vicinity of the focal point of the objective, the diffracted field

**e**

_{n}(

**x**,

*t*) arising from each.

*NA*lens [31

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

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

*kF*

_{o}>>1 and points

**x**near the focal point such that

**x-x**′≈

*F*

_{o}

**ê**

_{W}, the generalized Kirchoff diffraction integral of Eq. (1) simplifies to:

**x**″,

*t*) at the plane of the rear pupil to the resulting field [

**e**

_{W}(

**x**′

*t*)]

_{n}on

*W*, and evaluating the resulting integral.

*na*paraxial analysis leading to Eqs. (2–4), the angles of refraction were sufficiently small that [

**e**

_{W}(

**x**′

*t*)]

_{n}could be approximated by directly mapping

**x**′,

*t*) onto

*W*. Now, however, because the solid angle

**x**″ in the rear pupil is transformed into a convergent ray propagating towards the focal point in the direction:

**ê**

_{φ}(

**x**″) associated with the input ray is unchanged in this process (i.e.,

**ê**

_{φ}(

**x**′)=

**ê**

_{φ}(

**x**″)), the radial component

**ê**

_{ρ}(

**x**″) is rotated by refraction in the objective to the direction

**ê**

_{θ}(

**x**′), where:

*ψ*

_{n}(

*x′, y′*)=

*ψn*(

**x**″) gives the variation in the amplitude of the input beam across the rear pupil (i.e.,

**x**″,

*t*)=

*ψ*

_{n}(

**x**″)

*iωt*)), and

*χ*(

*x′,y′*) describes how this amplitude is transformed upon projection onto

*W*.

*dE*

_{n}(

**x**″)∝|

**x**″)|

^{2}

*dx″ dy*″ in area

*dx″dy*″ within the rear pupil maps onto the curved element of

*W*at

*dx′dy*′/(

**ê**

_{r}(

**x**′)·

**ê**

_{z}). Hence, energy conservation demands

**x**″

_{bc})

_{n}of Eq. (6) produces the central output ray along

**k**

_{n}having a field [

**e**

_{W}(

**x**′,

*t*)]

_{n}as given by Eq. (10) with

**ê**

_{r}(

**x**′)=

**k**

_{n}/

*k*in Eq. (9). This field must match the polarization

**e**

_{n}of the corresponding plane wave of the ideal lattice to achieve the desired basis. Therefore, working backwards through the objective:

**ê**

_{θ})

_{n}=

**ê**

_{θ}((

**x**″

_{bc})

_{n}), (

**ê**

_{ρ})

_{n}=

**ê**

_{φ}((

**x**″

_{bc})

_{n}), and (

**ê**

_{φ})

_{n}=

**ê**

_{φ}((

**x**″

_{bc})

_{n}). Combining these results with Eqs. (8–10), we finally arrive at the field of the convergent beam in the vicinity of the focal point of the objective:

*ψ*

_{n}(

**x**″)=

*e*

_{o}), Eq. (12) yields the single excitation maximum at the focal point of the confocal microscope, and described in terms of three integrals

*I*

_{0},

*I*

_{1},

*I*

_{2}, each over the single spherical coordinate variable

*θ*′, in Ref. [32

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

*b<A*) offset ((

**x**″

_{bc})n≠0) illumination considered here, the double integral in Eq. (12) must be evaluated numerically, using a mesh of |Δ

**x**′|≪

*F*

_{o}/(|

**x**|

_{max}/

*λ*) to accurately estimate the phase term exp(

*ik*|

**x-x**′|).

*ψ*

_{n}(

**x**″) that leads to a lattice bound to an excitation zone of the desired size, shape, and edge sharpness, yet that closely approximates the ideal lattice therein. Four possibilities are considered across the columns of Fig. 5, and evaluated for the particular case of a maximally symmetric composite simple cubic lattice of period

**e**(

**x**)|

^{2}, chosen because all eight constituent wavevectors

**k**

_{n}=

*k*(±

**ê**

_{x}±

**ê**

_{y}±

**ê**

_{z})/√3 lie within the illumination cones of opposed

*NA*=1.2 water immersion objectives.

*b*=0.055

*A*, expected to yield an excitation zone of width 2

*ρ*~10

*λ*, is used. The mask is placed a distance

*z*

_{mask}=2·10

^{4}

*λ*<<

*b*

^{2}/

*λ*from the rear pupil plane sufficiently remote to be outside the path of the collected signal, yet sufficiently near that the wavevector spread Δ

*k/k*~O(

*λ/b*) introduced by the aperture does not significantly influence the input beam field over the majority of its diameter at the rear pupil plane (i.e., so that (Δ

*k/k*)

*z*<<

*a*).

*ψ*

_{n}(

**x**″)| and phase arg[

*ψ*

_{n}(

**x**″)] plots in rows 1 and 2, respectively, of the particular input beam centered at (

**x**″

_{bc})

_{n}=0.64(

**ê**

_{x}+

**ê**

_{y})/

*A*demonstrate that the beam remains largely confined to the original aperture diameter at the pupil plane, with relatively uniform phase within this diameter. Consequently, the electric field energy density |

**e**

_{n}(

**x**

_{⊥})|

^{2}across the focal point of the resulting convergent beam is confined to approximately the expected width (row 3) in the

**ê**

_{φ}direction, and the phase arg[

**e**

_{n}(

**x**

_{⊥})·

**e**

_{n}] of its desired electric field component exhibits the uniformity (row 4) across this width necessary for the creation of a bound lattice of minimal distortion (rows 5, 6).

*z*

_{mask}<<

*b*

^{2}/

*λ*. Both models also indicate an elliptical compression of the convergent beam in the

**ê**

_{θ}direction (row 3) that distorts the shape of the final excitation zone (row 5). This occurs, as shown in Fig. 2(b), when the circular input beam is projected onto the curved surface

*W*, creating a convergent beam initially

*stretched*by a factor

*k*/(

**k**

_{n}·

**ê**

_{z}) along

**ê**

_{θ}based on its location (

**x**″

_{bc})

_{n}in the rear pupil. The effective

*na*of the beam is then

*kb*/(

*F*

_{o}

**k**

_{n}·

**ê**

_{z}) in this direction rather than

*b/F*

_{o}, creating an elliptical beam waist at the focus reduced by (

**k**

_{n}·

**ê**

_{z})/k along

**ê**

_{θ}according to Eq. (5).

**k**

_{n}·

**ê**

_{z})/k in the

**ê**

_{ρ}direction. Based on the above results an ideal elliptical input field model can be used (column 3, rows 1 and 2), provided

*z*

_{mask}<< [

*b*(

**k**

_{n}·

**ê**

_{z})/

*k*]

^{2}/

*λ*. The resulting convergent beam exhibits the expected circular waist with good phase uniformity (rows 3, 4), and a spherical excitation zone (rows 5, 6).

*ρ*increases, the required aperture diameter or major axis width 2

*b*≈0.52

*λF*

_{o}

*/ρ*becomes so small that the mask can no longer be placed outside the path of the collected signal while still satisfying

*z*

_{mask}<<

*b*

^{2}/

*λ*as required to achieve good amplitude confinement and phase uniformity of the input and convergent beams. For fluorescence microscopy, a patterned dichroic mirror or dichroic mask in the signal path could be used instead. More generally, a low

*na*lens, such as in Figs. 3(b), could be used to define each input beam and placed at a large distance from the rear pupil by increasing the focal length

*f*and radius

*a*in tandem. An excitation zone of width 2

*ρ*should then result for an input beam diameter 2

*b*≈0.52

*nλf/a*, or

*a/f=na=nρ/F*

_{o}. The case of 2

*ρ*≈10

*λ*, yielding

*a/f*≈0.001 for

*n*=1.33 and

*F*

_{o}=6.6.10

^{3}

*λ*, is considered in column 4 of Fig. 5.

*ψ*

_{n}(

**x**″)| (row 1) and excitation zone (row 5) are indeed confined to the expected dimensions. However, even when the lens is placed with its focal point coincident with the rear pupil plane, so that arg[

*ψ*

_{n}(

**x**″)] (row 2) is constant over the central maximum, there remains enough beam energy of opposite phase in the first annulus of |

*ψ*

_{n}(

**x**″)| that the convergent beam (row 3,4) is distorted, affecting both the excitation zone symmetry and lattice quality near its periphery (row 6). Therefore, the aperture method of defining the input beams is preferred whenever the condition

*z*

_{mask}<<

*b*

^{2}/

*λ*can be met.

## 4. Tailoring the lattice to the optical geometry

*via*precision, high

*NA*apochromatic microscope objectives permits compensation for chromatic and spherical aberration. The former is necessary to achieve diffraction-limited performance and correctable dispersion with ultrafast, pulsed excitation, and the latter is required to achieve the amplitude and phase characteristics for each convergent beam predicted by Eq. (12) and seen in Fig. 5. Cover glass correction can also be used when illuminating substrate mounted specimens to compensate for the rapidly varying optical path across oblique convergent beams of moderate

*na*.

*NA*=1.2) insufficient to encompass all beams of most maximally symmetric composite lattices, particularly those of the cubic crystal group that can produce particularly symmetric, well-confined excitation maxima. One solution, as shown in Fig. 6(a), is simply to omit those beams (translucent gold) whose ideal wavevectors

**k**

_{n}fall outside the illumination cones (translucent green) defined by the objectives. However, the transverse confinement [(

**k**

_{m}-

**k**

_{n})·

**ê**

_{x,y}]

_{max}of the maxima (Fig. 6(b)) and their contrast with respect to the remainder of each primitive cell (Figs. 6(c),(d)) is thereby reduced, while the weighting of spatial frequencies (

**k**

_{m}-

**k**

_{n})·êz along the optical axis is increased. This leads to a multi-lobed excitation profile (Figs. 6(b),(d)) similar to the 4

*π*microscope, and introduces uncertainty as to from which lobe a given signal photon originated.

**ê**

_{z}, and hence the number of wavevectors

**k**

_{n}that lie within the objective illumination cones can often be increased by choosing an appropriate orientation. With the omission of fewer beams, the transverse confinement of the maxima and their contrast is thereby improved. Alternatively, the number of beams within the illumination cones can be increased by changing the geometric properties of the ideal lattice and hence the distribution of its constituent wavevectors. Thus, all the wavevectors of any 2D lattice oriented with its invariant axis along

**ê**

_{z}can be included by reducing their common cone angle

*θ*

_{2D}=cos

^{-1}(

**k**

_{n}·

**ê**

_{z}

*/k*) to fall within the maximum illumination angle

*θ*

_{max}=sin

^{-1}(

*NA/n*) of the objective. Similarly, by decreasing the unit cell aspect ratio

*c/a*of any 3D lattice having symmetry (e.g., tetragonal) permitting such adjustment and oriented with

*c*-axis ||

**ê**

_{z}, the wavevector distribution can be altered to produce the highest possible transverse spatial frequency 2

*kNA/n*and resulting maxima confinement for the given objectives.

## 5. Hybrid excitation

**e**(

**x**)|

^{2}, consisting of 24 plane waves. Eight input beams are introduced at the rear pupil of each of two opposed

*NA*=1.2 water immersion objectives (Fig. 7(a)), and elliptically compressed by (

**k**

_{n}·

**ê**

_{z})/

*k*along

**ê**

_{ρ}to produce a spherical excitation zone. The extent of the beams is limited to yield an effective

*na*for the corresponding convergent beams of 0.02, insuring that the beams nearest the periphery fit entirely within the rear pupil (Fig. 7(b)). The polarization

**e**(

**x**)·

**ê**

_{x}| at the excitation maxima.

*na*= 0.02, each illuminated at the polarization

**e**

_{n}of the corresponding ideal plane wave to achieve the desired basis, and each confocal with the common focal point of the objectives (Fig. 7(c)).

*via*Eq. (12) and the latter

*via*Eq. (2b), is shown in Figs. 7(d),(e). The excitation zone is indeed spherically symmetric, near the expected diameter, and closely approximates the ideal lattice therein.

33. S.F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**, 1601–1613 (1991). [CrossRef]

*na*of the external beams or by phase compensating with a spatial light modulator (SLM), a simpler prescription may be to dispense with the substrate and immobilize the cell instead with a patch clamp pipette or a single or multiple point [34

34. E.R. Dufresne and D.G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. **69**, 1974–1977 (1998). [CrossRef]

*λ*

_{trap}.

## 6. Shaping the excitation zone

*na*of the convergent beams is left as the only controllable parameter. To achieve greater control over the shape, uniformity, and sharpness of the zone, similar control must be exerted over the properties of the convergent beams and how they intersect to define the zone.

**k**

_{n}(Fig. 8(a)). For every internal beam, the dimensions of the input beam at the rear pupil of the objective in the (

**ê**

_{φ})

_{n}=

**ê**

_{φ}((

**x**″

_{bc})

_{n}) and (

**ê**

_{ρ})

_{n}=

**ê**

_{ρ}((

**x**″

_{bc})

_{n}) directions of Eq. (9) are then determined by applying Eq. (5) to the cross-sectional widths

*c*

_{φ},

*c*

_{θ}of the zone in the (

**ê**

_{φ})

_{n}and (

**ê**

_{θ})

_{n}=

**ê**

_{θ}((

**x**″

_{bc})

_{n}) directions, respectively. An additional compression factor of (

**k**

_{n}·

**ê**

_{z})/k is applied along (

**ê**

_{ρ})n to compensate for the ellipticity effect seen in Figs. 2(b) and 5. For external beams, the effective lens

*na*in directions orthogonal to

**k**

_{n}is set inversely to the cross-sectional dimensions of the zone as per Eq. (5).

*u*

_{l}

*z*

_{l})/

*u*

_{l}

*z*

_{l}]

^{2}axial dependence of the intensity near the focus of each convergent beam, which serves to further restrict the excitation zone for beams of sufficiently high

*na*. Nevertheless, the

*xy*and

*yz*plots in Figs. 8(c) and 8(d), respectively, indicate that the lattice properties are well preserved within this envelope.

*Z*smaller, overlapping excitation sub-zones. The boundary sharpness is then dictated by the width of each sub-zone, and the shape of the overall zone is determined by their spatial distribution. The positions as well as the phases of the N beams comprising each sub-zone should be chosen relative to all others to create a uniform field of lattice excitation maxima within the composite zone.

*x*

_{t}in the vicinity of the focal point by illuminating the lens at an angle

*α*

_{t}

*=Δx*

_{t}

*/f*to the lens axis. Axial translation by Δ

*z*is then accomplished by illuminating the lens with a spherical wavefront of curvature

*s=f*

^{2}/Δz rather than a planar one.

**x**″

_{bc})

_{n}and having a complex confinement factor

*ψ*

_{n}(

**x**″). However, the beam is now considered to originate from a point source at position (

**x**″

_{s})

_{n}beyond the rear pupil plane prior to its confinement by means such in Fig. 5, introducing an additional phase term Φ

_{n}(

*x′,y′*)=

*k|x*′

**ê**

_{x}+

*y*′

**ê**

_{y}-[(

**x**″

_{s})

_{n}-(

**x**″

_{bc})

*n*] | characteristic of an offset spherical wave in Eq. (7):

*s*=(

**x**″

_{s})

_{n}-(

**x**″

_{bc})

_{n}, then:

*x*′

_{φ}≡

**x**′·(

**ê**

_{φ})

_{n},

**x**′

_{ρ}≡

**x**′·(

**ê**

_{ρ})

_{n},

*x*″

_{φ}≡(

**x**″

_{s})

_{n}·(

**ê**

_{φ})

_{n},

*x*″

_{ρ}≡(

**x**″

_{s})

_{n}·(

**ê**

_{ρ})

_{n}, (

**x**″

_{bc})

_{n}=-

*ρbc*(

**ê**

_{ρ})

_{n},

*ε*′

_{φ}≡[

**x**′-(

**x**″

_{bc})

_{n}]·(

**ê**

_{φ})

_{n}, and

*ε*′

_{ρ}≡[

**x**′-(

**x**″

_{bc})

*n*]·(

**ê**

_{ρ})

_{n}. However, for points x sufficiently close to the objective focal point to satisfy

*k*|

**x-x**′|≈

*k*|

**x**′|-

*k*

**x**·

**x**′/|

**x**′|can be expressed as:

*x*

_{φ}≡

**x**·(

**ê**

_{φ})

_{n},

*x*

_{θ}≡

**x**·(

**ê**

_{θ})

_{n},

*xk*≡

**x**·

**k**

_{n}

*/k*, and cosγ≡

**k**

_{n}·

**ê**

_{z}

*/k*. Thus, combining Eqs. (13–15) and comparing the result to Eq. (7), we see that Eq. (13) leads to a convergent beam field

**e**

_{n}(

**x**,

*t*) identical to that in Eq. (12), except offset by a distance ∆

**x**from the focal point of the objective, when the spherical wave generating the input beam is of radius:

**x**″

_{bc})

_{n}and centered beyond the rear pupil of the objective at:

*x*

_{φ}≡≡Δ

**x**·

**ê**

_{φ}, Δ

*x*

_{θ}≡Δ

**x**·

**ê**

_{θ}, and Δ

*z*≡Δ

*x*

_{θ}sin γ+(Δ

**x**·

**k**

_{n})cos γ/

*k*. In other words, the effect of the wavefront curvature is to displace the waist of the convergent beam along the wavevector

**k**

_{n}, and the effect of the source offsets

*x*″

_{φ},

*x″*

_{ρ}from (

**x**″

_{bc})

_{n}is to displace the waist along the (

**ê**

_{φ})n and (

**ê**

_{θ})

_{n}directions. In the low

*NA*limit with full pupil illumination, Eqs. (16) and (17) reduce to the standard paraxial result described above.

**e**(

**x**)|

^{2}, created with 16 internal beams through each of two opposed objectives, and 16 external beams between them. To yield a small sub-zone capable of contributing to a sharp composite zone boundary,

*na*=0.08 was chosen for all 48 convergent beams. Comparison (Fig. 9(b)) of the undisplaced lattice, created with input beams of flat phase, with a lattice displaced by Δ

**x**=(3

**ê**

_{x}+4

**ê**

_{x}+5

**ê**

_{x})

*λ*, created with confined input beams of phase as dictated by Eqs. (16) and (17) and shown in Fig. 9(a), demonstrates that the excitation zone can be translated without affecting the underlying lattice. This is confirmed by the xy plots through the centers of the two zones in Figs. 9(c),(d).

*ρ*≈7

*λ*in Fig. 9), and the phases of all beams in all sub-zones should be adjusted to yield maximal constructive interference at common lattice points.

## 7. Conclusions

*ψ*

_{n}(

**x**″) and superposition factor

*Π*

_{n}(

**x**″) of every input beam in a group. This should allow for rapid, programmable control of the lattice symmetry, Wigner-Seitz primitive cell shape, spatial frequency content, overall basis field

**e**(

**x**,

*t*), and excitation zone envelope. In turn, this may lead to an unprecedented level of dynamic control over the shape, width, depth, and eigenstate selectivity of the trapping potentials used in optical lattice based quantum simulators and quantum computers. In this regard, it should also be noted that the convergent beam displacement equations (Eqs. (16,17)) can be extended in the full pupil illumination limit to determine an SLM-mediated phase response yielding a steerable, high

*NA*single focus trap capable of loading single atoms into individual potential minima from a reservoir, thereby permitting the creation a defect-free atomic lattice of individually addressable sites.

35. G. Timp, *et al*., “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. **69**, 1636–1639(1992). [CrossRef] [PubMed]

37. M. Mützel, *et al*., “Atomic nanofabrication with complex light fields,” Appl. Phys. B. **77**, 1–9 (2003). [CrossRef]

*λ*, while the size of each simultaneously written copy of an arbitrarily complex pattern in each primitive cell can be large compared to

*λ*. Similarly, massively parallel fabrication of aperiodic 3D structures large compared to

*λ*should be possible with 3D sparse composite lattices in a thick photosensitive material. In fact, with basis fields chosen as described in [24], the contrast between each lattice maximum and the surrounding background should be superior to that seen near the focus of a confocal microscope, potentially leading to reduced proximity effects and hence the ability to write more closely packed features without fear of overexposure. Significantly higher axial resolution is also obtained. Two-photon absorption (TPA) photopolymerization [38

38. T. Tanaka, H.B. Sun, and S. Kawata, “Rapid sub-diffraction-limit laser micro/nanoprocessing in a threshold material system,” Appl. Phys. Lett. **80**, 312–314 (2002). [CrossRef]

*l*≈

*cτ*of the ultrafast pulses of temporal width t necessary to achieve TPA naturally leads to a bound lattice excitation zone of similar size, even without spatial confinement of the constituent beams. Finally, although large period sparse composite lattices would appear to be ill suited for the fabrication of photonic crystals with band gaps on the order the wavelength used for their creation, discrete exposure at many points in each primitive cell can create such structures. In fact, recent experiments have aimed at tailoring the optical properties of photonic crystals by controlling the properties of the basis field used in their creation [39

39. X.L. Yang, L.Z. Cai, and Y.R. Wang, “Larger bandgaps of two-dimensional triangular photonic crystals fabricated by holographic lithography can be realized by recording geometry design,” Opt. Express **12**, 5850–5856 (2004). [CrossRef] [PubMed]

*NA*objectives potentially addresses the issue of collection efficiency, but the magnified images of the individual maxima at an image plane become increasingly aberrated for maxima increasingly far from the objective focal plane, reducing the detection resolution and, for spatially filtered signals, the effective collection efficiency. Means to deal with these issues and achieve optimal 3D detection resolution and efficiency will be considered in an upcoming paper [25].

40. Perkin-Elmer UltraVIEW Live Cell Imager, http://las.perkinelmer.com/content/livecellimaging/nipkow.asp

41. M. Petran, M. Hadravsky, M.D. Egger, and R. Galambos, “Tandem-scanning reflected-light microscope,” J. Opt. Soc. Am. **58**, 661–664 (1968). [CrossRef]

*π*excitation and/or detection to conventional multifocal microscopy would be challenging, but is readily accomplished with opposed objectives in lattice microscopy, to yield a substantial improvement in axial resolution. Fourth, slight improvement in transverse resolution and elimination of the axial side-lobes plaguing 4

*π*microscopy is possible using hybrid excitation (Fig. 7). Fifth, since the polarization of every input beam contributing to a lattice can be individually optimized, greater polarization purity at the foci is possible than in multifocal microscopy, where constant polarization over the rear pupil leads to depolarization effects at the foci in a high

*NA*system [42

42. K. Bahlmann and S.W. Hell, “Electric field depolarization in high aperture focusing with emphasis on annular apertures,” J. Microsc. **200**, 59–67 (2000). [CrossRef] [PubMed]

## References and links

1. | J. Zhang, R.E. Campbell, A.Y. Ting, and R.Y. Tsien, “Creating new fluorescent probes for cell biology,” Nat. Rev. Mol. Cell Biol. |

2. | D. Axelrod, “Total internal fluorescence microscopy,” in |

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

4. | S.W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. |

5. | C.Y. Dong, P.T.C. So, C. Buehler, and E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik |

6. | LZ. Cai, X.L. Wang, and Y.R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. |

7. | L. Yuan, G.P. Wang, and X. Huang, “Arrangements of four beams for any Bravais lattice,” Opt. Lett.1769–1771 (2003). [CrossRef] [PubMed] |

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

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

10. | 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. |

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

12. | 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 |

13. | V. Berger, O. Gauthler-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. |

14. | M. Campbell, D.N. Sharp, M.T. Harrison, R.G. Denning, and A.J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature |

15. | D.C. Meisel, M. Wegener, and K. Busch, “Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: Symmetries and complete photonic band gaps,” Phys. Rev. B |

16. | P.S. Jessen, |

17. | A. Hemmerich and T.W. Hänsch, “Two-dimensional atomic crystal bound by light,” Phys. Rev. Lett. |

18. | G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. |

19. | M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature |

20. | J.I. Cirac and P. Zoller, “How to manipulate cold atoms,” Science |

21. | R. Dumke, |

22. | J.I. Cirac and P. Zoller, “New frontiers in quantum information with atoms and ions,” Phys. Today |

23. | K.I. Petsas, A.B. Coates, and G. Grynberg, “Crystallography of optical lattices,” Phys. Rev. A |

24. | E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 has submitted a paper entitled “Sparse and composite coherent lattices”. |

25. | E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 is preparing a paper to be called “Detection strategies for optical lattice microscopy”. |

26. | M.J. Booth, M.A.A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA |

27. | J.D. Jackson, |

28. | M. Born and E. Wolf, |

29. | M. Gu, |

30. | S. Hell and E.H.K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A |

31. | E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London Ser. A |

32. | B. Richards and E. Wolf, ““Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A |

33. | S.F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A |

34. | E.R. Dufresne and D.G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. |

35. | G. Timp, |

36. | J.J. McClelland, R.E. Scholten, E.C. Palm, and R.J. Celotta, “Laser-focused atomic deposition”, Science |

37. | M. Mützel, |

38. | T. Tanaka, H.B. Sun, and S. Kawata, “Rapid sub-diffraction-limit laser micro/nanoprocessing in a threshold material system,” Appl. Phys. Lett. |

39. | X.L. Yang, L.Z. Cai, and Y.R. Wang, “Larger bandgaps of two-dimensional triangular photonic crystals fabricated by holographic lithography can be realized by recording geometry design,” Opt. Express |

40. | Perkin-Elmer UltraVIEW Live Cell Imager, http://las.perkinelmer.com/content/livecellimaging/nipkow.asp |

41. | M. Petran, M. Hadravsky, M.D. Egger, and R. Galambos, “Tandem-scanning reflected-light microscope,” J. Opt. Soc. Am. |

42. | K. Bahlmann and S.W. Hell, “Electric field depolarization in high aperture focusing with emphasis on annular apertures,” J. Microsc. |

43. | E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 is preparing a paper to be called “Optical lattice microscopy: implications for live cell and molecular imaging”. |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(050.1960) Diffraction and gratings : Diffraction theory

(180.6900) Microscopy : Three-dimensional microscopy

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 31, 2005

Revised Manuscript: April 4, 2005

Published: April 18, 2005

**Citation**

Eric Betzig, "Excitation strategies for optical lattice microscopy," Opt. Express **13**, 3021-3036 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3021

Sort: Journal | Reset

### References

- J. Zhang, R.E. Campbell, A.Y. Ting, and R.Y. Tsien, �??Creating new fluorescent probes for cell biology,�?? Nat. Rev. Mol. Cell Biol. 3, 906-918 (2002). [CrossRef] [PubMed]
- D. Axelrod, �??Total internal fluorescence microscopy,�?? in Methods in Cellular Imaging, A. Periasamy, ed., American Physiological Society Book Series (Oxford Univ. Press, 2001).
- G.E. Cragg and P.T.C. So, �??Lateral resolution enhancement with standing evanescent waves,�?? Opt. Lett 25, 46-48 (2000). [CrossRef]
- S.W. Hell and J. Wichmann, �??Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,�?? Opt. Lett. 19, 780-782 (1994). [CrossRef] [PubMed]
- C.Y. Dong, P.T.C. So, C. Buehler, and E. Gratton, �??Spatial resolution in scanning pump-probe fluorescence microscopy,�?? Optik 106, 7-14 (1997).
- LZ. Cai, X.L. Wang, and Y.R. Wang, �??All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,�?? Opt. Lett. 27, 900-902 (2002). [CrossRef]
- L. Yuan, G.P. Wang, and X. Huang, �??Arrangements of four beams for any Bravais lattice,�?? Opt. Lett. 1769-1771 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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, 7232-7236 (2000). [CrossRef] [PubMed]
- V. Berger, O. Gauthler-Lafaye, and E. Costard, �??Photonic band gaps and holography,�?? J. Appl. Phys. 82, 60-64 (1997). [CrossRef]
- M. Campbell, D.N. Sharp, M.T. Harrison, R.G. Denning, and A.J. Turberfield, �??Fabrication of photonic crystals for the visible spectrum by holographic lithography,�?? Nature 404, 53-56 (2000). [CrossRef] [PubMed]
- D.C. Meisel, M. Wegener, and K. Busch, �??Three-dimensional photonic crystals by holographic lithography using the umbrella configuration: Symmetries and complete photonic band gaps,�?? Phys. Rev. B 70, 165104 (2004). [CrossRef]
- P.S. Jessen, et al., �??Observation of quantized motion of Rb atoms in an optical field,�?? Phys. Rev. Lett. 69, 49-52 (1992). [CrossRef] [PubMed]
- A. Hemmerich and T.W. Hänsch, �??Two-dimensional atomic crystal bound by light,�?? Phys. Rev. Lett. 70, 410-413 (1993). [CrossRef] [PubMed]
- G. Grynberg, B. Lounis, P. Verkerk, J.-Y. Courtois, and C. Salomon, �??Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,�?? Phys. Rev. Lett. 70, 2249-2252 (1993). [CrossRef] [PubMed]
- M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, I. Bloch, �??Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,�?? Nature 415, 39 (2002). [CrossRef] [PubMed]
- J.I. Cirac and P. Zoller, �??How to manipulate cold atoms,�?? Science 301, 176-177 (2003). [CrossRef] [PubMed]
- R. Dumke, et al, �??Micro-optical realization of arrays of selectively addressable dipole traps: A scalable configuration for quantum computation with atomic qubits,�?? Phys. Rev. Lett. 89, 097903 (2002). [CrossRef] [PubMed]
- J.I. Cirac and P. Zoller, "New frontiers in quantum information with atoms and ions," Phys. Today 57, No. 3, 38-44 (2004).
- K.I. Petsas, A.B. Coates, and G. Grynberg, �??Crystallography of optical lattices,�?? Phys. Rev. A 50, 5173-5189 (1994). [CrossRef] [PubMed]
- E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 has submitted a paper entitled �??Sparse and composite coherent lattices�??.
- E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 is preparing a paper to be called �??Detection strategies for optical lattice microscopy�??.
- M.J. Booth, M.A.A. Neil, R. Juškaitis, and T. Wilson, �??Adaptive aberration correction in a confocal microscope,�?? Proc. Natl. Acad. Sci. USA 99, 5788-5792 (2002). [CrossRef] [PubMed]
- J.D. Jackson, Classical Electrodynamics, second ed. (Wiley, New York, 1975), Secs. 9.8 and 9.9.
- M. Born and E. Wolf, Principles of Optics, sixth (corrected) ed. (Pergamon, Oxford, 1980), Sec. 8.8.
- M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996). [CrossRef]
- S. Hell and E.H.K. Stelzer, �??Properties of a 4Pi confocal fluorescence microscope,�?? J. Opt. Soc. Am. A 9, 2159-2166 (1992). [CrossRef]
- E. Wolf, �??Electromagnetic diffraction in optical systems I. An integral representation of the image field,�?? Proc. R. Soc. London Ser. A 253, 349-357 (1959). [CrossRef]
- B. Richards and E. Wolf, �??�??Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,�?? Proc. R. Soc. London Ser. A 253, 358-379 (1959). [CrossRef]
- S.F. Gibson and F. Lanni, �??Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,�?? J. Opt. Soc. Am. A8, 1601-1613 (1991). [CrossRef]
- E.R. Dufresne and D.G. Grier, �??Optical tweezer arrays and optical substrates created with diffractive optics,�?? Rev. Sci. Instrum. 69, 1974-1977 (1998). [CrossRef]
- G. Timp, et al., �??Using light as a lens for submicron, neutral-atom lithography,�?? Phys. Rev. Lett. 69, 1636 -1639(1992). [CrossRef] [PubMed]
- J.J. McClelland, R.E. Scholten, E.C. Palm, and R.J. Celotta, �??Laser-focused atomic deposition�??, Science 262, 877-880 (1993). [CrossRef] [PubMed]
- M. Mützel, et al., �??Atomic nanofabrication with complex light fields,�?? Appl. Phys. B. 77, 1-9 (2003). [CrossRef]
- T. Tanaka, H.B. Sun, and S. Kawata, �??Rapid sub-diffraction-limit laser micro/nanoprocessing in a threshold material system,�?? Appl. Phys. Lett. 80, 312-314 (2002). [CrossRef]
- X.L. Yang, L.Z. Cai, and Y.R. Wang, �??Larger bandgaps of two-dimensional triangular photonic crystals fabricated by holographic lithography can be realized by recording geometry design,�?? Opt. Express 12, 5850-5856 (2004). [CrossRef] [PubMed]
- Perkin-Elmer UltraVIEW Live Cell Imager, <a href="http://las.perkinelmer.com/content/livecellimaging/nipkow.asp">http://las.perkinelmer.com/content/livecellimaging/nipkow.asp</a>
- M. Petran, M. Hadravsky, M.D. Egger, R. Galambos, �??Tandem-scanning reflected-light microscope,�?? J. Opt. Soc. Am. 58, 661-664 (1968). [CrossRef]
- K. Bahlmann and S.W. Hell, �??Electric field depolarization in high aperture focusing with emphasis on annular apertures,�?? J. Microsc. 200, 59-67 (2000). [CrossRef] [PubMed]
- E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 is preparing a paper to be called �??Optical lattice microscopy: implications for live cell and molecular imaging�??.

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