## Analysis of focal-shift effect in planar metallic nanoslit lenses |

Optics Express, Vol. 20, Issue 2, pp. 1320-1329 (2012)

http://dx.doi.org/10.1364/OE.20.001320

Acrobat PDF (836 KB)

### Abstract

A theoretical analysis based on scalar diffraction theory about the recently reported focal-shift phenomena in planar metallic nanoslit lenses is presented. Under Fresnel approximation, an axial intensity formula is obtained, which is used to analyze the focal performance in the far field zone of lens. The relative focal shift is totally dependent on the Fresnel number only. The influences of the lens size, preset focal length and incident wavelength can be attributed to the change of Fresnel number. The total phase difference of the lens is approximately equal to the Fresnel number multiplied by π. Numerical simulations performed using finite-difference time-domain (FDTD) and near-far field transformation method are in agreement with the theoretical analysis. Using the theoretical formula assisted by simple numerical method, we provide predictions on the focal shift for the previous literatures.

© 2012 OSA

## 1 Introduction

1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

3. H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express **13**(18), 6815–6820 (2005). [CrossRef] [PubMed]

4. T. Xu, C. Wang, C. Du, and X. Luo, “Plasmonic beam deflector,” Opt. Express **16**(7), 4753–4759 (2008). [CrossRef] [PubMed]

5. Q. Zhu, J. Ye, D. Wang, B. Gu, and Y. Zhang, “Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm,” Opt. Express **19**(10), 9512–9522 (2011). [CrossRef] [PubMed]

6. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. **95**(7), 071112 (2009). [CrossRef]

7. Y. J. Jung, D. Park, S. Koo, S. Yu, and N. Park, “Metal slit array Fresnel lens for wavelength-scale optical coupling to nanophotonic waveguides,” Opt. Express **17**(21), 18852–18857 (2009). [CrossRef] [PubMed]

12. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. **39**(4), 211–215 (1981). [CrossRef]

14. W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A **12**(9), 1947–1953 (1995). [CrossRef]

15. Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express **19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

8. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. **9**(1), 235–238 (2009). [CrossRef] [PubMed]

11. X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express **18**(11), 11683–11688 (2010). [CrossRef] [PubMed]

15. Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express **19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

## 2 Planar diffractive lenses

### 2.1 Theory

16. H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express **15**(3), 1240–1253 (2007). [CrossRef] [PubMed]

3. H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express **13**(18), 6815–6820 (2005). [CrossRef] [PubMed]

*xz*-plane denotes the incident plane and the structure is infinite along the

*y*-axis. The planar lens with lens size of 2

*a*is located at

*z*= 0. We use F to denote the geometrical focus of the incident wave and P as the observation point on the axis of the lens. The field distribution along the axis can be described using the Fresnel approximation [17]where

*λ*and

*k*are wavelength and wave number of light in vacuum, respectively;

*U*

_{1}is the field distribution at the aperture. When a plane wave with constant intensity profile illuminates the planar lens, the waveform at

*z*= 0 is the product of a constant amplitude factor

*A*with the phase function of the lens:

*x*for a planar lens can be obtained according to the equal optical length principle

15. Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express **19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

*a*/

*f*)

^{2}<<1 in order to simplify the derivation process. Therefore, the phase function

*φ*(

*x*) can be approximated using series expansion as

*p*is the relative position of the observation point P:and

*C*and

*S*are the Fresnel cosine integral and Fresnel sine integralwith

12. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. **39**(4), 211–215 (1981). [CrossRef]

*p*of the observation point. The Fresnel number

*N*is the only parameter left, which means that the axial field is totally dependent on the Fresnel number.

*a*can be obtained readily by letting

**19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

### 2.2 Focal shift

*p*

_{m}, can be obtained numerically for different Fresnel numbers

*N*. According to the definition of

*p*in Eq. (6),

*p*

_{m}stands for the relative focal shift of the lens. Figure 2 presents

*p*

_{m}as a function of the Fresnel number

*N*. The curve depicted in Fig. 2 is similar to that for the case of circular aperture in Ref [12

12. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. **39**(4), 211–215 (1981). [CrossRef]

*p*

_{m}are all negative, which means that the real focus shifts toward the lens. This is in accordance with the reported results in previous literatures [8

8. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. **9**(1), 235–238 (2009). [CrossRef] [PubMed]

11. X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express **18**(11), 11683–11688 (2010). [CrossRef] [PubMed]

**19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

*N*, the focal shift |

*p*

_{m}| become very small, being 1% when

*N*= 7.5. As

*N*decreases, the focus shift increases rapidly.

*φ*(

*a*)| in Eq. (4)) divided by a coefficient of π. As shown in Fig. 2, the lens with a Fresnel number of 2 still has a focal shift of more than 10%, which means that the focal-shift effect still exists, even when the total phase difference reaches 2π. This finding is a little different from the report in Ref [15

**19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

*N*. Other parameters, like lens size, preset focal length and the incident wavelength contribute to

*N*. However, in practical lens design, it is a common requirement to determine the lens size at a given wavelength and focal length. We plot curve of the relation between the focal shift and the lens size at different working wavelengths in Fig. 3 . As the lens size increases, the focal shift decreases to zero gradually, which is consistent with the numerical results in Ref [15

**19**(10), 9434–9444 (2011). [CrossRef] [PubMed]

*a*/

*f*)

^{2}<<1. We continue to discuss the validity of this treatment. First, (

*a*/

*f*)

^{2}can be rewriten as

*N*= 2, when the incident wavelength is 637 nm and preset focal length

*f*= 20 μm, the lens size 2

*a*is 10 μm which yields a very small ratio (

*a*/

*f*)

^{2}: 0.06.

### 2.3 Planar lens based on metallic nanoslit array

*xz*-plane denotes the incident plane and the structure is infinite along the

*y*-axis. F is the geometrical focus of the lens. The planar lens can be realized by carving nanoscale slits with varied width on a metal film. Since the propagation constants of the fundamental modes inside the slits are highly sensitive to the slit width, the metal film can be used to produce the required phase retardation. Ignoring the coupling between adjacent slits, we can obtain the propagation constant of the fundamental TM mode in the slit using the dispersive relationwith

*k*

_{1}= (

*β*

^{2}-

*ε*

_{d}

*k*

^{2})

^{1/2}and

*k*

_{2}= (

*β*

^{2}-

*ε*

_{m}

*k*

^{2})

^{1/2}, where

*d*is the slit width,

*k*is the wave number in free space,

*β*is the propagation constant of the plasmonic mode,

*ε*

_{m}and

*ε*

_{d}are relative permittivity of metal and air, respectively. The phase retardation of light transmitted through the slit can be approximated as

*φ*= Re(

*β*)

*h*.

*ε*= −11.04+ 0.78

_{m}*i*at wavelength of 637 nm [8

8. L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. **9**(1), 235–238 (2009). [CrossRef] [PubMed]

*h*= 600 nm and the slit interspacing is Δ = 200 nm which is large enough to prohibit coupling between adjacent slits. The half-size of the designed metallic lens is assumed to be

*a*= 2 μm. The focal lengths are chosen to be 2.8 μm, 6 μm, 10 μm and infinity to produce different Fresnel numbers: 2.24, 1.05, 0.63 and 0. After the focal length

*f*is determined, the required phase delay at each slit can be obtained using Eq. (3). The phase retardations of the designed lenses are depicted in Fig. 4(b). The total phase differences of these lenses are 2.01π, 1.02π, 0.62π and 0 for

*f*= 2.8 μm, 6 μm, 10 μm and infinity, respectively.

*x =*Δ

*z =*2 nm is used to obtain the field distribution around the metal structure. The electromagnetic field and its normal derivative along

*z*-direction are obtained on the surface

*z*= 0; the field is then projected to points in the zone

*z*> 0 by Green’ function approach [19

19. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A **14**(1), 34–43 (1997). [CrossRef]

*f*= 2.8 μm, the Fresnel number is

*N*= 2.24 and the designed total phase difference is 2.01π. The analytical Eq. (5) predicts a relative focal shift of 9.3%. The simulation results are shown in Fig. 5 . The real position of the peak irradiance is at 2.41, corresponding to 13.9% mismatch of the focal position. The mismatch means that even with a phase difference of 2π, the focal shift still exists.

*f*= 6 μm. In this case, the Fresnel number is

*N*= 1.05 and the designed total phase difference is 1.02π. The relative focal shift is 20.7% obtained from simulation and 27.1% from the analytical prediction (Eq. (5)).

*f*= 10 μm. In this case, the Fresnel number is

*N*= 0.63 and the designed total phase difference is 0.62π. The relative focal shift is 48.9% obtained from simulation and 44.0% from the analytical prediction (Eq. (5)).

*f*= infinity. In this case, the slits have equal widths, the Fresnel number is

*N*= 0 and the designed total phase difference is 0. Considering the diffraction effect of the limited aperture, a maximum intensity along the axis appears and locates at a distant about 8.67 μm. It can be concluded that there exists a maximal distance that the focal points of the lenses designed by traditional geometric method cannot exceed when the lens size and work wavelength are kept fixed.

*a*= 2 μm obtained using Eq. (11). It can be seen that the position of peak irradiance moves closer to the aperture. It can be estimated that the focal-shift effect may be more obvious in this case.

*a*= 3 μm are designed and checked by simulation. The preset focal lengths are chosen as 6.3 μm, 13.5 μm, 22.5 μm and infinity to produce the same Fresnel numbers as in the case of

*a*= 2. The slit-width sequences (from middle to the edges of lenses) are: 16, 16, 16, 16, 18, 18, 20, 20, 22, 24, 28, 32, 40, 50, 70 and 116 nm for

*f*= 6.3 μm; 26, 26, 26, 26, 28, 28, 30, 30, 32, 34, 38, 42, 46, 52, 62 and 74 nm for

*f*= 13.5 μm; 36, 36, 36, 36, 38, 38, 40, 40, 42, 44, 48, 50, 54, 60, 66 and 76 nm for

*f*= 22.5 μm. Other parameters remain the same as above. The simulation and analytical results along with those for

*a*= 2 and 3 μm are all listed in Table 1 . It is obvious that the focus shift is totally dependent on the Fresnel number, which agrees well with the analytical results shown in Fig. 2.

*N*. It can be used as a fast method to predict how much the focal point shifts. Table 2 lists the prediction results involving the recently reported structures in previous literatures. From the table, we can see a large difference in the focal length between the theoretical design and the simulation results. Based on the Fresnel number theory, the predictions of the real focal positions are very close to the numerical simulation results, except for Ref [11

11. X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express **18**(11), 11683–11688 (2010). [CrossRef] [PubMed]

**18**(11), 11683–11688 (2010). [CrossRef] [PubMed]

_{1}, an oscillatory wave mode. The slit width are large (412 nm ~616 nm) since the metallic slit imposes a cutoff width for the higher-order modes. The diffraction of the field from the slits cannot simply be treated using the point source model. Moreover, the interval between slits is relatively larger (nearly 0.83λ as shown in Table 2) than those in other literatures. The analytical formula presented above is not suitable for this case.

*β*)

*h*. The Fabry-Perot oscillations of light inside the slit and even the interference between adjacent slits are the reasons of the phase retardation. Nevertheless, all these influences are smaller than the main phase retardation and are not decisive in the focal shift effect. In the case of

*f*= infinity, as shown in Fig. 8(b), the analytical and the simulation results coincide since all these extra effects do not exist in this case.

20. Q. Chen and D. R. S. Cumming, “Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film,” Opt. Express **18**(14), 14788–14793 (2010). [CrossRef] [PubMed]

21. Q. Chen, “Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment,” Plasmonics **6**(1), 75–82 (2011). [CrossRef]

9. P. Ruffieux, T. Scharf, H. P. Herzig, R. Völkel, and K. J. Weible, “On the chromatic aberration of microlenses,” Opt. Express **14**(11), 4687–4694 (2006). [CrossRef] [PubMed]

## 3 Conclusions

## Acknowledgment

## References and links

1. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

2. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

3. | H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express |

4. | T. Xu, C. Wang, C. Du, and X. Luo, “Plasmonic beam deflector,” Opt. Express |

5. | Q. Zhu, J. Ye, D. Wang, B. Gu, and Y. Zhang, “Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm,” Opt. Express |

6. | L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. |

7. | Y. J. Jung, D. Park, S. Koo, S. Yu, and N. Park, “Metal slit array Fresnel lens for wavelength-scale optical coupling to nanophotonic waveguides,” Opt. Express |

8. | L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett. |

9. | P. Ruffieux, T. Scharf, H. P. Herzig, R. Völkel, and K. J. Weible, “On the chromatic aberration of microlenses,” Opt. Express |

10. | M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett. |

11. | X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express |

12. | Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. |

13. | Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.) |

14. | W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A |

15. | Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express |

16. | H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express |

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

18. | K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol. |

19. | D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A |

20. | Q. Chen and D. R. S. Cumming, “Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film,” Opt. Express |

21. | Q. Chen, “Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment,” Plasmonics |

**OCIS Codes**

(220.3630) Optical design and fabrication : Lenses

(240.6680) Optics at surfaces : Surface plasmons

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 14, 2011

Revised Manuscript: December 25, 2011

Manuscript Accepted: December 25, 2011

Published: January 6, 2012

**Citation**

Yang Gao, Jianlong Liu, Xueru Zhang, Yuxiao Wang, Yinglin Song, Shutian Liu, and Yan Zhang, "Analysis of focal-shift effect in planar metallic nanoslit lenses," Opt. Express **20**, 1320-1329 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1320

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### References

- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature424(6950), 824–830 (2003). [CrossRef] [PubMed]
- H. Shi, C. Wang, C. Du, X. Luo, X. Dong, and H. Gao, “Beam manipulating by metallic nano-slits with variant widths,” Opt. Express13(18), 6815–6820 (2005). [CrossRef] [PubMed]
- T. Xu, C. Wang, C. Du, and X. Luo, “Plasmonic beam deflector,” Opt. Express16(7), 4753–4759 (2008). [CrossRef] [PubMed]
- Q. Zhu, J. Ye, D. Wang, B. Gu, and Y. Zhang, “Optimal design of SPP-based metallic nanoaperture optical elements by using Yang-Gu algorithm,” Opt. Express19(10), 9512–9522 (2011). [CrossRef] [PubMed]
- L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett.95(7), 071112 (2009). [CrossRef]
- Y. J. Jung, D. Park, S. Koo, S. Yu, and N. Park, “Metal slit array Fresnel lens for wavelength-scale optical coupling to nanophotonic waveguides,” Opt. Express17(21), 18852–18857 (2009). [CrossRef] [PubMed]
- L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, “Planar lenses based on nanoscale slit arrays in a metallic film,” Nano Lett.9(1), 235–238 (2009). [CrossRef] [PubMed]
- P. Ruffieux, T. Scharf, H. P. Herzig, R. Völkel, and K. J. Weible, “On the chromatic aberration of microlenses,” Opt. Express14(11), 4687–4694 (2006). [CrossRef] [PubMed]
- M.-K. Chen, Y.-C. Chang, C.-E. Yang, Y. Guo, J. Mazurowski, S. Yin, P. Ruffin, C. Brantley, E. Edwards, and C. Luo, “Tunable terahertz plasmonic lenses based on semiconductor microslits,” Microw. Opt. Technol. Lett.52(4), 979–981 (2010). [CrossRef]
- X. M. Goh, L. Lin, and A. Roberts, “Planar focusing elements using spatially varying near-resonant aperture arrays,” Opt. Express18(11), 11683–11688 (2010). [CrossRef] [PubMed]
- Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun.39(4), 211–215 (1981). [CrossRef]
- Y. Li and H. Platzer, “An experimental investigation of diffraction patterns in low Fresnel-number focusing systems,” Opt. Acta (Lond.)30(11), 1621–1643 (1983). [CrossRef]
- W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A12(9), 1947–1953 (1995). [CrossRef]
- Y. Yu and H. Zappe, “Effect of lens size on the focusing performance of plasmonic lenses and suggestions for the design,” Opt. Express19(10), 9434–9444 (2011). [CrossRef] [PubMed]
- H. Kurt and D. S. Citrin, “Graded index photonic crystals,” Opt. Express15(3), 1240–1253 (2007). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 2002).
- K. D. Mielenz, “Computation of Fresnel integrals. II,” J. Res. Natl. Inst. Stand. Technol.105, 589–590 (2000).
- D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A14(1), 34–43 (1997). [CrossRef]
- Q. Chen and D. R. S. Cumming, “Visible light focusing demonstrated by plasmonic lenses based on nano-slits in an aluminum film,” Opt. Express18(14), 14788–14793 (2010). [CrossRef] [PubMed]
- Q. Chen, “Effect of the number of zones in a one-dimensional plasmonic zone plate lens: simulation and experiment,” Plasmonics6(1), 75–82 (2011). [CrossRef]

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