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  • Editor: Gregory W. Faris
  • Vol. 3, Iss. 6 — Jun. 17, 2008
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Structured illumination for the extension of imaging interferometric microscopy

Alexander Neumann, Yuliya Kuznetsova, and S. R. J. Brueck  »View Author Affiliations


Optics Express, Vol. 16, Issue 10, pp. 6785-6793 (2008)
http://dx.doi.org/10.1364/OE.16.006785


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Abstract

Structured illumination applied to imaging interferometric microscopy (IIM) allows extension of the resolution limit of low numerical aperture objective lenses to ultimate linear systems limits (≲λ/4 in air) without requiring a reference beam around the objective lens. Instead, the reference beam is provided by an illumination beam just at the edge of the optical system numerical aperture resulting in a shift of the recorded spatial frequencies (equivalent to an intermediate frequency). The restoration procedure is discussed. This technique is adaptable readily to existing microscopes, since extensive access to the imaging system pupil plane is not required.

© 2008 Optical Society of America

1. Introduction

It is not necessary to explain the importance of resolution enhancement for millions of existing optical microscopes. The classical diffraction limit [1

1. E. Abbe, Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413–420 (1873). [CrossRef]

] has been successfully exceeded in the impressive results of fluorescent microscopy [2–5

2. 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]

]. However, there are many applications where non-fluorescent transmission/reflection microscopy is of critical importance. Much work has been reported in resolution improvement beyond the classic Rayleigh limit by synthetic aperture approaches [6–11

6. W. Lukosz and M. Marchant, “Optischen Abbildung Unter Ueberschreitung der Beugungsbedingten Aufloesungsgrenze,” Opt. Acta 10, 241–255 (1963). [CrossRef]

].

2. Structured Illumination

Fig. 1. Optical arrangements for (a) conventional IIM with an interferometer that includes the objective lens, and (b) structured illumination with the interferometer in front of the object.

A0,0exp(iωoffx)eiγ0,0offz+k,l0T(kωxωoff;lωy)Ak,lexp[ix(kωxωoff)+ilωyy]eiγk,loffz
(1)

Where x, y and z are orthogonal spatial coordinates; ωoff=2πsin(θoff)/λ is the spatial frequency offset arising from the off-axis illumination at angle θoff (assumed in the x-direction), the prime on the A0,0 refers to the re-injected 0-order, ωx, ωy are the discrete spatial frequency increments of the Fourier summation; γoffk,l≡[(2πn/λ)2-(x-ωoff)2-(y)2]1/2 with n the refractive index of the transmission medium (=1 for air); {k,l} is the set of integers, for which (γoffk,l)2>0, that is the range of integers for which the diffracted beams are within the band pass of the medium and are propagating in the z-direction, away from the object. A scalar electromagnetic model is adequate since the NA of the microscope system is relatively modest.

The transmission function of the optical system T(X;Y) is a simple band pass function:

T(kωX;lωY)={1for(kωX)2+(lωY)2ωMAX=2πNAλ0else
(2)

Taking the square of expression (1) provides the intensity on the imaging camera:

A0,02+..............................................................................................(dcoffset)
k,l0A0,0Ak,l*T(kωxωoff;lωy)exp[ikωxx+ilωyy]ei(γ0,0offγk,loff)z+c.c.+...(imaging)
k.l0,k,l0Ak,lT(kωxωoff;lωy)An,l*T(kωxωoff;lωy)×
exp[i(kk)ωxx+i(ll)ωyy]ei(γk,loffγk,loff)z.........................(darkfield)
(3)

where the three terms on separate lines correspond to (top) a constant term, (middle) the imaging terms and (bottom) the dark field cross-correlation image. Subtracting out the dark field terms (by taking an image with the reference zero-order blocked so that only the third term survives) provides a sub-image that accurately captures the spatial frequency components that are transmitted through the optical system. Note that the imaging terms (middle line) are at the correct frequencies and that the offset illumination angle has cancelled out of the expression except for the filter transmission functions. Changing the illumination angle (and the angle of reintroduction) changes the offset allowing recording of a different region of frequency space.

A0,0exp(iωoffx)eiγ0,0offz+k,l0Ak,lexp[i(kωxωoff)x+ilωyy]eiγk,loffz+
B0,0exp(iωNAx)eiγ0,0NA+p,r0Bp,rexp[i(pωxωNA)x+irωyy]eiγp,rNAz
(4)

and squaring while taking advantage of the fact that the A 0,0 beam is not transmitted to the objective image plane while the B 0,0 beam is transmitted through the lens gives:

B0,02+
5(a)
{p,r0B0,0Bp,r*T(pωxωNA;rωy)exp[i(pωxx+rωyy)]ei(γ0,0NAγp,rNA)z+c.c.+p,r0p,r0Bp,rBp,r*T(pωxωNA;rωy)T(pωxωNA;rωy)exp[i(pp)x+i(rr)y]ei(γp,rNAγp,rNA)z}+
5(b)
{k,lB0,0Ak,l*T(lωxωoff;nωy)exp[i(kωxωoff+ωNA)xikωyy]ei(γ0,0NAγk,loff)z+c.c.}+
5(c)
k,lk,lAk,lAk,l′*T(kωxωoff;lωy)T(kωxωoff;lωy)exp[i(kk)ωxx+i(ll)ωyy]ei(γk,loffγk,loff)z+c.c.
k,lp,r0Ak,lBp,r*T(kωxωoff;lωy)T(pωxωNA;rωy)×
5(d)
exp[i(kp)ωx+i(ωNAωoff)x+i(lr)ωy]ei(γk,loffγp,rNA)z+c.c.
5(e)

The first and the second (in the upper bracket) terms [(5a), (5b)] are just the result of the off-axis illumination at the edge of the pupil. This sub-image can be measured independently by blocking the extreme off axis beam, and subtracted from the sub-image. The third term (5c) is the one we want, the image terms from the extreme off-axis illumination beating against a zero-order beam from the second illumination beam; because the zero-order beam is not at the correct angle to reset the frequencies to match the object frequencies (adjusted for magnification) there is a shift between the observed and the actual image plane frequencies that requires signal processing to reset (e.g. we are evaluating the Fourier components at an intermediate frequency). The forth term (5d) is the dark field from the extreme off-axis illumination. Finally the last term (5e) is the combined dark field from the two illumination beams.

Fig. 2. Schematic of structural illumination and restoration algorithms: a) the object is illuminated simultaneously by two coherent beams: one at an extreme off-axis angle (green) and one (local oscillator, orange) at an angle of ~sin-1(NA) to the normal. High frequencies diffracted from the extreme off-axis illumination mix with low frequencies from the local oscillator, the dark field of the image is obtained by blocking the 0-order beam in the image pupil plane, b) low frequency image/dark field obtained by local oscillator illumination only with and without the 0-order blocked c) the dark field of the image is subtracted as well as the low frequency image without the dark field. Then frequencies are shifted in Fourier space and the total image can be reconstructed by standard IIM procedures: combining high and low frequency images.

Two approaches to eliminating the unwanted dark-field terms are presented. The first one requires blocking just the zero-order beam without making an obstacle to the other diffracted information. This can be done by adding a moveable block at the edge the objective pupil as shown in Fig. 1. A flow chart schematic of the procedure is shown in Fig. 2. The object is illuminated by two beams: one at an extreme off-axis angle (beyond the objective sin-1(NA)) and one (local oscillator) with an angle close to sin-1(NA), such that its 0-order transmission is captured by the objective lens (Fig. 2(a)). By blocking only the 0-order we obtain the dark field image which can be subtracted from the image formed by interference of all of the orders from both beams [5(b), 5(d), and 5(e)]. Then we record the low frequency image obtained by the local oscillator object illumination with and without the 0-order blocked (Fig. 2(b)). We subtract the low frequency image without dark field from mixed image and restore high frequency image by shifting frequencies in Fourier space (Fig. 2(c)). The reconstructed image can be obtained by adding high frequency images with the low frequency images recorded in necessary directions or just with on-axis illumination image as was done in the current experiment. This procedure is repeated in the orthogonal spatial direction for the Manhattan geometry test object; additional images are necessary for objects with arbitrary orientations. Appropriate filtering to deal with overlaps in frequency space coverage should be applied as has been discussed previously [13

13. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy,” J Opt Soc Am A 25, 811–822 (2008). [CrossRef]

].

Fig. 3. Structured illumination with extreme off-axis illumination beam (green) and reference beam (orange) injected between object and objective lens.

A second method to obtain the same result is to use a zero-order beam reinjected before the objective using a beamsplitter (Fig. 3). The beamsplitter is located between the object and the objective lens, eliminating all of the diffracted beams associated with the local oscillator, Bp,r=0,∀p,r≠0, and simplifying Eq. (5). We subtract the image dark field (recorded with blocked reference beam) and reference beam image (recorded with blocked illumination beam) using the same procedure described in Ref [8

8. C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck, “Imaging interferometric microscopy,” Opt. Lett. 28, 1424–1426 (2003). [CrossRef] [PubMed]

] and then restore high frequency image by shifting frequencies in Fourier space (Fig. 2).

Both methods have advantages and issues. The first arrangement requires access to the pupil plane of the system, which can be nontrivial. The second method does not contain the first, second and fifth terms ((5a), (5b), (5e)), so no access to the pupil plane is required, but it does require a beamsplitter between the object and the optical system which reduces the system working distance. Also, there is a possibility of introducing aberrations, especially if the beam splitter is at an angle to the optic axis; aberrations can be minimized by using a thin pellicle beamsplitter. In this configuration, it is particularly straightforward to use phase-shifting dark-field retrieval [14

14. I. Tamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6185 (2001). [CrossRef]

,15

15. T. Kreis, Handbook of holographic interferometry: optical and digital methods (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005).

], as an alternative to blocking the reference zero-order, for eliminating the dark-field terms.

Interferometric methods require setting the phase relationship between the interferometer beams. This phase can be set in real time by observing the image of a reference object while adjusting the phase of one of the illumination beams. An alternative is to record images with an arbitrary phase shift and to evaluate the correct phase using signal processing approaches, again with the use of a reference object. MSE methods can be applied for higher precision in setting this phase. Even without the use of a reference object, the recorded image has higher contrast at the correct phase point, very analogous to the higher contrast observed in an image at focus, but this is a somewhat subjective evaluation and is certainly pattern dependent. The use of a reference object is a more reliable indicator of the correct phase.

3. Experimental results

For our experiments we use an NA=0.4 objective with a He-Ne laser illumination source (λ=633 nm) so that the Rayleigh resolution (above) is limited to ~950 nm. The results of a structural illumination experiment with 240 nm critical dimension (CD, equivalent to linewidth for these equal line:space structures) along with the corresponding simulations are shown in Fig. 4. The mixed image obtained as the result of the two-beam illumination and corresponding to the all terms of the Eq. (5) is shown in Figs. 4(a), 4(b); Figs. 4(c), 4(d) are the images after subtraction of the dark field and low frequency image, and Figs. 4(e), 4(f) are restored high frequency images.

Fig. 4. (a,b) the mixed image corresponding to the interference of the low and high images, (c,d) the image after subtraction dark field and low frequency image, and (e,f) restored high frequency image.
Fig. 5. (a). reconstructed image of 260- and 240-nm CD structures obtained using the optical configuration of Fig. 1 (b); b) crosscut of the image (green) compared with a crosscut of corresponding simulation (blue).

The reconstructed image of 260- and 240-nm CD structures, within the same image field, obtained by this method is shown in Fig. 5(a). Figure 5(b) is a crosscut of the image through 260 nm 240 nm structures compared with a crosscut of the corresponding simulation. A total of four offset images, two each in the x- and y-directions, with θill=53° and 80° were used along with a 0.4 NA objective. As discussed previously [13

13. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy,” J Opt Soc Am A 25, 811–822 (2008). [CrossRef]

], this configuration provides resolution to <∼240 nm CD. There is overlap in the frequency space coverage between these two exposures and frequency space filtering is used to assure a uniform coverage of frequency space. The present Manhattan geometry structure has spectral content concentrated along the x- and y-directions, so the offset illuminations were restricted to those directions. It would be a simple matter to add additional frequency-space coverage for arbitrarily shaped structures by taking additional sub-images with rotation of the object around the (x,y) axes. The spatial frequency content of the image covers a wide range as a result of the large box (at 10X the linewidth of the line:space structures).

The reconstructed image of the same structures obtained by the method with the beamsplitter is shown in Fig. 6(a) and a crosscut of the image with corresponding simulation is shown in Fig. 6(b).

Fig. 6. (a). reconstructed image of 260- and 240-nm CD structures (reinjection of zero-order between object and objective lens (Fig. 3), b) crosscut of the image (green) compared with a crosscut of corresponding simulation (blue).

The quality of the experimental results for both methods is quite comparable. The first method retains a long working distance, but requires access to the imaging system pupil for blocking 0-order. The second does not require any modification to the traditional microscopy components, but has reduced working distance due to the beamsplitter in front of the objective.

There are some extra features experimentally as compared to the model due to the lack of precision in mutual phase determination between the sub-images and speckle effects from the coherent illumination. These issues can be reduced by using improved arrangements and lower coherence sources. There are other possible alternatives; the optimum choice will depend on the specifics of the object and the constraints of specific optical systems.

4. Conclusions

Acknowledgments

We are grateful to Felix Jaeckel for e-beam writing of the mask with small features. Support for this work was provided by DARPA under the University Photonics Research Center Program.

References and links

1.

E. Abbe, Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413–420 (1873). [CrossRef]

2.

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]

3.

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. 102, 13081–13086, (2005). [CrossRef] [PubMed]

4.

G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, and S. W. Hell, “Macromolecular-scale resolution in biological fluorescence microscopy,” Proc. Natl. Acad. Soc. USA 103, 11440–11445 (2006). [CrossRef]

5.

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94, 143903 (2005). [CrossRef] [PubMed]

6.

W. Lukosz and M. Marchant, “Optischen Abbildung Unter Ueberschreitung der Beugungsbedingten Aufloesungsgrenze,” Opt. Acta 10, 241–255 (1963). [CrossRef]

7.

W. Lucosz, “Optical Systems with Resolving Powers Exceeding the Classical Limit,” J. Opt. Soc. Am. 57, 932–941 (1967). [CrossRef]

8.

C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck, “Imaging interferometric microscopy,” Opt. Lett. 28, 1424–1426 (2003). [CrossRef] [PubMed]

9.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102 (2006). [CrossRef] [PubMed]

10.

V. Mico, Z. Zalevsky, and J. Garcia, “Superresolution optical system by common-path interferometry,” Opt. Express 14, 5168–5177 (2006). [CrossRef] [PubMed]

11.

Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy — approaching the linear systems limits of optical resolution,” Opt. Express 15, 6651 (2007). [CrossRef] [PubMed]

12.

X. Chen and S. R. J. Brueck, “Imaging interferometric lithography - approaching the resolution limits of optics,” Opt. Lett. 24, 124–126 (1999). [CrossRef]

13.

Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, “Imaging interferometric microscopy,” J Opt Soc Am A 25, 811–822 (2008). [CrossRef]

14.

I. Tamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177–6185 (2001). [CrossRef]

15.

T. Kreis, Handbook of holographic interferometry: optical and digital methods (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005).

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(110.3175) Imaging systems : Interferometric imaging

ToC Category:
Microscopy

History
Original Manuscript: January 22, 2008
Revised Manuscript: April 17, 2008
Manuscript Accepted: April 18, 2008
Published: April 28, 2008

Virtual Issues
Vol. 3, Iss. 6 Virtual Journal for Biomedical Optics

Citation
Alexander Neumann, Yuliya Kuznetsova, and S. R. Brueck, "Structured illumination for the extension of imaging interferometric microscopy," Opt. Express 16, 6785-6793 (2008)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-10-6785


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References

  1. E. Abbe, Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413 - 420 (1873). [CrossRef]
  2. 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]
  3. M. G. L. Gustafsson, "Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution," Proc. Natl. Acad. Sci. 102, 13081-13086, (2005). [CrossRef] [PubMed]
  4. G. Donnert, J. Keller, R. Medda, M. A. Andrei, S. O. Rizzoli, R. Lührmann, R. Jahn, C. Eggeling, AND S. W. Hell, "Macromolecular-scale resolution in biological fluorescence microscopy," Proc. Natl. Acad. Soc. USA 103, 11440-11445 (2006). [CrossRef]
  5. V. Westphal and S. W. Hell, "Nanoscale resolution in the focal plane of an optical microscope," Phys. Rev. Lett. 94, 143903 (2005). [CrossRef] [PubMed]
  6. W. Lukosz and M. Marchant, "Optischen Abbildung Unter Ueberschreitung der Beugungsbedingten Aufloesungsgrenze," Opt. Acta 10, 241-255 (1963). [CrossRef]
  7. W. Lucosz, "Optical Systems with Resolving Powers Exceeding the Classical Limit," J. Opt. Soc. Am. 57, 932-941 (1967). [CrossRef]
  8. C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck, "Imaging interferometric microscopy," Opt. Lett. 28, 1424-1426 (2003). [CrossRef] [PubMed]
  9. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, "Synthetic aperture Fourier holographic optical microscopy," Phys. Rev. Lett. 97, 168102 (2006). [CrossRef] [PubMed]
  10. V. Mico, Z. Zalevsky, and J. Garcia, "Superresolution optical system by common-path interferometry," Opt. Express 14, 5168-5177 (2006). [CrossRef] [PubMed]
  11. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, "Imaging interferometric microscopy - approaching the linear systems limits of optical resolution," Opt. Express 15, 6651 (2007). [CrossRef] [PubMed]
  12. X. Chen and S. R. J. Brueck, "Imaging interferometric lithography - approaching the resolution limits of optics," Opt. Lett. 24, 124-126 (1999). [CrossRef]
  13. Y. Kuznetsova, A. Neumann, and S. R. J. Brueck, "Imaging interferometric microscopy," J. Opt. Soc. Am. A 25, 811-822 (2008). [CrossRef]
  14. I. Tamaguchi, J. Kato, S. Ohta, and J. Mizuno, "Image formation in phase-shifting digital holography and applications to microscopy," Appl. Opt. 40, 6177-6185 (2001). [CrossRef]
  15. T. Kreis, Handbook of holographic interferometry: optical and digital methods (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2005).

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