myjournal manuscript No.
(will be inserted by the editor)
Polarization properties of subwavelength hole arrays consisting of
rectangular holes
Xi-Feng Ren, Pei Zhang, Guo-Ping Guo⋆, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of
China
Received: date / Revised version: date
Abstract Influence of hole shape on extraordinary op-
tical transmission was investigated using hole arrays con-
sisting of rectangular holes with different aspect ratio.
It was found that the transmission could be tuned con-
tinuously by rotating the hole array. Further more, a
phase was generated in this process, and linear polar-
ization states could be changed to elliptical polarization
states. This phase was correlated with the aspect ratio of
the holes. An intuitional model was presented to explain
these results.
PACS numbers:78.66.Bz,73.20.MF, 71.36.+c
1 introduction
In metal films perforated with a periodic array of sub-
wavelength apertures, it has long been observed that
there is an unusually high optical transmission[1]. It is
⋆ E-mail: e-mail: :gpguo@ustc.edu.cn
believed that metal surface plays a crucial role and the
phenomenon is mediated by surface plasmon polaritons
(SPPs) and there is a process of transforming photon
to SPP and back to photon[2,3,4]. This phenomenon
can be used in various applications, for example, sen-
sors, optoelectronic device, etc[5,6,7,8,9,10]. Polariza-
tion properties of nanohole arrays have been studied in
many works[11,12,13]. Recently, orbital angular momen-
tum of photons was explored to investigate the spatial
mode properties of surface plasmon assisted transmis-
sion [14,15]. It is also showed that entanglement of pho-
ton pairs can be preserved when they respectively travel
through a hole array [15,16,17]. Therefore, the macro-
scopic surface plasmon polarizations, a collective excita-
tion wave involving typically 1010 free electrons propa-
gating at the surface of conducting matter, have a true
quantum nature. However, the increasing use of EOT
requires further understanding of the phenomenon.
http://arxiv.org/abs/0704.0854v2
2 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
The polarization of the incident light determines the
mode of excited SPP which is also related to the periodic
structure. For the manipulation of light at a subwave-
length scale with periodic arrays of holes, two ingredi-
ents exist: shape and periodicity[2,3,4,11,18,19,20]. In-
fluence of unsymmetrical periodicity on EOT was dis-
cussed in [21]. Influence of the hole shape on EOT was
also observed recently[18,20], in which the authors mainly
focused on the transmission spectra. In this work, we
used rectangle hole arrays to investigate the influence of
hole shape on the polarization properties of EOT. It is
found that linear polarization states could be changed
to elliptical polarization states and a phase could be
added between two eigenmode directions. The phase was
changed when the aspect ratio of the rectangle holes was
varied. The hole array was also rotated in the plane per-
pendicular to the illuminate beam. The optical transmis-
sion was changed in this process. It strongly depended
on the rotation angle, in other words, the angle between
polarization of incident light and axis of hole array, as
in the case with unsymmetrical hole array structure[21].
2 experimental results and modeling
2.1 Relation between transmission efficiency and
photon polarization
Fig. 1(a) is a scanning electron microscope picture of
part of our hole arrays. The hole arrays are produced
as follows: after subsequently evaporating a 3-nm tita-
nium bonding layer and a 135-nm gold layer onto a 0.5-
mm-thick silica glass substrate, a focused ion beam etch-
ing system is used to produce rectangle holes (100nm×
100nm, 100nm × 150nm, 100nm × 200nm, 100nm ×
300nm respectively) arranged as a square lattice (520nm
period). The area of the hole array is 10µm× 10µm.
Transmission spectra of the hole arrays were recorded
by a silicon avalanche photodiode single photon counter
couple with a spectrograph through a fiber. White light
from a stabilized tungsten-halogen source passed though
a single mode fiber and a polarizer (only vertical polar-
ized light can pass), then illuminated the sample. The
hole arrays were set between two lenses of 35mm focal
length, so that the light was normally incident on the
hole array with a cross sectional diameter about 10µm
and covered hundreds of holes. The light exiting from
the hole array was launched into the spectrograph. The
hole arrays were rotated anti-clockwise in the plane per-
pendicular to the illuminating light, as shown in Fig.
            (a)                                  (b)  
Fig. 1 (Color online)The rectangle hole arrays. (a) Scanning
electron microscope pictures. (b) Rotation direction. S (L)
is the axis of short (long) edge of rectangle hole; H(V) is
horizontal (vertical) axis.
Polarization properties of subwavelength hole arrays consisting of rectangular holes 3
1(b). Transmission spectra of the hole arrays for rota-
tion angle θ = 0o and 90o were given in Fig. 2. There
were large difference between the two cases, which was
also observed in [18].
Further, the typical hole array(100nm×300nm holes)
was rotated anti-clockwise in the plane perpendicular to
the illuminating light(see Fig.1 (b)). Transmission effi-
ciencies of H and V photons(702nm wavelength) were
measured with rotation angle θ = 0o, 30o, 45o, 60o, and
90o respectively, as shown in Fig. 3. They were varied
with θ. To explain the results, we gave a simple model.
For our sample, photons with 702nm wavelength will
excite the SPP eigenmodes (0,±1) and (±1, 0). Since
the SPPs were excited in the directions of long (L) and
short (S) edges of rectangle holes, we suspected that this
550 600 650 700 750 800 850
550 600 650 700 750 800 850
550 600 650 700 750 800 850
550 600 650 700 750 800 850
Wavelength(nm)
Fig. 2 (Color online)Hole array transmittance as a function
of wavelength for rotation angle θ = 0o(black square dots)
and 90o(red round dots)(holes for a, b, c, and d are 100nm×
100nm, 100nm × 150nm, 100nm × 200nm, and 100nm ×
300nm respectively). The dashed vertical lines indicate the
wavelength of 702nm used in the experiment.
two directions were eigenmode-directions for our sample.
The polarization of illuminating light was projected into
the two eigenmode-directions to excite SPPs. After that,
the two kinds of SPPs transmitted the holes and irritated
light with different transmission efficiencies TL and TS
respectively. For light whose polarization had an angle θ
with the S direction, the transmission efficiency Tθ will
Tθ = TS cos
2(θ) + TL sin
2(θ). (1)
This equation was also given in the works[20,21]. Due
to the unequal values of TL and TS, the whole transmis-
sion efficiency was varied with angle θ. So if we know
the transmission spectra for enginmode-directions (here
L and S), we can calculate out the transmission spectra
(including the heights and locations of peaks) for any
0 15 30 45 60 75 90
10000
20000
30000
40000
50000
60000
70000
Tilt angle  (degree)
Fig. 3 (Color online)Transmittance as a function rotation
angle θ for photons in 702nm wavelength(100nm × 300nm
holes). Red round dots and black square dots are the counts
for V and H photons respectively. The lines come from the-
oretical calculation.
4 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
θ. The theoretical calculations were also given in Fig.
3, which agreed well with the experimental data. The
similar results were also observed when the hole arrays
(100nm× 150nm and 100nm× 200nm) were used. With
this model, the transmission efficiency can be continu-
ously tuned in a certain range.
2.2 Influence of hole shape on photon polarization
To investigate the polarization property of the hole ar-
ray, we used the method of polarization state tomog-
raphy. Experimental setup was shown in Fig. 4. White
light from a stabilized tungsten-halogen source passed
though single mode fiber and 4nm filter (center wave-
length 702 nm) to generate 702nm wavelength photons.
Polarization of input light was controlled by a polarizer,
a HWP (half wave plate, 702nm) and a QWP (quar-
ter wave plate, 702nm). The hole array was set between
two lenses of 35mm focal length. Symmetrically, a QWP,
a HWP and a polarizer were combined to analyze the
polarization of transmitted photons. For arbitrary in-
put states, the output states were measured in the four
bases: H , V , 1/
2(|H〉 + |V 〉), and 1/
2(|H〉 + i|V 〉).
With these experimental data, we could get the density
matrix of output states, which gave the full polarization
characters of transmitted photons. For example, in the
case of θ = 0o, for input state 1/
2(|H〉 + eI∗0.5π|V 〉),
four counts (8943, 31079, 3623 and 21760) were recorded
when we used the four detection bases. The density ma-
trix was calculated as:
0.223 −0.410− 0.043i
−0.410 + 0.043i 0.777
, (2)
which had a fidelity of 0.997 with the pure state 0.472|H〉+
0.882eI∗0.967π|V 〉. Compared this state with the input
state, we found that not only the ratio of |H〉 and |V 〉
was changed, but also a phase ϕ = 0.467π was added
between them. The similar phenomenon was also ob-
served when the input state was 1/
2(|H〉 + |V 〉) and
in this case ϕ = 0.442π. We also considered the cases
for θ = 30o, 45o, 60o, and 90o. The experimental density
matrices had the fidelities all larger than 0.960 with the
theoretical calculations, where ϕ = (0.462 ± 0.053)π. It
can be seen that the phase ϕ was hardly influenced by
the rotation.
To study the dependence of phase ϕ with the hole
shape, we performed the same measurements on other
hole arrays which were shown in Fig. 1. It was found
that ϕ was changed with the aspect ratio of the rectan-
Filter Detector 
HA SMF Source SMF 
Polarization Controller 
Polarization Analyzer 
Fig. 4 (Color online)Experimental setup to investigate the
polarization property of our rectangle hole array. Polariza-
tion of input light was controlled by a polarizer, a HWP and
a QWP. The hole array was set between two lenses of 35mm
focal length. Symmetrically, a QWP, a HWP and a polar-
izer were combined to analyze the polarization of transmitted
photons.
Polarization properties of subwavelength hole arrays consisting of rectangular holes 5
gle holes. Fig. 5 gave the relation between ϕ and aspect
ratio. The phases are 0, (0.227±0.032)π, (0.357±0.020)π
and (0.462±0.053)π for aspect ratio 1, 1.5, 2.0 and 3.0 re-
spectively. As mentioned above, period is another impor-
tant parameter in the EOT experiments. Since no similar
result was observed for hole arrays with symmetrical pe-
riods, a special quadrate hole array(see Fig. 1 of [21]) was
also investigated to show the influence of the hole period.
We found that even the periods were different in two di-
rections, there was no birefringent phenomenon(ϕ = 0).
This birefringent phenomenon might be explained
with the propagating of SPPs on the metal surface. As
we know, the interaction of the incident light with sur-
face plasmon is made allowed by coupling through the
grating momentum and obeys conservation of momen-
k sp =
k 0 ± i
Gx ± j
Gy, (3)
1.0 1.5 2.0 2.5 3.0
Aspect ratio
Fig. 5 (Color online)Relation between birefringent phase ϕ
and hole shape aspect ratio. ϕ becomes lager when the aspect
ratio increases.
where
k sp is the surface plasmon wave vector,
k 0 is the
component of the incident wave vector that lies in the
plane of the sample,
Gx and
Gy are the reciprocal lattice
vectors, and i, j are integers. Usually, Gx = Gy = 2π/d
for a square lattice, and relation
k sp ∗ d = mπ was sat-
isfied, where m was the band index[22]. While for our
rectangle hole arrays, the length of holes in L direction
was changed form 150nm to 300nm, which was not as
same as it in S direction. Though Gx = Gy = 2π/d for
our rectangle hole array, the time for surface plasmon
polariton propagating in the L direction must be influ-
enced by the aspect ratio of hole shape, which could not
be same as that in the S direction. A phase difference
ϕ was generated between the two directions, leading the
birefringent phenomenon. Due to the absorption or scat-
tering of the SPPs and scattering at the hole edges, it
is hard to give the accurate value of the phase or the
exact relation between the phase and aspect ratio of
holes. Even so, ϕ could be controlled by changing the
hole shape. As a contrast, there was no birefringent phe-
nomenon observed when the quadrate hole array(see Fig.
1 of [21]) was used. The reason was that phase Gx ∗ dx
always equal to Gy ∗ dy, even Gx 6= Gy for the quadrate
hole array.
3 conclusion
In conclusion, rectangle hole array was explored to study
the influence of hole shape on EOT, especially the prop-
erties of photon polarization. Because of the unsymmet-
6 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
rical of the hole shape, a birefringent phenomenon was
observed. The phase was determined by the hole shape,
which gave us a potential method to control this bire-
fringent process. It was also found that the transmission
efficiency can be tuned continuously by rotating the hole
array. These results might be explained using an intu-
itional model based on surface plasmon eigenmodes.
This work was funded by the National Fundamental
Research Program, National Natural Science Foundation
of China (10604052), Program for New Century Excel-
lent Talents in University, the Innovation Funds from
Chinese Academy of Sciences, the Program of the Educa-
tion Department of Anhui Province (Grant No.2006kj074A).
Xi-Feng Ren also thanks for the China Postdoctoral Sci-
ence Foundation (20060400205) and the K. C. Wong Ed-
ucation Foundation, Hong Kong.
References
1. T.W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and
P. A. Wolff, Nature 391, 667 (1998).
2. H. Raether, Surface Plasmons on Smooth and Rough Sur-
faces and on Gratings, Vol. 111 of Springer Tracts in Mod-
ern Physics, Springer, Berlin, (1988).
3. D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin,
and Tineke Thio, Appl. Phys. Lett. 77 1569 (2000).
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T. Thio, J. B. Pendry, and T. W. Ebbesen, Phys. Rev.
Lett. 86, 1114 (2001).
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D. Stafford, S. R. Bishop, U. K. Lincoln, and J. V. Coe,
J. Phys. Chem. B. 108, 11833 (2004).
6. A. G. Brolo, R. Gordon, B. Leathem, and K. L. Kavanagh,
Langmuir. 20, 4813 (2004).
7. A. Nahata, R. A. Linke, T. Ishi, and K. Ohashi, Opt. Lett.
28, 423 (2003).
8. X. Luo and T. Ishihara, Appl. Phys. Lett. 84, 4780 (2004).
9. S. Shinada, J. Hasijume and F. Koyama, Appl. Phys. Lett.
83, 836 (2003).
10. C. Genet and T. W. Ebbeson, Nature, 445, 39 (2007).
11. J. Elliott, I. I. Smolyaninov, N. I. Zheludev, and A. V.
Zayats, Opt. Lett. 29, 1414 (2004).
12. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B.
Leathem, and K. L. Kavanagh, Phys. Rev. Lett. 92,
037401 (2004).
13. E. Altewischer, C. Genet, M. P. van Exter, and J. P.
Woerdman, Opt. Lett. 30, 90 (2005).
14. X. F. Ren, G. P. Guo, Y. F. Huang, Z. W. Wang, and
G. C. Guo, Opt. Lett. 31, 2792, (2006).
15. X. F. Ren, G. P. Guo, Y. F. Huang, C. F. Li, and G. C.
Guo, Europhys. Lett. 76, 753 (2006).
16. E. Altewischer, M. P. van Exter and J. P. Woerdman
Nature 418 304 (2002).
17. S. Fasel, F. Robin, E. Moreno, D. Erni, N. Gisin and H.
Zbinden, Phys. Rev. Lett. 94 110501 (2005).
18. K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F.
van Hulst and L. Kuipers, Phys. Rev. Lett. 92 183901
(2004).
19. Zhichao Ruan and Min Qiu, Phys. Rev. Lett. 96 233901
(2006).
Polarization properties of subwavelength hole arrays consisting of rectangular holes 7
20. M. Sarrazin, J. P. Vigneron, Opt. Commun. 240 89
(2004) .
21. X. F. Ren, G. P. Guo, Y. F. Huang, Z. W. Wang, and
G. C. Guo, Appl. Phys. Lett. 90, 161112 (2007).
22. F. L. Tejeira, S. G. Rodrigo, L. M. Moreno, F. J. G. Vi-
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	introduction
	experimental results and modeling
	conclusion

  Influence of hole shape on extraordinary optical transmission was
investigated using hole arrays consisting of rectangular holes with different
aspect ratio. It was found that the transmission could be tuned continuously by
rotating the hole array. Further more, a phase was generated in this process,
and linear polarization states could be changed to elliptical polarization
states. This phase was correlated with the aspect ratio of the holes. An
intuitional model was presented to explain these results.

myjournal manuscript No.
(will be inserted by the editor)
Polarization properties of subwavelength hole arrays consisting of
rectangular holes
Xi-Feng Ren, Pei Zhang, Guo-Ping Guo⋆, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of
China
Received: date / Revised version: date
Abstract Influence of hole shape on extraordinary op-
tical transmission was investigated using hole arrays con-
sisting of rectangular holes with different aspect ratio.
It was found that the transmission could be tuned con-
tinuously by rotating the hole array. Further more, a
phase was generated in this process, and linear polar-
ization states could be changed to elliptical polarization
states. This phase was correlated with the aspect ratio of
the holes. An intuitional model was presented to explain
these results.
PACS numbers:78.66.Bz,73.20.MF, 71.36.+c
1 introduction
In metal films perforated with a periodic array of sub-
wavelength apertures, it has long been observed that
there is an unusually high optical transmission[1]. It is
⋆ E-mail: e-mail: :gpguo@ustc.edu.cn
believed that metal surface plays a crucial role and the
phenomenon is mediated by surface plasmon polaritons
(SPPs) and there is a process of transforming photon
to SPP and back to photon[2,3,4]. This phenomenon
can be used in various applications, for example, sen-
sors, optoelectronic device, etc[5,6,7,8,9,10]. Polariza-
tion properties of nanohole arrays have been studied in
many works[11,12,13]. Recently, orbital angular momen-
tum of photons was explored to investigate the spatial
mode properties of surface plasmon assisted transmis-
sion [14,15]. It is also showed that entanglement of pho-
ton pairs can be preserved when they respectively travel
through a hole array [15,16,17]. Therefore, the macro-
scopic surface plasmon polarizations, a collective excita-
tion wave involving typically 1010 free electrons propa-
gating at the surface of conducting matter, have a true
quantum nature. However, the increasing use of EOT
requires further understanding of the phenomenon.
http://arxiv.org/abs/0704.0854v2
2 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
The polarization of the incident light determines the
mode of excited SPP which is also related to the periodic
structure. For the manipulation of light at a subwave-
length scale with periodic arrays of holes, two ingredi-
ents exist: shape and periodicity[2,3,4,11,18,19,20]. In-
fluence of unsymmetrical periodicity on EOT was dis-
cussed in [21]. Influence of the hole shape on EOT was
also observed recently[18,20], in which the authors mainly
focused on the transmission spectra. In this work, we
used rectangle hole arrays to investigate the influence of
hole shape on the polarization properties of EOT. It is
found that linear polarization states could be changed
to elliptical polarization states and a phase could be
added between two eigenmode directions. The phase was
changed when the aspect ratio of the rectangle holes was
varied. The hole array was also rotated in the plane per-
pendicular to the illuminate beam. The optical transmis-
sion was changed in this process. It strongly depended
on the rotation angle, in other words, the angle between
polarization of incident light and axis of hole array, as
in the case with unsymmetrical hole array structure[21].
2 experimental results and modeling
2.1 Relation between transmission efficiency and
photon polarization
Fig. 1(a) is a scanning electron microscope picture of
part of our hole arrays. The hole arrays are produced
as follows: after subsequently evaporating a 3-nm tita-
nium bonding layer and a 135-nm gold layer onto a 0.5-
mm-thick silica glass substrate, a focused ion beam etch-
ing system is used to produce rectangle holes (100nm×
100nm, 100nm × 150nm, 100nm × 200nm, 100nm ×
300nm respectively) arranged as a square lattice (520nm
period). The area of the hole array is 10µm× 10µm.
Transmission spectra of the hole arrays were recorded
by a silicon avalanche photodiode single photon counter
couple with a spectrograph through a fiber. White light
from a stabilized tungsten-halogen source passed though
a single mode fiber and a polarizer (only vertical polar-
ized light can pass), then illuminated the sample. The
hole arrays were set between two lenses of 35mm focal
length, so that the light was normally incident on the
hole array with a cross sectional diameter about 10µm
and covered hundreds of holes. The light exiting from
the hole array was launched into the spectrograph. The
hole arrays were rotated anti-clockwise in the plane per-
pendicular to the illuminating light, as shown in Fig.
            (a)                                  (b)  
Fig. 1 (Color online)The rectangle hole arrays. (a) Scanning
electron microscope pictures. (b) Rotation direction. S (L)
is the axis of short (long) edge of rectangle hole; H(V) is
horizontal (vertical) axis.
Polarization properties of subwavelength hole arrays consisting of rectangular holes 3
1(b). Transmission spectra of the hole arrays for rota-
tion angle θ = 0o and 90o were given in Fig. 2. There
were large difference between the two cases, which was
also observed in [18].
Further, the typical hole array(100nm×300nm holes)
was rotated anti-clockwise in the plane perpendicular to
the illuminating light(see Fig.1 (b)). Transmission effi-
ciencies of H and V photons(702nm wavelength) were
measured with rotation angle θ = 0o, 30o, 45o, 60o, and
90o respectively, as shown in Fig. 3. They were varied
with θ. To explain the results, we gave a simple model.
For our sample, photons with 702nm wavelength will
excite the SPP eigenmodes (0,±1) and (±1, 0). Since
the SPPs were excited in the directions of long (L) and
short (S) edges of rectangle holes, we suspected that this
550 600 650 700 750 800 850
550 600 650 700 750 800 850
550 600 650 700 750 800 850
550 600 650 700 750 800 850
Wavelength(nm)
Fig. 2 (Color online)Hole array transmittance as a function
of wavelength for rotation angle θ = 0o(black square dots)
and 90o(red round dots)(holes for a, b, c, and d are 100nm×
100nm, 100nm × 150nm, 100nm × 200nm, and 100nm ×
300nm respectively). The dashed vertical lines indicate the
wavelength of 702nm used in the experiment.
two directions were eigenmode-directions for our sample.
The polarization of illuminating light was projected into
the two eigenmode-directions to excite SPPs. After that,
the two kinds of SPPs transmitted the holes and irritated
light with different transmission efficiencies TL and TS
respectively. For light whose polarization had an angle θ
with the S direction, the transmission efficiency Tθ will
Tθ = TS cos
2(θ) + TL sin
2(θ). (1)
This equation was also given in the works[20,21]. Due
to the unequal values of TL and TS, the whole transmis-
sion efficiency was varied with angle θ. So if we know
the transmission spectra for enginmode-directions (here
L and S), we can calculate out the transmission spectra
(including the heights and locations of peaks) for any
0 15 30 45 60 75 90
10000
20000
30000
40000
50000
60000
70000
Tilt angle  (degree)
Fig. 3 (Color online)Transmittance as a function rotation
angle θ for photons in 702nm wavelength(100nm × 300nm
holes). Red round dots and black square dots are the counts
for V and H photons respectively. The lines come from the-
oretical calculation.
4 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
θ. The theoretical calculations were also given in Fig.
3, which agreed well with the experimental data. The
similar results were also observed when the hole arrays
(100nm× 150nm and 100nm× 200nm) were used. With
this model, the transmission efficiency can be continu-
ously tuned in a certain range.
2.2 Influence of hole shape on photon polarization
To investigate the polarization property of the hole ar-
ray, we used the method of polarization state tomog-
raphy. Experimental setup was shown in Fig. 4. White
light from a stabilized tungsten-halogen source passed
though single mode fiber and 4nm filter (center wave-
length 702 nm) to generate 702nm wavelength photons.
Polarization of input light was controlled by a polarizer,
a HWP (half wave plate, 702nm) and a QWP (quar-
ter wave plate, 702nm). The hole array was set between
two lenses of 35mm focal length. Symmetrically, a QWP,
a HWP and a polarizer were combined to analyze the
polarization of transmitted photons. For arbitrary in-
put states, the output states were measured in the four
bases: H , V , 1/
2(|H〉 + |V 〉), and 1/
2(|H〉 + i|V 〉).
With these experimental data, we could get the density
matrix of output states, which gave the full polarization
characters of transmitted photons. For example, in the
case of θ = 0o, for input state 1/
2(|H〉 + eI∗0.5π|V 〉),
four counts (8943, 31079, 3623 and 21760) were recorded
when we used the four detection bases. The density ma-
trix was calculated as:
0.223 −0.410− 0.043i
−0.410 + 0.043i 0.777
, (2)
which had a fidelity of 0.997 with the pure state 0.472|H〉+
0.882eI∗0.967π|V 〉. Compared this state with the input
state, we found that not only the ratio of |H〉 and |V 〉
was changed, but also a phase ϕ = 0.467π was added
between them. The similar phenomenon was also ob-
served when the input state was 1/
2(|H〉 + |V 〉) and
in this case ϕ = 0.442π. We also considered the cases
for θ = 30o, 45o, 60o, and 90o. The experimental density
matrices had the fidelities all larger than 0.960 with the
theoretical calculations, where ϕ = (0.462 ± 0.053)π. It
can be seen that the phase ϕ was hardly influenced by
the rotation.
To study the dependence of phase ϕ with the hole
shape, we performed the same measurements on other
hole arrays which were shown in Fig. 1. It was found
that ϕ was changed with the aspect ratio of the rectan-
Filter Detector 
HA SMF Source SMF 
Polarization Controller 
Polarization Analyzer 
Fig. 4 (Color online)Experimental setup to investigate the
polarization property of our rectangle hole array. Polariza-
tion of input light was controlled by a polarizer, a HWP and
a QWP. The hole array was set between two lenses of 35mm
focal length. Symmetrically, a QWP, a HWP and a polar-
izer were combined to analyze the polarization of transmitted
photons.
Polarization properties of subwavelength hole arrays consisting of rectangular holes 5
gle holes. Fig. 5 gave the relation between ϕ and aspect
ratio. The phases are 0, (0.227±0.032)π, (0.357±0.020)π
and (0.462±0.053)π for aspect ratio 1, 1.5, 2.0 and 3.0 re-
spectively. As mentioned above, period is another impor-
tant parameter in the EOT experiments. Since no similar
result was observed for hole arrays with symmetrical pe-
riods, a special quadrate hole array(see Fig. 1 of [21]) was
also investigated to show the influence of the hole period.
We found that even the periods were different in two di-
rections, there was no birefringent phenomenon(ϕ = 0).
This birefringent phenomenon might be explained
with the propagating of SPPs on the metal surface. As
we know, the interaction of the incident light with sur-
face plasmon is made allowed by coupling through the
grating momentum and obeys conservation of momen-
k sp =
k 0 ± i
Gx ± j
Gy, (3)
1.0 1.5 2.0 2.5 3.0
Aspect ratio
Fig. 5 (Color online)Relation between birefringent phase ϕ
and hole shape aspect ratio. ϕ becomes lager when the aspect
ratio increases.
where
k sp is the surface plasmon wave vector,
k 0 is the
component of the incident wave vector that lies in the
plane of the sample,
Gx and
Gy are the reciprocal lattice
vectors, and i, j are integers. Usually, Gx = Gy = 2π/d
for a square lattice, and relation
k sp ∗ d = mπ was sat-
isfied, where m was the band index[22]. While for our
rectangle hole arrays, the length of holes in L direction
was changed form 150nm to 300nm, which was not as
same as it in S direction. Though Gx = Gy = 2π/d for
our rectangle hole array, the time for surface plasmon
polariton propagating in the L direction must be influ-
enced by the aspect ratio of hole shape, which could not
be same as that in the S direction. A phase difference
ϕ was generated between the two directions, leading the
birefringent phenomenon. Due to the absorption or scat-
tering of the SPPs and scattering at the hole edges, it
is hard to give the accurate value of the phase or the
exact relation between the phase and aspect ratio of
holes. Even so, ϕ could be controlled by changing the
hole shape. As a contrast, there was no birefringent phe-
nomenon observed when the quadrate hole array(see Fig.
1 of [21]) was used. The reason was that phase Gx ∗ dx
always equal to Gy ∗ dy, even Gx 6= Gy for the quadrate
hole array.
3 conclusion
In conclusion, rectangle hole array was explored to study
the influence of hole shape on EOT, especially the prop-
erties of photon polarization. Because of the unsymmet-
6 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo
rical of the hole shape, a birefringent phenomenon was
observed. The phase was determined by the hole shape,
which gave us a potential method to control this bire-
fringent process. It was also found that the transmission
efficiency can be tuned continuously by rotating the hole
array. These results might be explained using an intu-
itional model based on surface plasmon eigenmodes.
This work was funded by the National Fundamental
Research Program, National Natural Science Foundation
of China (10604052), Program for New Century Excel-
lent Talents in University, the Innovation Funds from
Chinese Academy of Sciences, the Program of the Educa-
tion Department of Anhui Province (Grant No.2006kj074A).
Xi-Feng Ren also thanks for the China Postdoctoral Sci-
ence Foundation (20060400205) and the K. C. Wong Ed-
ucation Foundation, Hong Kong.
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	introduction
	experimental results and modeling
	conclusion
