## Single-step superresolution by interferometric imaging

Optics Express, Vol. 12, Issue 12, pp. 2589-2596 (2004)

http://dx.doi.org/10.1364/OPEX.12.002589

Acrobat PDF (750 KB)

### Abstract

The use of vertical-cavity surface-emitting laser (VCSEL) arrays for implementation of incoherent source superresolution is presented. The method uses an interferometer setup to obtain superresolution in a single step. The novelty of the method relies on the use of a VCSEL array as the light source, which provides a set of coherent sources which are mutually incoherent. The technique accomplishes the transmission of several spatial frequency bands of the object’s spectrum in parallel by use of spatial multiplexing that occurs because of the tilted illumination of the source array. The recording process is done by interference of each frequency band with a complementary set of reference plane waves. After the reconstruction process, the resolution of any optical system can approach the natural λ/2 limit. The benefit of our system is improved modulation speed and hence more rapid image synthesis. Moreover, any desired synthetic coherent transfer function can be realized at ultrafast rates if we simply change the electrical drive of the VCSEL array.

© 2004 Optical Society of America

## 1. Introduction

1. G. Toraldo di Francia, “Resolving
power and information,” J. Opt. Soc. Am. **45**, 497–501 (1955). [CrossRef]

9. E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system using three fixed generalized gratings: experimental
results,” J. Opt. Soc. Am. A. **18**, 514–520 (2001). [CrossRef]

2. C. S. Chung and H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod.
Opt. **35**, 1485–1511 (1988). [CrossRef]

3. J. Campos and M. J. Yzuel, “Axial and extra-axial responses in aberrated optical systems with apodizers.
Optimization of the Strehl ratio,” J. Mod. Opt. **36**, 733–749 (1989). [CrossRef]

*a priori*information is added, the gain in resolution is modest.

*et al*. [14

14. E. N. Leith, D. Angell, and C.-P. Kuei,
“Superresolution by incoherent-to-coherent conversion,” J. Opt. Soc. Am. A **4**, 1050–1054 (1987). [CrossRef]

15. P. C. Sun and E. N. Leith, “Superresolution by spatial-temporal encoding methods,”
Appl. Opt. **31**, 4857–4862 (1992). [CrossRef] [PubMed]

16. A. Cunha and E. N. Leith, “Generalized one-way phase-conjugation systems,” J. Opt.
Soc. Am. B **6**,1803–1812 (1989). [CrossRef]

17. P. Naulleau and E. Leith, “Imaging through optical fibers by spatial coherence encoding
methods,” J. Opt. Soc. Am. A **13**, 2096–2101 (1996). [CrossRef]

## 2. Theoretical analysis of the optical setup

_{c}) and produces a set of parallel beams with different orientations that illuminate the object. The object is imaged through two identical lenses (L

_{1}and L

_{2}) in a 4

*f*configuration onto a CCD camera. Two beam splitters (BS

_{1}and BS

_{2}) bend the optical path and allow the separation and later recombination of a reference beam for each VCSEL. The reference arm of the system must have the same angular magnification as the imaging arm. With an image magnification of -1, one can use a dove prism to invert the angles in the reference so that they equal the angles in both paths. A bias in the carrier frequency of the interference pattern can be achieved by slightly tilting mirror M

_{1}or M

_{2}.

_{1}. We define the input as

*f*(

*x*) and

*f̃*(

*ν*) is the Fourier transform, where

*x*and

*ν*are the spatial and spatial-frequency coordinates, respectively. A rectangular pupil of size Δ

*ν*acts as the CTF at the Fourier plane, so the amplitude distribution in the CCD from the upper branch of our optical system is a low-pass version of input

*ν*in the Fourier domain (all the computations are done in normalized units of

*λF*, where

*λ*is the wavelength of the illumination and

*F*is the focal length of various lenses). Then VCSEL number

*m*illuminates the input with an inclined plane wave

*exp*[

*j*2

*πm*Δ

*νx*] and so the complex amplitude at the Fourier plane that corresponds to the

*m*th VCSEL is

*m*VCSEL source creates a different plane wave from the reference arm of the setup. The advantages of this system in comparison with previous systems [11

11. F. Le Clerc, M. Gross, and L. Collot,
“Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography,” Opt. Lett. **26**, 1550–1552 (2001). [CrossRef]

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

*m*VCSEL in a single step. No need to add the

*m*distributions

*a posteriori*.

*m*is

*exp*[-

*j*2

*π*(

*m*Δ

*ν*+

*Q*)

*x*], where

*Q*is the bias carrier frequency. This

*Q*value must be larger than half of the size of the pupil plus the size of one slot, where

*N*is the number of slots. So,

*Q*≥Δ

*ν*×(

*N*/2+1).

*T*

_{1}(

*x*),

*T*

_{2}(

*x*),

*T*

_{2}(

*x*), and

*T*

_{4}(

*x*). After the intensity

*I*(

_{m}*x*) is stored, we perform digitally an inverse Fourier transformation. The Fourier transform of the first term [

*T̃*

_{1}(

*ν*)] is a delta function centered at the origin. The second term, also centered at the origin, is the centered autocorrelation of the bandpass slot (with width 2 Δ

*ν*):

*T̃*

_{3}(

*ν*) is the

*m*th slot at its left position plus an offset -

*Q*:

*f*(-

*x*), so the input object can be completely reconstructed. Also, note that the fourth term [Eq. (8)] appropriately centered and Fourier transformed will give the complex conjugate of the inverted image. Note that Eq. (9) can be rewritten as

14. E. N. Leith, D. Angell, and C.-P. Kuei,
“Superresolution by incoherent-to-coherent conversion,” J. Opt. Soc. Am. A **4**, 1050–1054 (1987). [CrossRef]

*f*(

*x*) is made, so for any complex input distribution we would always reconstruct that input. This advantage is important for application in microscopy, three-dimensional imaging, or invariant pattern recognition.

## 3. Experimental results

*λ*=850

*nm*. Without the slit in the Fourier plane that limits the aperture, the system has unity magnification and a NA of 0.12. Owing to the unity magnification, the resolution of the unobstructed system is higher than that of the CCD detector. Thus, to demonstrate the method, the resolution capability of the system is reduced by means of the Fourier plane slit. We used two test objects: the first was a 25-µm-wide slit and the second was a resolution test. As a reference, high-resolution coherent images of the test objects, without limitation in the Fourier plane, are shown in Fig. 2.

*T̃*

_{1}(

*ν*),

*T̃*

_{2}(

*ν*),

*T̃*

_{3}(

*ν*), and

*T̃*

_{4}(

*ν*). The region of interest is marked in Fig. 3(b). For the reconstruction the values outside the region of interest are not considered.

## 4. Conclusion and discussion

## Acknowledgments

## References and links

1. | G. Toraldo di Francia, “Resolving
power and information,” J. Opt. Soc. Am. |

2. | C. S. Chung and H. H. Hopkins, “Influence of non-uniform amplitude on PSF,” J. Mod.
Opt. |

3. | J. Campos and M. J. Yzuel, “Axial and extra-axial responses in aberrated optical systems with apodizers.
Optimization of the Strehl ratio,” J. Mod. Opt. |

4. | R. W. Gerchberg, “Super-resolution
through error energy reduction,” Opt. Acta |

5. | W. Lukosz, “Optical sytems with
resolving powers exceeding the classical limits. II,” J. Opt. Soc. Am |

6. | M. A. Grimm and A. W. Lohmann, “Superresolution image for one-dimensional objects,”
J. Opt. Soc. Am. |

7. | A. W. Lohmann and D. P. Paris, “Superresolution for nonbirefringent objects,” Appl.
Opt. |

8. | A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia-Martinez, “Superresolving optical system with time multiplexing and computer
decoding,” Appl. Opt. |

9. | E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system using three fixed generalized gratings: experimental
results,” J. Opt. Soc. Am. A. |

10. | J. Salomon, Z. Zalevsky, and D. Mendlovic,
“Superresolution by use of code division multiplexing,” Appl. Opt. |

11. | F. Le Clerc, M. Gross, and L. Collot,
“Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography,” Opt. Lett. |

12. | X. Chen and S. R. J Brueck, “Imaging interferometric lithography: approaching the resolution limits of
optics,” Opt. Lett. |

13. | C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck,
“Imaging interferometric microscopy,” Opt. Lett. |

14. | E. N. Leith, D. Angell, and C.-P. Kuei,
“Superresolution by incoherent-to-coherent conversion,” J. Opt. Soc. Am. A |

15. | P. C. Sun and E. N. Leith, “Superresolution by spatial-temporal encoding methods,”
Appl. Opt. |

16. | A. Cunha and E. N. Leith, “Generalized one-way phase-conjugation systems,” J. Opt.
Soc. Am. B |

17. | P. Naulleau and E. Leith, “Imaging through optical fibers by spatial coherence encoding
methods,” J. Opt. Soc. Am. A |

**OCIS Codes**

(090.0090) Holography : Holography

(100.0100) Image processing : Image processing

(100.2000) Image processing : Digital image processing

(100.6640) Image processing : Superresolution

(110.0110) Imaging systems : Imaging systems

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 13, 2004

Revised Manuscript: May 25, 2004

Published: June 14, 2004

**Citation**

Vicente Mico, Zeev Zalevsky, Pascuala Garcia-Martinez, and Javier Garcia, "Single-step superresolution by interferometric imaging," Opt. Express **12**, 2589-2596 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-12-2589

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### References

- G. Toraldo di Francia, �??Resolving power and information,�?? J. Opt. Soc. Am. 45, 497�?? 501 (1955). [CrossRef]
- C. S. Chung and H. H. Hopkins, �??Influence of non-uniform amplitude on PSF,�?? J. Mod. Opt. 35, 1485�??1511 (1988). [CrossRef]
- J. Campos and M. J. Yzuel, �??Axial and extra-axial responses in aberrated optical systems with apodizers Optimization of the Strehl ratio,�?? J. Mod. Opt. 36, 733�??749 (1989). [CrossRef]
- R. W. Gerchberg, �??Super-resolution through error energy reduction,�?? Opt. Acta 21 709�??720 (1974). [CrossRef]
- W. Lukosz, �??Optical sytems with resolving powers exceeding the classical limits. II,�?? J. Opt. Soc. Am 57, 932�??941 (1967). [CrossRef]
- M. A. Grimm and A. W. Lohmann, �??Superresolution image for one-dimensional objects,�?? J. Opt. Soc. Am. 56, 1151�??1156 (1966). [CrossRef]
- A. W. Lohmann and D. P. Paris, �??Superresolution for nonbirefringent objects,�?? Appl. Opt. 3, 1037�??1043 (1964). [CrossRef]
- A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. Garcia-Martinez, �??Superresolving optical system with time multiplexing and computer decoding,�?? Appl. Opt. 38, 7245�??7251 (1999). [CrossRef]
- E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, �??Superresolution optical system using three fixed generalized gratings: experimental results,�?? J. Opt. Soc. Am. A. 18, 514�??520 (2001). [CrossRef]
- J. Salomon, Z. Zalevsky, and D. Mendlovic, �??Superresolution by use of code division multiplexing,�?? Appl. Opt. 42, 1451�??1462 (2003). [CrossRef]
- F. Le Clerc, M. Gross, and L. Collot, �??Synthetic-aperture experiment in the visible with on-axis digital heterodyne holography,�?? Opt. Lett. 26, 1550�??1552 (2001). [CrossRef]
- X. Chen and S. R. J Brueck, �??Imaging interferometric lithography: approaching the resolution limits of optics,�?? Opt. Lett. 24, 124�??126 (1999). [CrossRef]
- C. J. Schwarz, Y. Kuznetsova, and S. R. J. Brueck, �??Imaging interferometric microscopy,�?? Opt. Lett. 28, 1424�??1426 (2003). [CrossRef] [PubMed]
- E. N. Leith, D. Angell, and C.-P. Kuei, �??Superresolution by incoherent-to-coherent conversion,�?? J. Opt. Soc. Am. A 4, 1050�??1054 (1987). [CrossRef]
- P. C. Sun and E. N. Leith, �??Superresolution by spatial-temporal encoding methods,�?? Appl. Opt. 31, 4857�?? 4862 (1992). [CrossRef] [PubMed]
- A. Cunha and E. N. Leith, �??Generalized one-way phase-conjugation systems,�?? J. Opt. Soc. Am. B 6,1803�?? 1812 (1989). [CrossRef]
- P. Naulleau and E. Leith, �??Imaging through optical fibers by spatial coherence encoding methods,�?? J. Opt. Soc. Am. A 13, 2096�??2101 (1996). [CrossRef]

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