## Quantum lithography by coherent control of classical light pulses

Optics Express, Vol. 12, Issue 26, pp. 6600-6605 (2004)

http://dx.doi.org/10.1364/OPEX.12.006600

Acrobat PDF (593 KB)

### Abstract

The smallest spot in optical lithography and microscopy is generally limited by diffraction. Quantum lithography, which utilizes interference between groups of *N* entangled photons, was recently proposed to beat the diffraction limit by a factor *N*. Here we propose a simple method to obtain *N* photons interference with classical pulses that excite a narrow multiphoton transition, thus shifting the “quantum weight” from the electromagnetic field to the lithographic material. We show how a practical complete lithographic scheme can be developed and demonstrate the underlying principles experimentally by two-photon interference in atomic Rubidium, to obtain focal spots that beat the diffraction limit by a factor of 2.

© 2004 Optical Society of America

1. T. A. Brunner, “Why optical lithography will live forever,” J. Vac. Sci. Technol. B **21**, 2632–2637 (2003). [CrossRef]

*d*

_{1}~

*λNA*, where

*λ*is the optical wavelength and

*NA*is the numerical aperture of the imaging setup [2]. Essentially, the minimal spot is generated by the interference between many plane waves arriving to the substrate at different angles (as high as allowed by the numerical aperture).

*N*-photon absorption in the lithographic medium leads to an improvement of factor √

*N*in the spot size (

*d*

_{N}=

*d*

_{1}/√

*N*) due to the enhanced contrast of the material response, as was indeed demonstrated [3

3. S. Kawata, H. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature **412**, 697–698 (2001). [CrossRef] [PubMed]

*N*-fold over the diffraction limit was suggested by use of “Quantum Lithography” [4

4. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. **85**, 2733–2736 (2000). [CrossRef] [PubMed]

6. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de-Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**, 213601 (2002). [CrossRef] [PubMed]

*N*entangled photons that arrive at different angles. This multi-photon interference is equivalent to single-photon interference with a wavelength of

*λ/N*, leading to the desired improvement in resolution. In order to obtain a spot size

*d*

_{1}/

*N*with quantum lithography, two conditions must be met: First, the absorption in the lithographic material should be purely

*N*-photonic. Second, the photons should interfere only as

*N*-photons entangled groups, but not as single photons; i.e. the absorption of the

*N*photons should be either from one direction or another, excluding possibilities of absorbing some photons from one direction and the rest from another [5

5. M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. **87**, 013602 (2001). [CrossRef]

7. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” quant-ph/p. 0411023 (2004), http://xxx.lanl.gov/abs/quant-ph/0411023

*N*-photon groups arrive “one at a time”) [7

7. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” quant-ph/p. 0411023 (2004), http://xxx.lanl.gov/abs/quant-ph/0411023

*N*-photons transition). As a result, higher powers can be utilized according to standard practical constraints without any inherent limitation. The main principle in our approach is that when a short pulse excites a narrow transition in the material, the excitation lifetime is much longer than the pulse duration. Thus, if the transition is excited again by another pulse within the excitation lifetime of the first pulse, these two excitations can interfere even if the two pulses do not (i.e. are mutually incoherent). Since this interference occurs through the medium, the relative phase that affects it is dictated by the transition frequency

*ω*

_{A}and the relative delay

*τ*between the pulses (

*ϕ=ω*

_{A}

*τ*). If this excitation is non-linear of order

*N*, the center frequency of the exciting pulse is

*ω*

_{0}=

*ω*

_{A}/

*N*. As a result, this “quantum interference” is equivalent to one-photon interference with a wavelength shorter by a factor of

*N*, as was indeed demonstrated for

*N*=2 [8

8. V. Blanchet, C. Nicole, M. A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transition: from optical interferences to quantum interferences,” Phys. Rev. Lett. **78**, 2716–2719 (2002). [CrossRef]

*N*-photon coherence between mutually incoherent (non-overlapping) pulses. In the following we theoretically analyze and experimentally demonstrate how control over the spatial dependence of the “quantum phase”

*ϕ*(

*r*)=

*Nω*

_{0}

*τ*(

*r*) leads to a complete lithographic scheme with a resolution that is improved by a factor of

*N*.

*d*

_{1}≈

*λf/D*, where f is the focal length of the lens, D is the beam diameter and paraxial optics is assumed. We wish to obtain an additional factor of

*N*in resolution using quantum interference of a train of pulses.

*N*th power of the electric field

*E*

^{N}(

*ωA*) [9

9. D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature **396**, 239 (1998). [CrossRef]

10. S.A. Hosseini and D. Goswami, “Coherent control of multiphoton transitions with femtosecond pulse shaping,” Phys. Rev. A **64**, 033410 (2001). [CrossRef]

*E*

_{1}and

*E*

_{2}), the intensity of excitation will be

*x*is the spatial coordinate. Note that since the pulses do not overlap in time, mixed terms (e.g.,

10. S.A. Hosseini and D. Goswami, “Coherent control of multiphoton transitions with femtosecond pulse shaping,” Phys. Rev. A **64**, 033410 (2001). [CrossRef]

*M*pulse fields at the lens surface {

*ε*

_{k}(

*x*

_{1})} whose corresponding focal fields {

*E*

_{k}(

*x*

_{f})} fulfill

*I*(

*x*

_{f}) is the desired narrow lithographic spot. Two constraints must be considered. First, the focal spots cannot be smaller than the diffraction limit imposed by the aperture of the lens. Second, due to the non-linearity, spreading the energy either in space or in time is not desired, so the number of pulses

*M*and the spatial extent of the focal fields should be minimized (i.e., maximize the spatial extent at the lens). While a general solution is currently unknown to us, a simple practical approach is to divide the lens into two non overlapping segments and delay the pulse in one of the segments (e.g., with a piece of glass) as schematically sketched in Fig. 1. Since each segment can be considered as an off axis lens, the two segments generate overlapping spots with linear phase fronts of opposite slopes. If the delay between the pulses is tuned correctly, this will lead to the desired constructive interference at the center. It is interesting to note that a Fourier equivalent of the segmentation scheme of Fig. 1 was suggested for doubling the resolution with two-photon absorption [12

12. E. Yablonovitch and R. B. Vrijen, “Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure,” Opt. Eng. **38**, 334–338 (1999). [CrossRef]

13. D. V. Korobkin and E. Yablonovitch, “Twofold spatial resolution enhancement by two-photon exposure of photographic film,” Opt. Eng. **41**, 1729–1732(2002). [CrossRef]

*spectral*components are separated spatially to two segments on the lens. While such an approach can yield similar results for the two-photon case, generalization to the

*N*photon case is not straight forward and its implementation is more complicated.

12. E. Yablonovitch and R. B. Vrijen, “Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure,” Opt. Eng. **38**, 334–338 (1999). [CrossRef]

*N*-photon spots. However, any use of non-linearity inherently favors scanning because of power considerations. Note that even though the intensity of the non-linear lithographic response with entangled photons depends

*linearly*on the incoming photon flux, the above power considerations also apply, since the sensitivity to spatial expansion remains non-linear [5

5. M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. **87**, 013602 (2001). [CrossRef]

7. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” quant-ph/p. 0411023 (2004), http://xxx.lanl.gov/abs/quant-ph/0411023

*π*phase shift to frequencies above the resonance and below it, in order to maximize the excitation [14

14. N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. **86**, 47–50 (2001). [CrossRef] [PubMed]

9. D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature **396**, 239 (1998). [CrossRef]

15. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Two-photon absorption and coherent control with broadband down-converted light,” Phys. Rev. Lett. **93**, 023005 (2004). [CrossRef] [PubMed]

## References and links

1. | T. A. Brunner, “Why optical lithography will live forever,” J. Vac. Sci. Technol. B |

2. | J. W. Goodman, |

3. | S. Kawata, H. Sun, T. Tanaka, and K. Takada, “Finer features for functional microdevices,” Nature |

4. | A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling., “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. |

5. | M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. |

6. | K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de-Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. |

7. | B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” quant-ph/p. 0411023 (2004), http://xxx.lanl.gov/abs/quant-ph/0411023 |

8. | V. Blanchet, C. Nicole, M. A. Bouchene, and B. Girard, “Temporal coherent control in two-photon transition: from optical interferences to quantum interferences,” Phys. Rev. Lett. |

9. | D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature |

10. | S.A. Hosseini and D. Goswami, “Coherent control of multiphoton transitions with femtosecond pulse shaping,” Phys. Rev. A |

11. | N. Dudovich, T. Polack, A. Pe’er, and Y. Silberberg, “Coherent control with real optical fields: a simple route to strong field control,” Submitted to Phys. Rev. Lett (2004). |

12. | E. Yablonovitch and R. B. Vrijen, “Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure,” Opt. Eng. |

13. | D. V. Korobkin and E. Yablonovitch, “Twofold spatial resolution enhancement by two-photon exposure of photographic film,” Opt. Eng. |

14. | N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, “Transform-limited pulses are not optimal for resonant multiphoton transitions,” Phys. Rev. Lett. |

15. | B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Two-photon absorption and coherent control with broadband down-converted light,” Phys. Rev. Lett. |

**OCIS Codes**

(020.4180) Atomic and molecular physics : Multiphoton processes

(110.5220) Imaging systems : Photolithography

(180.0180) Microscopy : Microscopy

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 19, 2004

Revised Manuscript: December 14, 2004

Published: December 27, 2004

**Citation**

Avi Pe'er, Barak Dayan, Marija Vucelja, Yaron Silberberg, and Asher Friesem, "Quantum lithography by coherent control of classical light pulses," Opt. Express **12**, 6600-6605 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-26-6600

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### References

- T. A. Brunner, "Why optical lithography will live forever," J. Vac. Sci. Technol. B 21, 2632-2637 (2003). [CrossRef]
- J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (McGraw-Hill, 1996).
- S. Kawata, H. Sun, T. Tanaka and K. Takada, "Finer features for functional microdevices," Nature 412, 697-698 (2001). [CrossRef] [PubMed]
- A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams and J. P. Dowling., "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000). [CrossRef] [PubMed]
- M. D'Angelo, M. V. Chekhova and Y. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001). [CrossRef]
- K. Edamatsu, R. Shimizu and T. Itoh, "Measurement of the photonic de-Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion," Phys. Rev. Lett. 89, 213601 (2002). [CrossRef] [PubMed]
- B. Dayan, A. Pe'er, A. A. Friesem, and Y. Silberberg, "Nonlinear interactions with an ultrahigh flux of broadband entangled photons," quant-ph/ p. 0411023 (2004), <a href="http://xxx.lanl.gov/abs/quant-ph/0411023">http://xxx.lanl.gov/abs/quant-ph/0411023</a>
- V. Blanchet, C. Nicole, M. A. Bouchene and B. Girard, "Temporal coherent control in two-photon transition: from optical interferences to quantum interferences," Phys. Rev. Lett. 78, 2716-2719 (2002). [CrossRef]
- D. Meshulach and Y. Silberberg, "Coherent quantum control of two-photon transitions by a femtosecond laser pulse," Nature 396, 239 (1998). [CrossRef]
- S.A. Hosseini and D. Goswami, "Coherent control of multiphoton transitions with femtosecond pulse shaping," Phys. Rev. A 64, 033410 (2001). [CrossRef]
- N. Dudovich, T. Polack, A. Pe'er and Y. Silberberg, "Coherent control with real optical fields: a simple route to strong field control," Submitted to Phys. Rev. Lett. (2004).
- E. Yablonovitch and R. B. Vrijen, "Optical projection lithography at half the Rayleigh resolution limit by two-photon exposure," Opt. Eng. 38, 334-338 (1999). [CrossRef]
- D. V. Korobkin and E. Yablonovitch, "Twofold spatial resolution enhancement by two-photon exposure of photographic film," Opt. Eng. 41, 1729-1732 (2002). [CrossRef]
- N. Dudovich, B. Dayan, S. M. Gallagher Faeder, and Y. Silberberg, "Transform-limited pulses are not optimal for resonant multiphoton transitions," Phys. Rev. Lett. 86, 47-50 (2001). [CrossRef] [PubMed]
- B. Dayan, A. Pe'er, A. A. Friesem, and Y. Silberberg, "Two-photon absorption and coherent control with broadband down-converted light," Phys. Rev. Lett. 93, 023005 (2004). [CrossRef] [PubMed]

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