## Young’s experiment with a double slit of sub-wavelength dimensions |

Optics Express, Vol. 21, Issue 16, pp. 18805-18811 (2013)

http://dx.doi.org/10.1364/OE.21.018805

Acrobat PDF (1303 KB)

### Abstract

We report that the interference pattern of Young’s double-slit experiment changes as a function of polarization in the sub-wavelength diffraction regime. Experiments carried out with terahertz time-domain spectroscopy reveal that diffracted waves from sub-wavelength-scale slits exhibit either positive or negative phase shift with respect to Gouy phase depending on the polarization. Theoretical explanation based on the induction of electric current and magnetic dipole in the vicinity of the slits shows an excellent agreement with the experimental results.

© 2013 OSA

## 1. Introduction

3. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. **90**, 213901 (2003). [CrossRef] [PubMed]

11. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

3. F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. **90**, 213901 (2003). [CrossRef] [PubMed]

4. C. Wang, C. Du, and X. Luo, “Refining the model of light diffraction from a subwavelength slit surrounded by grooves on a metallic film,” Phys. Rev. B **74**, 245403 (2006). [CrossRef]

5. Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. **86**, 5601 (2001). [CrossRef] [PubMed]

6. F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. **89**, 063901 (2002). [CrossRef] [PubMed]

9. E. H. Khoo, E. P. Li, and K. B. Crozier, “Plasmonic wave plate based on subwavelength nanoslits,” Opt. Lett. **36**, 2498–2500 (2011). [CrossRef] [PubMed]

10. P. F. Chimento, N. V. Kuzmin, J. Bosman, P. F. A. Alkemade, G. W.’t Hooft, and E. R. Eliel, “A subwavelength slit as a quarter-wave retarder,” Opt. Express **19**, 24219–24227 (2011). [CrossRef] [PubMed]

12. M. Yi, K. Lee, J. D. Song, and J. Ahn, “Terahertz phase microscopy in the sub-wavelength regime,” Appl. Phys. Lett. **100**, 161110 (2012). [CrossRef]

13. K. Lee, M. Yi, S. E. Park, and J. Ahn, “Phase-shift anomaly caused by subwavelength-scale metal slit or aperture diffraction,” Opt. Lett. **38**, 166–168 (2013). [CrossRef] [PubMed]

## 2. Young’s double-slit interference pattern in the sub-wavelength regime

*d*

_{1}and

*d*

_{2}(

*d*

_{1}<

*d*

_{2}) and with spacing

*a*, as shown in Fig. 1. When electromagnetic wave of angular frequency

*ω*diffracts from the slits and we assume that

*D*

_{1,2}and

*ϕ*

_{1,2}are respectively the amplitude and phase of the transmitted wave from each slit, the electric field sum in the far-field diffraction region is given by where

*δ*(

*x*) is Dirac delta function,

*f*is the focal length of the imaging lens, and

*k*is the wave number. Note that

*D*

_{1,2}could not just be proportional to

*d*

_{1,2}and

*ϕ*

_{1,2}could be different from 90 degeee Gouy phase shift, in the sub-wavelength regime [13

13. K. Lee, M. Yi, S. E. Park, and J. Ahn, “Phase-shift anomaly caused by subwavelength-scale metal slit or aperture diffraction,” Opt. Lett. **38**, 166–168 (2013). [CrossRef] [PubMed]

*ϕ*=

*ϕ*

_{2}−

*ϕ*

_{1}, is given as a function of

*d*

_{1}and

*d*

_{2}. So, for a non-zero Δ

*ϕ*the interference pattern is not given symmetrical to the optical axis, different from what is expected from Kirchhoff’s scalar diffraction theory. Furthermore, Δ

*ϕ*is also strongly coupled with the polarization angle

*θ*(to be explained in Sec. 4). So, it is expected that the location of the interference pattern changes as a function of

*θ*, in the sub-wavelength regime.

## 3. Experimental description

14. D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B **7**, 2006–2015 (1990). [CrossRef]

16. P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B **18**, 313–317 (2001). [CrossRef]

17. Y. Kim, M. Yi, B. G. Kim, and J. Ahn, “Investigation of THz birefringence measurement and calculation in Al2O3 and LiNbO3,” Appl. Opt. **50**, 2906–2910 (2011). [CrossRef] [PubMed]

*μ*m, a length of

*L*= 3 mm, and various widths of 30, 40, 50, 100, 150, and 200

*μ*m, respectively. The slits were inserted at the focus of the propagating THz waves in a one-dimensional 4-

*f*or 8-

*f*geometry THz beam delivery system composed of two or four teflon lenses with focal length

*f*= 100 mm. For the double-slit experiments to be explained in Sec. 5, double-slits were fabricated via laser micro-machining on an aluminum sheet of thickness 18

*μ*m, and the sizes of the slits were

*d*

_{1}= 100

*μ*m and

*d*

_{2}= 300

*μ*m, respectively, and the slit spacing was

*a*= 10 mm. Note that the incoming THz beam was collimated near the slit structure with a uniform intensity region over 20×20 mm

^{2}[13

13. K. Lee, M. Yi, S. E. Park, and J. Ahn, “Phase-shift anomaly caused by subwavelength-scale metal slit or aperture diffraction,” Opt. Lett. **38**, 166–168 (2013). [CrossRef] [PubMed]

## 4. Polarization dependence of sub-wavelength-scale single-slit diffraction

*θ*with respect to the slit orientation diffracts through a slit of width

*d*and of length

*L*as shown in Fig. 2(a). Note that the electric field within the slit strongly depends on

*θ*, while the magnetic field does not, due to the symmetric boundary consideration [2, 11

11. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

*Perpendicular polarization case (θ*=

*π*/2

*):*Waveguide theory predicts in this polarization case that the propagation mode of electric field exists, regardless of the size of

*d*[2], and the electric field becomes even strongly enhanced in small

*d*limit [7

7. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photon. **3**, 152–156 (2009). [CrossRef]

8. J. H. Kang, D. S. Kim, and Q. Park, “Local Capacitor Model for Plasmonic Electric Field Enhancement,” Phys. Rev. Lett. **102**, 093906 (2009). [CrossRef] [PubMed]

*E*(

_{t}*R*) is the ordinary

*d/λ*-dependent diffraction field from the scalar diffraction theory, and the second one

*E*(

_{i}*R*) is the induced field from the enhanced electric field minus the incident electric field on the slit. Recent experiment shows that

*E*(

_{i}*R*) has

*λ*-dependent field enhancement compared to the first one [7

7. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photon. **3**, 152–156 (2009). [CrossRef]

*E*(

_{i}*R*) should be a constant of

*d/λ*due to scale invariance [8

8. J. H. Kang, D. S. Kim, and Q. Park, “Local Capacitor Model for Plasmonic Electric Field Enhancement,” Phys. Rev. Lett. **102**, 093906 (2009). [CrossRef] [PubMed]

**38**, 166–168 (2013). [CrossRef] [PubMed]

*π*/2 between

*E*(

_{t}*R*) and

*E*(

_{i}*R*) [13

**38**, 166–168 (2013). [CrossRef] [PubMed]

*α*is a proportional constant. Note that the phase shift compared from the phase factor,

*e*

^{i(kR−ωt)}, changes from the usual

*π*/2 advancement, or Gouy phase [18–22

22. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. **26**, 485–487 (2001). [CrossRef]

*ϕ*

_{1}>

*ϕ*

_{2}in Eq. (2).

*Parallel polarization case (θ*= 0

*):*The electric field is negligibly small in this polarization geometry due to the boundary condition on metal, so the diffracted field is mainly contributed by the remaining magnetic field. Based on the Bethe’s effective magnetic dipole model [11

11. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

*L*long enough, the magnetic surface charge density

*ρ*(

*y*) across the slit (

*y*-direction) is given by where

*H*

_{0}is the incident magnetic field amplitude and the proportional coefficient (1/2

*π*) can be obtained from Gauss law ∇ ·

*H*= 4

*πσ*, where

*σ*is the effective magnetic density [23]. The effective magnetic dipole moment then becomes

*m*

_{eff}= −

*H*

_{0}

*d*

^{2}

*L*/16 so that the diffracted field along the optical axis is given by where

*E*

_{0}is the incident electric field amplitude and

*d*

^{2}/

*λ*

^{2}contrary to the linear

*d/λ*-dependence expected in the scalar diffraction theory. So, the diffracted field from the slit is much smaller than what is expected from Kirchhoff’s diffraction theory in the small

*d*limit. Note that this

*d/λ*-ratio reduction from Kirchhoff’s theory is similar to the case of sub-wavelength-scale hole diffraction [11

11. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

*θ*= 0 is

*π*/2 more time-advanced than usual Gouy phase shift, so the net phase shift is

*π*(time advancement) from

*e*

^{i}^{(kR−ωt)}. Note that both this polarization case and the sub-wavelength circular aperture case [11

11. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**, 163–182 (1944). [CrossRef]

**38**, 166–168 (2013). [CrossRef] [PubMed]

*π*/2 to

*π*like sub-wavelength hole diffraction due to the optical theorem [13

**38**, 166–168 (2013). [CrossRef] [PubMed]

*θ*=

*π*/2 case, or

*ϕ*

_{1}<

*ϕ*

_{2}in the notation of Eq. (2).

*Diffraction phase:*The phase shifts of the diffracted waves from sub-wavelength-scale slits were directly measured in our THz waveform detection. Figures 2(b)–2(g) represent the normalized time-signals of the diffracted THz waves from 30, 40, 50, 100, 150, and 200

*μ*m slits, respectively. The blue and red lines in Fig. 2 represent

*θ*= 0 and

*θ*=

*π*/2 cases, respectively, while the reference signals without a slit are plotted in dotted lines. Phase advancement and retardation of nearly

*π*/2 are clearly observed in 30

*μ*m-slit diffraction shown in Fig. 2(b) so the fields in

*θ*=

*π*/2 and

*θ*= 0 have opposite phases to each other. For intermediate size, phase difference between

*θ*=

*π*/2 and

*θ*= 0 becomes smaller as shown in Figs. 2(c)–2(g).

*Diffraction amplitude:*The amplitudes of the diffracted waves from sub-wavelength-scale slits are shown in Fig. 3, where the measured fields through sub-wavelength-scale slits for

*θ*=

*π*/2 and

*θ*= 0 are respectively in Figs. 3(a) and 3(c). The scaling behaviors of the diffracted wave amplitude in Eqs. (3) and (5) are verified respectively in Fig. 3(b) for

*θ*=

*π*/2 and in Fig. 3(d) for

*θ*= 0. As expected, significant amount of fields are transmitted for

*θ*=

*π*/2 even in the smallest slit of

*d*= 30

*μ*m, while negligible amount transmitted for

*θ*= 0. Note that the

*d*

^{2}/

*λ*

^{2}dependence of the transmission amplitude for

*θ*= 0 in Fig. 3(d) breaks off from the cutoff wavelength condition at

*d/λ*= 0.5 in waveguide theories [2].

## 5. Results of double-slit interference pattern in the sub-wavelength regime

*θ*=

*π*/2). The measured THz time signal

*E*(

*x,t*), the interference pattern spectrum

*I*(

*x*,

*ω*), and the interference pattern

*I*(

*x*,

*ω*) at

_{o}*ω*= 0.7 THz are shown in Figs. 4(a)–4(c), respectively. In this polarization, the interference pattern is shifted to the direction of the first (narrow) slit, because the transmitted field from the first slit has time retardation compared to the second (wide) slit (

_{o}*i.e.*,

*ϕ*

_{1}>

*ϕ*

_{2}) as shown in Figs. 4(b) and 4(c). Based on the measurement in Sec. 4, the diffraction phase difference Δ

*ϕ*=

*ϕ*

_{2}−

*ϕ*

_{1}≈ −30°, which is consistent with the experimental result in Fig. 4(c). The maximum time peak also shifts to the narrow-slit direction in Fig. 4(a). The visibility of the pattern is given from Eq. (2) by where

*I*

_{max}and

*I*

_{min}are the maximum and minimum of the intensity, respectively. In Kirchhoff’s theory, it is given simply by

*D*

_{1}should be larger than what is expected from Kirchhoff’s theory for the perpendicular polarized wave. So, the interference pattern should show clearer visibility in this case, which is verified in Fig. 4(c) where the experimental data (crosses) exhibits a better visibility than Kirchhoff’s theory line (dash-dot).

*θ*= 0) are shown in Figs. 4(d)–4(f). In this field polarization, the diffraction pattern is shifted to the direction of the second (wide) slit as shown in Figs. 4(e) and 4(f), which can be easily understood because the diffracted field from the narrow slit has phase advancement compared to the wide slit (

*i.e.*,

*ϕ*

_{1}<

*ϕ*

_{2}). Based on the measurement in Sec. 4, Δ

*ϕ*≈ 60°. The maximum time peak also appears near the wide slit in Fig. 4(d). In addition, the experimental data (circles) exhibits poorer visibility than Kirchhoff’s theory line (dash-dot) in Fig. 4(f), which can be also understood from Table 1 that

*D*

_{1}in

*θ*= 0 is smaller than what is expected from Kirchhoff’s theory.

## 6. Conclusion

## Acknowledgments

## References and links

1. | E. Hecht, |

2. | J. D. Jackson, |

3. | F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. |

4. | C. Wang, C. Du, and X. Luo, “Refining the model of light diffraction from a subwavelength slit surrounded by grooves on a metallic film,” Phys. Rev. B |

5. | Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. |

6. | F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett. |

7. | M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photon. |

8. | J. H. Kang, D. S. Kim, and Q. Park, “Local Capacitor Model for Plasmonic Electric Field Enhancement,” Phys. Rev. Lett. |

9. | E. H. Khoo, E. P. Li, and K. B. Crozier, “Plasmonic wave plate based on subwavelength nanoslits,” Opt. Lett. |

10. | P. F. Chimento, N. V. Kuzmin, J. Bosman, P. F. A. Alkemade, G. W.’t Hooft, and E. R. Eliel, “A subwavelength slit as a quarter-wave retarder,” Opt. Express |

11. | H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. |

12. | M. Yi, K. Lee, J. D. Song, and J. Ahn, “Terahertz phase microscopy in the sub-wavelength regime,” Appl. Phys. Lett. |

13. | K. Lee, M. Yi, S. E. Park, and J. Ahn, “Phase-shift anomaly caused by subwavelength-scale metal slit or aperture diffraction,” Opt. Lett. |

14. | D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B |

15. | Y.-S. Lee, |

16. | P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B |

17. | Y. Kim, M. Yi, B. G. Kim, and J. Ahn, “Investigation of THz birefringence measurement and calculation in Al2O3 and LiNbO3,” Appl. Opt. |

18. | L. G. Gouy, “Sur une propriete nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris |

19. | A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. |

20. | A. E. Siegman, |

21. | A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett. |

22. | S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. |

23. | K. Lee, “Fourier optical phenomena and applications using ultra broadband terahertz waves,” Ph. D. Thesis, KAIST (2013). |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(050.5080) Diffraction and gratings : Phase shift

(300.6495) Spectroscopy : Spectroscopy, teraherz

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: June 28, 2013

Revised Manuscript: July 23, 2013

Manuscript Accepted: July 23, 2013

Published: July 31, 2013

**Citation**

Kanghee Lee, Jongseok Lim, and Jaewook Ahn, "Young’s experiment with a double slit of sub-wavelength dimensions," Opt. Express **21**, 18805-18811 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-16-18805

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### References

- E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).
- J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
- F. J. García-Vidal, H. J. Lezec, T. W. Ebbesen, and L. Martín-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett.90, 213901 (2003). [CrossRef] [PubMed]
- C. Wang, C. Du, and X. Luo, “Refining the model of light diffraction from a subwavelength slit surrounded by grooves on a metallic film,” Phys. Rev. B74, 245403 (2006). [CrossRef]
- Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett.86, 5601 (2001). [CrossRef] [PubMed]
- F. Yang and J. R. Sambles, “Resonant transmission of microwaves through a narrow metallic slit,” Phys. Rev. Lett.89, 063901 (2002). [CrossRef] [PubMed]
- M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photon.3, 152–156 (2009). [CrossRef]
- J. H. Kang, D. S. Kim, and Q. Park, “Local Capacitor Model for Plasmonic Electric Field Enhancement,” Phys. Rev. Lett.102, 093906 (2009). [CrossRef] [PubMed]
- E. H. Khoo, E. P. Li, and K. B. Crozier, “Plasmonic wave plate based on subwavelength nanoslits,” Opt. Lett.36, 2498–2500 (2011). [CrossRef] [PubMed]
- P. F. Chimento, N. V. Kuzmin, J. Bosman, P. F. A. Alkemade, G. W.’t Hooft, and E. R. Eliel, “A subwavelength slit as a quarter-wave retarder,” Opt. Express19, 24219–24227 (2011). [CrossRef] [PubMed]
- H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163–182 (1944). [CrossRef]
- M. Yi, K. Lee, J. D. Song, and J. Ahn, “Terahertz phase microscopy in the sub-wavelength regime,” Appl. Phys. Lett.100, 161110 (2012). [CrossRef]
- K. Lee, M. Yi, S. E. Park, and J. Ahn, “Phase-shift anomaly caused by subwavelength-scale metal slit or aperture diffraction,” Opt. Lett.38, 166–168 (2013). [CrossRef] [PubMed]
- D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B7, 2006–2015 (1990). [CrossRef]
- Y.-S. Lee, Principles of Terahertz Science and Technology (Springer, 2009).
- P. C. M. Planken, H.-K. Nienhuys, H. J. Bakker, and T. Wenckebach, “Measurement and calculation of the orientation dependence of terahertz pulse detection in ZnTe,” J. Opt. Soc. Am. B18, 313–317 (2001). [CrossRef]
- Y. Kim, M. Yi, B. G. Kim, and J. Ahn, “Investigation of THz birefringence measurement and calculation in Al2O3 and LiNbO3,” Appl. Opt.50, 2906–2910 (2011). [CrossRef] [PubMed]
- L. G. Gouy, “Sur une propriete nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris110, 1251–1253 (1890).
- A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev.54, 931–936 (1938). [CrossRef]
- A. E. Siegman, Lasers (University Science Books, 1986).
- A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single-cycle Terahertz pulses,” Phys. Rev. Lett.83, 3410–3413 (1999). [CrossRef]
- S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett.26, 485–487 (2001). [CrossRef]
- K. Lee, “Fourier optical phenomena and applications using ultra broadband terahertz waves,” Ph. D. Thesis, KAIST (2013).

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