2028
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
Characteristics of surface plasmon polaritons at a chiral–metal interface Guangcan Mi* and Vien Van Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada *Corresponding author: [email protected] Received October 31, 2013; revised February 21, 2014; accepted February 23, 2014; posted February 24, 2014 (Doc. ID 200314); published March 26, 2014 The characteristics of surface plasmon polaritons at a chiral–metal interface are analyzed in detail. Compared to conventional surface plasmon waves at a dielectric–metal interface, it is shown that chiral surface plasmon waves have distinguishing features such as the presence of an s-wave at the metal surface, the existence of a cutoff frequency and chirality value, and the dependence of the propagation length on the chiral parameter. These properties of chiral surface plasmon waves can be exploited for on-chip chiral sensing and enantiometric detection applications. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (280.4788) Optical sensing and sensors. http://dx.doi.org/10.1364/OL.39.002028
Surface plasmon polaritons (SPPs) are electromagnetically coupled electron density oscillations that have been explored for various applications from sensors to subwavelength waveguiding devices. A common configuration of surface plasmon waveguide sensors consists of a thin metal film which is functionalized with a biorecognition layer [1]. The evanescent field of the SP mode is used to sense the refractive index change near the metal surface caused by a binding reaction with the target analyte. Besides refractive index sensing, another important method of optical sensing is based on the detection of the optical activity of the analyte. Biochemical molecules typically exhibit chirality (enantiomerism) which strongly influences their functionalities. The detection of chirality is thus important for enantiomer purity test, concentration measurement of chiral substance, and identification of enantiometry of proteins. Recently, it has been suggested that circular dichroism from chiral molecules can be enhanced by surface plasmon resonances if the chiral molecules are attached to metallic nanoparticles [2,3]. However, the use of surface plasmon waves at an interface between a metal and a chiral medium has not been investigated for chirality detection. The propagation of electromagnetic waves in chiral waveguides has been investigated for a chiral slab bounded by one or two perfectly conducting parallel plates [4,5] and for dielectric slab waveguides in which either the cladding or core or both is a chiral medium [6,7]. In these works it was shown that the modes in the waveguides are no longer purely TE or TM but are hybrid modes due to the coupling between the electric and magnetic fields in the chiral medium. Typically, two modes exist which correspond predominantly to the left-circularly polarized (LCP) and right-circularly polarized (RCP) waves. Few, however, have studied the interaction of SPPs with a chiral medium. The work in [8] examines SP waves that are supported at the interface of a metal and a chiral sculptured metamaterial composed of spiral nanorods, where it was suggested that a hybrid SP wave also exists at such an interface. It is the objective of this Letter to examine in detail the characteristics of SPPs at a chiral–metal surface 0146-9592/14/072028-04$15.00/0
for the purpose of chiral sensing applications. Specifically we find that the decay length of the SPPs depends on the chirality strength and prove that there exists a cutoff condition in terms of either the frequency or chiral parameter for which the SP wave is no longer supported. In addition, we show that the SP waves are degenerate and elliptically polarized, with the sense of rotation depending on the sign of the chiral parameter. Finally, we suggest that these unique properties of SPPs at a chiral–metal surface can be exploited to develop new techniques for chiral sensing and enantiometric detection. We consider an interface between a metal and a chiral medium characterized by electric permittivity εc , vacuum magnetic permeability μ0 , and chiral parameter ξ, as shown in Fig. 1. The constitutive relations for the electric and magnetic fields in the chiral medium are given by D εE − jξB and H −jξE B∕μ [9]. These two sets of constitutive relations are equivalent, with the chiral parameters ξ and κ related by κ μcξ. Due to the coupling between the E and H fields, all three components of the electric and magnetic fields exist in both metal and chiral media. In the chiral medium the fields satisfy the coupled wave equations [4] ∇
2
E H
1 2χ
2
n2c k20
E H
−2jξk20 ∕η20 H 2jξ1 χ 2 n2c k20 E
0; (1)
p where nc εc ∕ε0 is the refractive index of the chiral p material, η0 ε0 ∕μ0 is the admittance of free space, and χ ξ∕ηc is the normalized chirality admittance (ηc nc η0 ). In a bulk chiral medium, light propagates as RCP and LCP waves with propagation constants k and k− , prespectively, where k n k0 and n nc χ 1 χ 2 . For wave propagation along the chiral–metal surface, assuming propagating fields with z dependence E, H ∼ e−jβz , we can express the solutions for the E z and H z field components in the chiral medium as the superposition of two modes U 1 and U 2 [4] © 2014 Optical Society of America
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
x (εc, µ0, ξ) z
Fig. 1. Interface between a metal and a chiral medium.
E z U 1 x U 2 xe−jβz ;
(2)
q H z jηc 1 χ 2 U 1 x − U 2 xe−jβz ;
(3)
where U 1 and U 2 are solutions to d2 U 12 k2− − β2 U 12 0: dx2
(4)
−kx1 x Writing the solutions to Eq. (4) as U 1 and C1e q −kx2 x 2 2 U 2 C2 e , where kx1x2 β − k− , the solutions
for the other field components in the chiral medium can be obtained from Eq. (1) as (omitting e−jβz for clarity) E x −jβ∕kx1 C 1 e−kx1 x − jβ∕kx2 C 2 e−kx2 x ; q β β H x ηc 1 χ 2 C 1 e−kx1 x − C 2 e−kx2 x ; kx1 kx2 E y −k ∕kx1 C 1 e−kx1 x k− ∕kx2 C 2 e−kx2 x ; q k k H y −jηc 1 χ 2 C 1 e−kx1 x − C 2 e−kx2 x : kx1 kx2
(5) 1
(6)
(7)
(a)
(8)
k± n+
0.8 1.8
0.6 1.7
Neff
(εm, µ0)
waves (p-waves). In the limit k k− , the above relation reduces to the familiar dispersion of SPPs at a dielectric– metal interface, which supports only the TM polarization. In Fig. 2(a) we plot the dispersion relation for SPPs at the interface between gold and a chiral medium with parameters nc 1.5, ξ 10−5 Ω−1 . We assume the permittivity of gold is described by the Drude model with plasma frequency ωp 1.30 × 1016 rad∕s and damping constant γ 2.80 × 1013 rad∕s [10]. Also shown in the plot are the light lines for k as well as the dispersion curve (green-dashed line) of SPPs at an interface between gold and a dielectric medium with index nc (i.e., ξ 0). We observe that the dispersion curve of the chiral SPPs follows closely that of SPPs at a metal–dielectric interface, with both curves asymptotically approaching p the surface plasmon resonance fre quency ωsp ωp ∕ 1 n2c at large values of β. The dispersion curve of chiral SPPs lies completely to the right of the light lines for k and k− , indicating that the chiral SP waves travel more slowly than both circularly polarized waves in the bulk chiral medium. Another distinguishing feature of the plasmonic chiral–metal surface is that it supports only one degenerate SPP mode which, as will be shown later, is either left- or rightelliptically polarized. This is in contrast to chiral
Normalized frequency ω ωsp
0
2029
0.4
0.2
0 0
1.6 1.5 0
2
0.2
4
0.4
0.6
ω ωsp
6
8
10
12
Normalized propagation constant β ksp
q ωεm p kx2 k− kxm ωμ0 ηc 1 χ 2 kx1 k kxm ηc 1 χ 2 q ωεm 2 p kx1 k kxm ωμ0 ηc 1 χ kx2 k− kxm ηc 1 χ 2 0:
(9)
In the above expression, the first and third terms in the brackets are dispersion relations associated with the TE waves (s-waves) at an interface between the metal and a dielectric with refractive index n . The second and fourth terms are the dispersion relations for the TM
1
(b)
Normalized frequency ω ωsp
In the metal the fields are given by E z A1 ekxm x ; H z B1 ekxm x , E x jβ∕kxm E z ; H x jβ∕kxm H z , and Ey −jωμ0 ∕kxm H z ; H y −jωεm ∕kxm E z : In the p above, kxm β2 − ω2 μ0 εm is the transverse SP decay constant in the metal. By matching the tangential fields Ez , H z , E y , and H y at the interface x 0, we obtain the dispersion relation
0.8
0.6
0.4
0.2
0 -2 10
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Propagation length Lp (µm)
Fig. 2. (a) Dispersion curve (black line) of SPPs at an interface between gold and a chiral medium with parameters nc 1.5 and ξ 10−5 Ω−1 . The frequency and propagation constant are normalized by the SP resonance frequency ωsp and ksp ωsp ∕c, respectively. The inset shows the effective index N eff versus frequency for different ξ values. (b) Dependence of the SPP propagation length on frequency for various ξ values. The green dashed lines in both plots are the dispersion curves of SPPs at a dielectric–metal interface.
q ε2m 2εc εm 1 2χ 2 1 1∕χ 2 q 2 2 2 εc 1 χ − 1 1∕χ 2 0:
(10)
ωc
ω2P − γ2 ε∞ − εm ∕ε0
(11)
Figure 3 shows plots of the cutoff chirality value ξc versus the cutoff frequency ωc for Au and Al metal layers and different values of the chiral index nc . It is seen that a larger chiral value leads to a higher cutoff frequency. For typical ξ values of inorganic and organic chiral materials (10−7 − 10−5 Ω−1 ), the cutoff frequency falls in the infrared wavelength range. Alternatively, the plot in Fig. 3 shows that at a fixed wavelength/frequency, the SP mode is bounded for low values of ξ and becomes cut off when ξ approaches ξc . Also, as seen in Eq. (11), the cutoff frequency depends on the plasma frequency ωp and damping constant γ of the metal. For Al (ωp 2.32 × 1016 rad∕s, γ 9.09 × 1014 rad∕s [10]), the plot in Fig. 3 indicates that there is no cutoff frequency for a chiral medium with ξ below 10−5 Ω−1 . Figure 4(a) shows the dependence of the real part of the effective index of the chiral–Au SP mode on the chirality value ξ at various fixed wavelengths. For small ξ values near zero, the chiral SP mode behaves like SPPs
Au 10
10
-4
Al
-5
nc= 1.4 nc = 1.5 nc = 1.6
-6
0
100
200
300
400
500
600
700
Fig. 3. Relationship between the cutoff chirality parameter ξc and the cutoff frequency ωc for Au and Al metal layers and different values of the chiral index nc .
at the interface between the metal and a dielectric with index nc . As ξ gets larger and approaches the cutoff value, the effective index of the chiral SP mode approaches the value n for ξ > 0 (n− for ξ < 0). In Fig. 4(b) we plot the propagation loss of the chiral SP mode as a function of ξ for various wavelengths. It is seen that the propagation loss of chiral SPPs is lower than that of SPPs at a nonchiral–metal interface (ξ 0) and exhibits a steep drop-off near the cutoff value ξc . The strong dependence of the propagation loss of chiral SPPs on the parameter ξ near cutoff can be exploited for chiral sensing applications. Specifically, by simply measuring the transmitted power of the SP mode along the surface of a metal film coated with a layer of chiral material, one can determine the chiral strength of the material. This technique is much simpler than current methods of chiral detection based on circular dichroism or optical rotations which require measurements of the absorption of both left- and right-circularly polarized waves or the
1∕2 :
-3
Cut-off frequency, ωc/2π (THz)
For a given chiral parameter ξ, we solve the above equation for the value of εm at cutoff which allows us to determine the cutoff frequency from the real part of the Drude permittivity
10
10
(a)
Real part of effective index
dielectric slab waveguides [4] and chiral parallel-plate metal waveguides [2,3], where in general two nondegenerate left- and right-rotating modes are supported. The difference between the dispersion curves for chiral SPPs and conventional SPPs at a dielectric–metal interface is more evident if we examine the real part of the effective index, N eff Refβg∕k0 , at low frequencies. The inset in Fig. 2(a) shows the plot of N eff versus the normalized frequency for different values of ξ. It is seen that the effective index of the chiral SPPs begins to deviate from the value of conventional SPPs at low frequencies and approaches the value n shown by the black dashed line. As discussed below, this lowfrequency limit of N eff is due to the cutoff condition of chiral SPPs. Figure 2(b) shows the frequency dependence of the propagation length, Lp 1∕2 Imfβg, of the chiral SPPs for various chirality values ξ. It is seen that the Lp curves follow that of SPPs at a dielectric–metal surface fairly closely until at low frequencies, where the propagation length increases rapidly and the curves begin to level off at different frequency limits for different chirality values. These low-frequency limits signify a cutoff condition of chiral SPP modes which is not observed for SPPs at a dielectric–metal interface. Assuming n > n− (i.e., ξ > 0) this cutoff occurs when β k , in which case kx1 0 and kx2 k2 − k2− 1∕2 . Substituting these values into the dispersion relation in Eq. (9) gives the equation for the cutoff condition
Cut-off chiral value, ξc (Ω-1)
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
1.75 1.7 1.65
λ = 0.6µm
1.6
n+
0.8µm
1.55 1.5 0
1.0µm 1.55µm 1
2
3
4
5 -1
Chirality value ξ (Ω )
(b)
Propagation loss (dB/mm)
2030
6 x 10
-4
3
10
2
10
1.55µm 1.0µm
1
10
0.8µm
λ = 0.6µm
0
10
0
1
2
3
4
5 -1
Chirality value ξ (Ω )
6
7 -4
x 10
Fig. 4. (a) Real part of the effective index and (b) propagation loss of chiral–Au SPPs as functions of the chirality parameter ξ at various fixed wavelengths.
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
degree of rotation of linearly polarized waves. Also, the much longer decay length of chiral SPPs compared to conventional dielectric–metal SPPs suggests that chirality can provide a potential route toward mitigating loss in plasmonic devices. In Figs. 5(a)–5(d) we plot the amplitude distributions of the E z , E x , and E y fields of chiral–Au SP modes at λ 600 nm for two chirality values, ξ 1 × 10−4 Ω−1 and ξ 6.68 × 10−4 Ω−1 , the latter being close to the cutoff value ξc at this wavelength. The presence of both E x and E y fields [Figs. 5(c) and 5(d)] indicates that the mode is a hybrid combination of both TM (E x ) and TE (Ey ) components. For small values of ξ, the mode is SPP-like with a dominant TM component, as seen in Fig. 5(c). As ξ approaches the cutoff value ξc , the amplitude of E y grows indicating that more power is transferred to the TE component [Fig. 5(d)]. Since the TE wave is not bound to the metal surface, less ohmic loss occurs in the metal resulting in a decrease of the propagation loss with increasing ξ, as seen in Fig. 4(b). Near the metal surface, electromagnetic boundary conditions pin the E y field close to zero while the E x field is strongly enhanced. Far from the metal surface, the two fields are equal to each other. For ξ ≈ ξc , the transverse field decay constant kx1 in the chiral medium approaches zero, indicating that the mode is no longer bounded, as can be seen in Fig. 5(d). Thus, the cutoff condition occurs when enough power is transferred from the TM polarization to the TE polarization so that the mode is no longer bounded to the metal surface. To examine the polarization characteristics of the SP mode, we plot in Figs. 5(e)–5(h) the ratio of the ξ < ξc
|Ez|
0.4
0.4
0
0.5
1
1.5
2
2.5
3
|Ex|, |Ey|
0 -0.5
0.5
|Ex / Ey|
0.5
1
1.5
2
2.5
3
|Ey| 0
0.5
1
1.5
2
0
0.5
1
1.5
2
2.5
3
2.5
3
0 -0.5
0
0.5
8 6 4
20
2 0 -0.5
0
0.5
π
1
1.5
2
2.5
3
0 -0.5
π
ξ>0
1
1.5
2
2.5
3
1.5
2
2.5
3
ξ>0
0 −π
-0.5
ξ 0, the phase angle difference between the two fields is identically −π∕2 inside the metal and π∕2 in the chiral medium, indicating that the wave is elliptically polarized with opposite senses of rotation in the metal and the chiral medium. More specifically, for ξ > 0, the field is right-elliptically polarized in the chiral medium and left-elliptically polarized in the metal. Since jE x ∕E y j > 1, the major axis of the polarization ellipse is oriented along the direction transverse to the metal surface, although far from the metal surface the field becomes circularly polarized since jE x ∕E y j ≈ 1. The opposite directions of polarization rotation in the metal and chiral medium are a consequence of the fact that the dielectric constants of the metal and chiral material have opposite signs which causes a π-phase change in the E x field at the interface. Changing the sign of the chirality parameter does not affect the magnitudes of the field components but reverses the senses of polarization rotation in the metal and chiral medium, as seen by the plots for ξ < 0 in Figs. 5(g) and 5(h). This polarization property of chiral SPPs is useful for enantiometric detection applications. For example, by measuring the direction of polarization rotation of the transmitted chiral SP mode, one can determine whether the chiral medium is “left-” or “right-handed,” while the mode attenuation can be used to determine the strength of the chirality. In summary, we analyzed the characteristics of SPPs at a chiral–metal interface and showed that the chiral SP mode is a hybrid TM–TE mode with the degree of hybridization depending on the chirality parameter ξ. The SP mode is supported only up to a cutoff value of ξc , at which point the TE component becomes sufficiently large that the mode is no longer bounded. We also showed that the propagation loss of chiral SPPs depends strongly on the chirality strength while the sense of polarization rotation is determined by the sign of the chiral parameter. These unique properties of SPPs at a chiral–metal surface can be exploited to develop a new and simple method for on-chip chiral sensing and enantiometric detection.
0.5
|Ex|
40
φx - φy
0
1
1
0 -0.5
ξc
0.2
0.2
0 -0.5
ξ
2031
2.5
3
−π
-0.5
ξ
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
Characteristics of surface plasmon polaritons at a chiral–metal interface Guangcan Mi* and Vien Van Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada *Corresponding author: [email protected] Received October 31, 2013; revised February 21, 2014; accepted February 23, 2014; posted February 24, 2014 (Doc. ID 200314); published March 26, 2014 The characteristics of surface plasmon polaritons at a chiral–metal interface are analyzed in detail. Compared to conventional surface plasmon waves at a dielectric–metal interface, it is shown that chiral surface plasmon waves have distinguishing features such as the presence of an s-wave at the metal surface, the existence of a cutoff frequency and chirality value, and the dependence of the propagation length on the chiral parameter. These properties of chiral surface plasmon waves can be exploited for on-chip chiral sensing and enantiometric detection applications. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (280.4788) Optical sensing and sensors. http://dx.doi.org/10.1364/OL.39.002028
Surface plasmon polaritons (SPPs) are electromagnetically coupled electron density oscillations that have been explored for various applications from sensors to subwavelength waveguiding devices. A common configuration of surface plasmon waveguide sensors consists of a thin metal film which is functionalized with a biorecognition layer [1]. The evanescent field of the SP mode is used to sense the refractive index change near the metal surface caused by a binding reaction with the target analyte. Besides refractive index sensing, another important method of optical sensing is based on the detection of the optical activity of the analyte. Biochemical molecules typically exhibit chirality (enantiomerism) which strongly influences their functionalities. The detection of chirality is thus important for enantiomer purity test, concentration measurement of chiral substance, and identification of enantiometry of proteins. Recently, it has been suggested that circular dichroism from chiral molecules can be enhanced by surface plasmon resonances if the chiral molecules are attached to metallic nanoparticles [2,3]. However, the use of surface plasmon waves at an interface between a metal and a chiral medium has not been investigated for chirality detection. The propagation of electromagnetic waves in chiral waveguides has been investigated for a chiral slab bounded by one or two perfectly conducting parallel plates [4,5] and for dielectric slab waveguides in which either the cladding or core or both is a chiral medium [6,7]. In these works it was shown that the modes in the waveguides are no longer purely TE or TM but are hybrid modes due to the coupling between the electric and magnetic fields in the chiral medium. Typically, two modes exist which correspond predominantly to the left-circularly polarized (LCP) and right-circularly polarized (RCP) waves. Few, however, have studied the interaction of SPPs with a chiral medium. The work in [8] examines SP waves that are supported at the interface of a metal and a chiral sculptured metamaterial composed of spiral nanorods, where it was suggested that a hybrid SP wave also exists at such an interface. It is the objective of this Letter to examine in detail the characteristics of SPPs at a chiral–metal surface 0146-9592/14/072028-04$15.00/0
for the purpose of chiral sensing applications. Specifically we find that the decay length of the SPPs depends on the chirality strength and prove that there exists a cutoff condition in terms of either the frequency or chiral parameter for which the SP wave is no longer supported. In addition, we show that the SP waves are degenerate and elliptically polarized, with the sense of rotation depending on the sign of the chiral parameter. Finally, we suggest that these unique properties of SPPs at a chiral–metal surface can be exploited to develop new techniques for chiral sensing and enantiometric detection. We consider an interface between a metal and a chiral medium characterized by electric permittivity εc , vacuum magnetic permeability μ0 , and chiral parameter ξ, as shown in Fig. 1. The constitutive relations for the electric and magnetic fields in the chiral medium are given by D εE − jξB and H −jξE B∕μ [9]. These two sets of constitutive relations are equivalent, with the chiral parameters ξ and κ related by κ μcξ. Due to the coupling between the E and H fields, all three components of the electric and magnetic fields exist in both metal and chiral media. In the chiral medium the fields satisfy the coupled wave equations [4] ∇
2
E H
1 2χ
2
n2c k20
E H
−2jξk20 ∕η20 H 2jξ1 χ 2 n2c k20 E
0; (1)
p where nc εc ∕ε0 is the refractive index of the chiral p material, η0 ε0 ∕μ0 is the admittance of free space, and χ ξ∕ηc is the normalized chirality admittance (ηc nc η0 ). In a bulk chiral medium, light propagates as RCP and LCP waves with propagation constants k and k− , prespectively, where k n k0 and n nc χ 1 χ 2 . For wave propagation along the chiral–metal surface, assuming propagating fields with z dependence E, H ∼ e−jβz , we can express the solutions for the E z and H z field components in the chiral medium as the superposition of two modes U 1 and U 2 [4] © 2014 Optical Society of America
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
x (εc, µ0, ξ) z
Fig. 1. Interface between a metal and a chiral medium.
E z U 1 x U 2 xe−jβz ;
(2)
q H z jηc 1 χ 2 U 1 x − U 2 xe−jβz ;
(3)
where U 1 and U 2 are solutions to d2 U 12 k2− − β2 U 12 0: dx2
(4)
−kx1 x Writing the solutions to Eq. (4) as U 1 and C1e q −kx2 x 2 2 U 2 C2 e , where kx1x2 β − k− , the solutions
for the other field components in the chiral medium can be obtained from Eq. (1) as (omitting e−jβz for clarity) E x −jβ∕kx1 C 1 e−kx1 x − jβ∕kx2 C 2 e−kx2 x ; q β β H x ηc 1 χ 2 C 1 e−kx1 x − C 2 e−kx2 x ; kx1 kx2 E y −k ∕kx1 C 1 e−kx1 x k− ∕kx2 C 2 e−kx2 x ; q k k H y −jηc 1 χ 2 C 1 e−kx1 x − C 2 e−kx2 x : kx1 kx2
(5) 1
(6)
(7)
(a)
(8)
k± n+
0.8 1.8
0.6 1.7
Neff
(εm, µ0)
waves (p-waves). In the limit k k− , the above relation reduces to the familiar dispersion of SPPs at a dielectric– metal interface, which supports only the TM polarization. In Fig. 2(a) we plot the dispersion relation for SPPs at the interface between gold and a chiral medium with parameters nc 1.5, ξ 10−5 Ω−1 . We assume the permittivity of gold is described by the Drude model with plasma frequency ωp 1.30 × 1016 rad∕s and damping constant γ 2.80 × 1013 rad∕s [10]. Also shown in the plot are the light lines for k as well as the dispersion curve (green-dashed line) of SPPs at an interface between gold and a dielectric medium with index nc (i.e., ξ 0). We observe that the dispersion curve of the chiral SPPs follows closely that of SPPs at a metal–dielectric interface, with both curves asymptotically approaching p the surface plasmon resonance fre quency ωsp ωp ∕ 1 n2c at large values of β. The dispersion curve of chiral SPPs lies completely to the right of the light lines for k and k− , indicating that the chiral SP waves travel more slowly than both circularly polarized waves in the bulk chiral medium. Another distinguishing feature of the plasmonic chiral–metal surface is that it supports only one degenerate SPP mode which, as will be shown later, is either left- or rightelliptically polarized. This is in contrast to chiral
Normalized frequency ω ωsp
0
2029
0.4
0.2
0 0
1.6 1.5 0
2
0.2
4
0.4
0.6
ω ωsp
6
8
10
12
Normalized propagation constant β ksp
q ωεm p kx2 k− kxm ωμ0 ηc 1 χ 2 kx1 k kxm ηc 1 χ 2 q ωεm 2 p kx1 k kxm ωμ0 ηc 1 χ kx2 k− kxm ηc 1 χ 2 0:
(9)
In the above expression, the first and third terms in the brackets are dispersion relations associated with the TE waves (s-waves) at an interface between the metal and a dielectric with refractive index n . The second and fourth terms are the dispersion relations for the TM
1
(b)
Normalized frequency ω ωsp
In the metal the fields are given by E z A1 ekxm x ; H z B1 ekxm x , E x jβ∕kxm E z ; H x jβ∕kxm H z , and Ey −jωμ0 ∕kxm H z ; H y −jωεm ∕kxm E z : In the p above, kxm β2 − ω2 μ0 εm is the transverse SP decay constant in the metal. By matching the tangential fields Ez , H z , E y , and H y at the interface x 0, we obtain the dispersion relation
0.8
0.6
0.4
0.2
0 -2 10
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Propagation length Lp (µm)
Fig. 2. (a) Dispersion curve (black line) of SPPs at an interface between gold and a chiral medium with parameters nc 1.5 and ξ 10−5 Ω−1 . The frequency and propagation constant are normalized by the SP resonance frequency ωsp and ksp ωsp ∕c, respectively. The inset shows the effective index N eff versus frequency for different ξ values. (b) Dependence of the SPP propagation length on frequency for various ξ values. The green dashed lines in both plots are the dispersion curves of SPPs at a dielectric–metal interface.
q ε2m 2εc εm 1 2χ 2 1 1∕χ 2 q 2 2 2 εc 1 χ − 1 1∕χ 2 0:
(10)
ωc
ω2P − γ2 ε∞ − εm ∕ε0
(11)
Figure 3 shows plots of the cutoff chirality value ξc versus the cutoff frequency ωc for Au and Al metal layers and different values of the chiral index nc . It is seen that a larger chiral value leads to a higher cutoff frequency. For typical ξ values of inorganic and organic chiral materials (10−7 − 10−5 Ω−1 ), the cutoff frequency falls in the infrared wavelength range. Alternatively, the plot in Fig. 3 shows that at a fixed wavelength/frequency, the SP mode is bounded for low values of ξ and becomes cut off when ξ approaches ξc . Also, as seen in Eq. (11), the cutoff frequency depends on the plasma frequency ωp and damping constant γ of the metal. For Al (ωp 2.32 × 1016 rad∕s, γ 9.09 × 1014 rad∕s [10]), the plot in Fig. 3 indicates that there is no cutoff frequency for a chiral medium with ξ below 10−5 Ω−1 . Figure 4(a) shows the dependence of the real part of the effective index of the chiral–Au SP mode on the chirality value ξ at various fixed wavelengths. For small ξ values near zero, the chiral SP mode behaves like SPPs
Au 10
10
-4
Al
-5
nc= 1.4 nc = 1.5 nc = 1.6
-6
0
100
200
300
400
500
600
700
Fig. 3. Relationship between the cutoff chirality parameter ξc and the cutoff frequency ωc for Au and Al metal layers and different values of the chiral index nc .
at the interface between the metal and a dielectric with index nc . As ξ gets larger and approaches the cutoff value, the effective index of the chiral SP mode approaches the value n for ξ > 0 (n− for ξ < 0). In Fig. 4(b) we plot the propagation loss of the chiral SP mode as a function of ξ for various wavelengths. It is seen that the propagation loss of chiral SPPs is lower than that of SPPs at a nonchiral–metal interface (ξ 0) and exhibits a steep drop-off near the cutoff value ξc . The strong dependence of the propagation loss of chiral SPPs on the parameter ξ near cutoff can be exploited for chiral sensing applications. Specifically, by simply measuring the transmitted power of the SP mode along the surface of a metal film coated with a layer of chiral material, one can determine the chiral strength of the material. This technique is much simpler than current methods of chiral detection based on circular dichroism or optical rotations which require measurements of the absorption of both left- and right-circularly polarized waves or the
1∕2 :
-3
Cut-off frequency, ωc/2π (THz)
For a given chiral parameter ξ, we solve the above equation for the value of εm at cutoff which allows us to determine the cutoff frequency from the real part of the Drude permittivity
10
10
(a)
Real part of effective index
dielectric slab waveguides [4] and chiral parallel-plate metal waveguides [2,3], where in general two nondegenerate left- and right-rotating modes are supported. The difference between the dispersion curves for chiral SPPs and conventional SPPs at a dielectric–metal interface is more evident if we examine the real part of the effective index, N eff Refβg∕k0 , at low frequencies. The inset in Fig. 2(a) shows the plot of N eff versus the normalized frequency for different values of ξ. It is seen that the effective index of the chiral SPPs begins to deviate from the value of conventional SPPs at low frequencies and approaches the value n shown by the black dashed line. As discussed below, this lowfrequency limit of N eff is due to the cutoff condition of chiral SPPs. Figure 2(b) shows the frequency dependence of the propagation length, Lp 1∕2 Imfβg, of the chiral SPPs for various chirality values ξ. It is seen that the Lp curves follow that of SPPs at a dielectric–metal surface fairly closely until at low frequencies, where the propagation length increases rapidly and the curves begin to level off at different frequency limits for different chirality values. These low-frequency limits signify a cutoff condition of chiral SPP modes which is not observed for SPPs at a dielectric–metal interface. Assuming n > n− (i.e., ξ > 0) this cutoff occurs when β k , in which case kx1 0 and kx2 k2 − k2− 1∕2 . Substituting these values into the dispersion relation in Eq. (9) gives the equation for the cutoff condition
Cut-off chiral value, ξc (Ω-1)
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
1.75 1.7 1.65
λ = 0.6µm
1.6
n+
0.8µm
1.55 1.5 0
1.0µm 1.55µm 1
2
3
4
5 -1
Chirality value ξ (Ω )
(b)
Propagation loss (dB/mm)
2030
6 x 10
-4
3
10
2
10
1.55µm 1.0µm
1
10
0.8µm
λ = 0.6µm
0
10
0
1
2
3
4
5 -1
Chirality value ξ (Ω )
6
7 -4
x 10
Fig. 4. (a) Real part of the effective index and (b) propagation loss of chiral–Au SPPs as functions of the chirality parameter ξ at various fixed wavelengths.
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
degree of rotation of linearly polarized waves. Also, the much longer decay length of chiral SPPs compared to conventional dielectric–metal SPPs suggests that chirality can provide a potential route toward mitigating loss in plasmonic devices. In Figs. 5(a)–5(d) we plot the amplitude distributions of the E z , E x , and E y fields of chiral–Au SP modes at λ 600 nm for two chirality values, ξ 1 × 10−4 Ω−1 and ξ 6.68 × 10−4 Ω−1 , the latter being close to the cutoff value ξc at this wavelength. The presence of both E x and E y fields [Figs. 5(c) and 5(d)] indicates that the mode is a hybrid combination of both TM (E x ) and TE (Ey ) components. For small values of ξ, the mode is SPP-like with a dominant TM component, as seen in Fig. 5(c). As ξ approaches the cutoff value ξc , the amplitude of E y grows indicating that more power is transferred to the TE component [Fig. 5(d)]. Since the TE wave is not bound to the metal surface, less ohmic loss occurs in the metal resulting in a decrease of the propagation loss with increasing ξ, as seen in Fig. 4(b). Near the metal surface, electromagnetic boundary conditions pin the E y field close to zero while the E x field is strongly enhanced. Far from the metal surface, the two fields are equal to each other. For ξ ≈ ξc , the transverse field decay constant kx1 in the chiral medium approaches zero, indicating that the mode is no longer bounded, as can be seen in Fig. 5(d). Thus, the cutoff condition occurs when enough power is transferred from the TM polarization to the TE polarization so that the mode is no longer bounded to the metal surface. To examine the polarization characteristics of the SP mode, we plot in Figs. 5(e)–5(h) the ratio of the ξ < ξc
|Ez|
0.4
0.4
0
0.5
1
1.5
2
2.5
3
|Ex|, |Ey|
0 -0.5
0.5
|Ex / Ey|
0.5
1
1.5
2
2.5
3
|Ey| 0
0.5
1
1.5
2
0
0.5
1
1.5
2
2.5
3
2.5
3
0 -0.5
0
0.5
8 6 4
20
2 0 -0.5
0
0.5
π
1
1.5
2
2.5
3
0 -0.5
π
ξ>0
1
1.5
2
2.5
3
1.5
2
2.5
3
ξ>0
0 −π
-0.5
ξ 0, the phase angle difference between the two fields is identically −π∕2 inside the metal and π∕2 in the chiral medium, indicating that the wave is elliptically polarized with opposite senses of rotation in the metal and the chiral medium. More specifically, for ξ > 0, the field is right-elliptically polarized in the chiral medium and left-elliptically polarized in the metal. Since jE x ∕E y j > 1, the major axis of the polarization ellipse is oriented along the direction transverse to the metal surface, although far from the metal surface the field becomes circularly polarized since jE x ∕E y j ≈ 1. The opposite directions of polarization rotation in the metal and chiral medium are a consequence of the fact that the dielectric constants of the metal and chiral material have opposite signs which causes a π-phase change in the E x field at the interface. Changing the sign of the chirality parameter does not affect the magnitudes of the field components but reverses the senses of polarization rotation in the metal and chiral medium, as seen by the plots for ξ < 0 in Figs. 5(g) and 5(h). This polarization property of chiral SPPs is useful for enantiometric detection applications. For example, by measuring the direction of polarization rotation of the transmitted chiral SP mode, one can determine whether the chiral medium is “left-” or “right-handed,” while the mode attenuation can be used to determine the strength of the chirality. In summary, we analyzed the characteristics of SPPs at a chiral–metal interface and showed that the chiral SP mode is a hybrid TM–TE mode with the degree of hybridization depending on the chirality parameter ξ. The SP mode is supported only up to a cutoff value of ξc , at which point the TE component becomes sufficiently large that the mode is no longer bounded. We also showed that the propagation loss of chiral SPPs depends strongly on the chirality strength while the sense of polarization rotation is determined by the sign of the chiral parameter. These unique properties of SPPs at a chiral–metal surface can be exploited to develop a new and simple method for on-chip chiral sensing and enantiometric detection.
0.5
|Ex|
40
φx - φy
0
1
1
0 -0.5
ξc
0.2
0.2
0 -0.5
ξ
2031
2.5
3
−π
-0.5
ξ
