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

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References

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  24. E. Betzig, New Millennium Research, LLC, Okemos, MI 48864 has submitted a paper entitled �??Sparse and composite coherent lattices�??.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  40. 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>
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Appl. Phys. B. (1)

M. Mützel, et al., �??Atomic nanofabrication with complex light fields,�?? Appl. Phys. B. 77, 1-9 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

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]

J. Appl. Phys. (1)

V. Berger, O. Gauthler-Lafaye, and E. Costard, �??Photonic band gaps and holography,�?? J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

J. Microsc. (3)

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]

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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nat. Rev. Mol. Cell Biol. (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. 3, 906-918 (2002).
[CrossRef] [PubMed]

Nature (3)

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. 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]

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]

Opt. Express. (1)

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]

Opt. Lett (1)

G.E. Cragg and P.T.C. So, �??Lateral resolution enhancement with standing evanescent waves,�?? Opt. Lett 25, 46-48 (2000).
[CrossRef]

Opt. Lett. (4)

Optik (1)

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).

Phys. Rev. (1)

K.I. Petsas, A.B. Coates, and G. Grynberg, �??Crystallography of optical lattices,�?? Phys. Rev. A 50, 5173-5189 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

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]

Phys. Rev. Lett. (5)

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]

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]

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

Phys. Today (1)

J.I. Cirac and P. Zoller, "New frontiers in quantum information with atoms and ions," Phys. Today 57, No. 3, 38-44 (2004).

Proc. Natl. Acad. Sci (1)

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]

Proc. Natl. Acad. Sci. USA (1)

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]

Proc. R. Soc. London Ser. A (2)

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]

Rev. Sci. Instrum. (1)

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]

Science (2)

J.I. Cirac and P. Zoller, �??How to manipulate cold atoms,�?? Science 301, 176-177 (2003).
[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]

Other (8)

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�??.

D. Axelrod, �??Total internal fluorescence microscopy,�?? in Methods in Cellular Imaging, A. Periasamy, ed., American Physiological Society Book Series (Oxford Univ. Press, 2001).

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>

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]

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�??.

Supplementary Material (6)

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Figures (10)

Fig. 1.
Fig. 1.

A body-centered cubic sparse composite lattice of period 62 λ , with a basis that optimizes |e(xê z|2 at the excitation maxima: (a) wavevectors k n (green) and electric field vectors e n (different colors for tq=to +2πq/(8ω), q=0…7 for the 96 plane waves comprising the lattice; (b) isosurfaces of 0.5max(|e(x)|2) over (10λ)3 ; (c, d) plots of | e(x) |2 over (10λ)2 in the xy and yz planes shown in (b). Accompanying slide show (1.24 MB) presents plane wave properties and isosurfaces for 29 cubic lattices of differing periodicity, and includes six final frames indicating the fields e n that optimize different desired polarization states for a given lattice.

Fig. 2.
Fig. 2.

Methods for the generation of confined beams comprising a bound lattice. (a) Uniform plane wave illumination along the optical axis ê z of a lens of low numerical aperture (a/f<<1), oriented in the direction k n of the corresponding plane wave of the ideal lattice. (b) Localized, offset illumination in the rear pupil of a high numerical aperture microscope objective at the location (x bc ) n yielding a convergent beam propagating along k n . A third method, illumination through an aperture in an opaque screen, is the a/f→0 limit of case (a).

Fig. 3.
Fig. 3.

Comparison of the amplitude (translucent red) and phase (translucent green) deviation from ideal plane wave behavior for the field diffracted from: a) a 30λ radius aperture and b) a lens of a/f=0.012. Phase data is truncated after passing through ±180° for clarity. Animation accompanying (b) illustrates the evolution with increasing a/f (1.79 MB).

Fig. 4.
Fig. 4.

(a) Creation of a simple cubic bound lattice (period 11 λ 2 in |e(x)|2) with 24 individual low numerical aperture lenses (orange) focusing 24 convergent beams (translucent blue) to a common focal point. b) Resultant lattice for a/f=0.08, as seen via isosurfaces of 50% (yellow), 20% (aqua), and 10% (magenta) of max(|e(x)|2) over (14.9λ)3. c) Surface plot of |e(x)|2 across the green xy slice plane in b). Corresponding animation (2.42 MB) illustrates the increase in lattice confinement with increasing a/f.

Fig. 5.
Fig. 5.

Comparison of four different input beam models (as characterized by the confinement factor ψn (x″)) for the generation of convergent beams e n (x,t) through a high NA objective, as applied to a specific beam k n =-k(ê x+ê y +ê s )/√3 of a simple cubic lattice of period 3 λ / 2 . Also compared are the resulting excitation zones of |e(x)|2 when each model is applied to all beams of the lattice.

Fig. 6.
Fig. 6.

(a) Geometric arrangement of the beams comprising a maximally symmetric simple cubic bound lattice of period 35 λ / 2 in | e(x) |2 with c-axis ||ê z , shown in relation to the illumination cones (translucent green) of opposed objectives (NA=1.2, n=1.33). (b) Isosurfaces of 0.4max[| e(x) |2] for both the maximally symmetric lattice (opaque red) comprised of all 48 beams, and the subset lattice (translucent blue) comprised of only the 32 beams (blue in (a)) internal to the objectives. The basis is chosen to optimize | e(xê x |2 at the excitation maxima. (c,d) Plots of | e(x) |2 for the maximally symmetric lattice and subset lattice, respectively, in the yz-plane in (b).

Fig. 7.
Fig. 7.

Hybrid excitation of a bound lattice: a) Overall view, showing opposed objectives; b) Expanded view near one rear pupil, showing input beams (blue) defined with an aperture mask, and an output signal beam (red) isolated with a dichroic mirror (green); (c) Expanded view between the objectives, showing 8 convergent beams (blue) generated internal to each objective as well 8 beams (purple) generated externally with individual low na lenses; d) Resulting lattice over (±21λ)3, exhibiting a spherical excitation zone, as seen via isosurfaces of 0.5 (yellow) and 0.2 (aqua) of max[| e(x) |2] ; e) Surface plot of | e(x) |2 through the xy slice plane (red) in (d).

Fig. 8.
Fig. 8.

Shaping the excitation zone by shaping the cross-section of each constituent beam: (a) Geometry and relevant parameters; (b) Ellipsoidal excitation zone over (±12λ)3 for a hybrid excited, maximally symmetric BCC lattice of period 2 λ in | e(x) |2, comprised of twelve shaped beams; (c,d) Surface plots of | e(x) |2 through the xy (red) and yz (green) slice planes, respectively, in (b). Accompanying movie (1.05 MB) illustrates the shapes of the four input beams at the rear pupil of one objective (NA=1.2) that yield various ellipsoidal excitation zones as shown when superimposed with all other appropriately shaped beams of the lattice.

Fig. 9.
Fig. 9.

Translating the excitation zone by phase-steering each constituent beam to a common offset point: a) Phase variation across the 16 input beams at the rear pupil of one of the objectives (NA=1.2), as required to produce an offset of Δx=(3ê x +4ê x +5ê x )λ;b) Isosurfaces of 0.5max[| e(x) |2] and 0.2max[| e(x) |2] for the original, centered bound lattice (yellow and aqua, respectively) and the resulting offset bound lattice (magenta and orange, respectively); c,d) Surface plots of | e(x) |2 for the centered and offset lattices, respectively, through the xy slice planes in (b). Accompanying animation (1.0 MB) shows the changes in phase variation across each input beam at the rear pupil of one objective as the lattice is translated to different points.

Fig. 10.
Fig. 10.

Composite excitation zone comprised of an 11×11×7 array of offset sub-zones of the type shown in Fig. 9: a) Superposition factor Π n (x″) applied to the 16 input beams at one rear pupil; b) Isosurfaces of 0.5max[| e(x) |2] (yellow) and 0.2max[| e(x) |2] (aqua) over (37λ)3 ; c,d) Surface plots of | e(x) |2 through the xy (red) and yz (green) slice planes, respectively, in (b). Accompanying animation (0.53 MB) shows the variation in Π n (x″) as the distribution of sub-zones contributing to the overall composite zone is changed.

Equations (24)

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e n ( x , t ) = ik 2 π [ e w ( x , t ) ] n exp ( ik x x ) x x ' ( 1 + i k x x ) e ̂ w · ( x x ) x x d 2 x
e n ( x , t ) { i a z ka exp ( ik ρ 2 2 z ) ( C ( u , v ) + i S ( u , v ) ) } e n exp [ i ( kz ω t ) ]
C ( u , v ) = 0 1 J 0 ( v η ) cos ( u η 2 2 ) η d η S ( u , v ) = 0 1 J 0 ( v η ) sin ( u η 2 2 ) η d η
u ( z ) = a z k a v ( ρ , z ) = a z k ρ
e n ( x , t ) { i a f ka exp ( ik ρ 2 2 f ) exp ( ikf ) ( C ( u l , v l ) i S ( u l , v l ) ) } e n exp [ i ( kz ω t ) ] ( lens only )
u l ( z ) = ( a f ) 2 · k ( z f ) v l ( x , y ) = a f k ρ ( lens only )
2 ρ 0.52 λ ( a f )
( x bc ) n = [ ( k n · e ̂ x ) e ̂ x + ( k n · e ̂ y ) e ̂ y ] F o k = [ ( k n · e ̂ x ) e ̂ x + ( k n · e ̂ y ) e ̂ y ] n k NA A
e n ( x , t ) = ik 2 π [ e w ( x , t ) ] n exp ( ik x x ) x x d 2 x .
e ̂ r ( x ) = [ ( e ̂ x · x ) e ̂ x ( e ̂ y · x ) e ̂ y + F 0 2 x 2 e ̂ z ] / F o
= [ ( e ̂ x · x ) e ̂ x ( e ̂ y · x ) e ̂ y + F 0 2 ( e ̂ x · x ) 2 ( e ̂ y · x ) 2 e ̂ z ] / F o
e ̂ φ ( x ) = [ e ̂ r ( x ) × e ̂ z ] / 1 ( e ̂ r ( x ) · e ̂ z ) 2 , e ̂ ρ ( x ) = e ̂ z × e ̂ φ ( x ) , e ̂ θ ( x ) = e ̂ r ( x ) × e ̂ φ ( x )
[ e w ( x , t ) ] n = χ ( x , y ) ψ n ( x , y ) [ ( e ̂ ρ ( x ) · e n i ) e ̂ θ ( x ) + ( e ̂ φ ( x ) · e n i ) e ̂ φ ( x ) ]
e n i = [ ( e ̂ θ ) n · e n ] ( e ̂ ρ ) n + [ ( e ̂ φ ) n · e n ] ( e ̂ φ ) n
e n ( x , t ) = ik 2 π A A A A Θ ( x 2 + y 2 A ) { [ ( ζ ( x , y ) x 2 + y 2 ) ( e ̂ x · e n i ) + ( ζ ( x , y ) 1 ) x y ( e ̂ y · e n i ) ] e ̂ x
+ [ ( ζ ( x , y ) 1 ) x y ( e ̂ x · e n i ) + 1 ( x 2 + ζ ( x , y ) y 2 ) ( e ̂ y · e n i ) ] e ̂ y
+ ( x 2 + y 2 ) [ x ( e ̂ x · e n i ) + y ( e ̂ y · e n i ) ] e ̂ z } e ̂ r ( x ) · e ̂ z x 2 + y 2 ψ ( x , y ) exp ( ik x x ) x x dx dy
e n ( x , t ) = ik 2 π exp [ i Φ n ( x , y ) ] [ e w ( x , t ) ] n exp ( ik x x ) x x d 2 x
Φ n ( x , y ) = k s 2 + ( x φ ) 2 + ( x ρ + ρ bc ) 2 2 x φ x φ 2 ( x ρ + ρ bc ) ( x ρ + ρ bc )
ks k x φ x ε φ k ( x ρ + ρ bc ) s ε ρ + k 2 s [ ( ε φ ) 2 + ( ε ρ ) 2 ]
k x x k F o + k x k k x φ F o ε φ + k x θ F o cos γ ε ρ k ( x θ tan γ + x k ) 2 F o 2 [ ( ε φ ) 2 + ( ε ρ ) 2 ]
s = F o 2 cos γ / Δ z
( x s ) n = ( x bc ) n [ Δ x φ cos γ ( e ̂ φ ) n Δ x θ ( e ̂ ρ ) n + ( F o 2 Δ x φ 2 ) cos 2 γ Δ x θ 2 e ̂ z ] F o / Δ z
e n i ( x , t ) = m = 1 z e n i ψ n ( x ) exp [ i Φ n ( x , y , Δ x m ) i ω t ] e n i ψ n ( x ) Π n ( x ) exp ( i ω t )

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