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Sub-wavelength microscopy with surface plasmons

Xiaodong Zeng, M. Al-Amri, M. Suhail Zubairy

Xiaodong Zeng, M. Al-Amri, M. Suhail Zubairy, "Sub-wavelength microscopy with surface plasmons," Proc. SPIE 10548, Steep Dispersion Engineering and Opto-Atomic Precision Metrology XI, 105480N (22 February 2018); doi: 10.1117/12.2299172 Event: SPIE OPTO, 2018, San Francisco, California, United States Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 2/26/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use

Invited Paper

Sub-wavelength microscopy with surface plasmons Xiaodong Zenga , M. Al-Amrib , and M. Suhail Zubairya † a b

Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA; The National Center for Applied Physics, KACST, P.O Box 6068, Riyadh 11442, Saudi Arabia ABSTRACT

An attractive feature of surface plasmons (SPs) is the sub-wavelength characteristics, especially the SPs in two dimensional Dirac systems. In mid-infrared region, the wave vectors of graphene plasmons (GPs) can be two orders larger than that in vacuum, which have potential applications in optical imaging. Here, we propose a scheme that combining the GPs and structured illumination microscopy to realize a nanometer-scale microscopy. This scheme also takes advantage of the other two exciting properties of GPs, i.e., tunability and low loss. The microscopy works in the linear regime and can be used in bioimaging. Keywords: Graphene plasmons, sub-wavelength microscopy, structure illumination microscopy

1. INTRODUCTION In a traditional far field fluorescence microscopy, the image of a single atom or molecule is a spot, called Airy spot. The half radius of the spot can be expressed as λ/n sin θ. Here λ is the fluorescence wavelength and n sin θ is the numerical aperture of the objective lens. This means that it is hard to obtain image information with a resolution better than half the wavelength, which is called the classical Abbe diffraction limit. However, as the frontiers of science and technology approach the nanoscale, defeating the diffraction limit is prerequisite, especially in biological area. The past two decades have seen several methods that are capable of resolving structure beyond the diffraction limit.1–7 However, all of these schemes suffer from certain shortcomings, i.e., limited resolution ability, low speed, strong laser intensity and so on. For example photo-activated localization microscopy (PALM) and stochastic optical reconstruction microscopy (STORM) methods require the generation of a large amount of raw images and have low imaging speed. Similarly, stimulated-emission depletion (STED) and saturated structured illumination microscopy (SIM) require strong driving laser field to realize high order nonlinearity. Surface plasmons (SPs) are electromagnetic excitations associated with charge density waves on the surface of a conduction or semi-conduction object. An attractive character of SPs is the possibility of obtaining subwavelength resolution, which means high ability to localize light and has lots of important applications, such as surface enhanced Raman scattering, biosensor and sub-wavelength lithography. However, traditional SPs on noble metals have relatively small wave number and large loss, which limited their usages. Recently, a new kind of SPs found on monolayer graphene, called graphene plasmons (GPs),8–14 have emerged as a hot topic due to its exciting properties. Firstly, the wave vectors of GPs are huge and can be two orders larger than that in vacuum. Secondly, the plasmonic wavelength can be manipulated easily by tuning the doping level, such as by electric gate. At last, due to Pauli-blocking, doped graphene has a low absorption in the mid-infrared region. Here, we combine the GPs and linear SIM to propose a nanometer-scale microscopy scheme in which the GPs act as the illumination light. The field pattern constructed by GPS has an extremely small period. By tuning the Fermi level of the graphene, all the spatial frequencies of the sample can be obtained and an increase of more than one hundred times resolution becomes possible. In addition, due to the microscopy works in the linear regime, it has important application in bioimaging. The paper is organized as follows: In Sec. 2, we give our model and scheme. Our simulations and discussion are present in Sec. 3. The last part is the conclusion. †

[email protected] Steep Dispersion Engineering and Opto-Atomic Precision Metrology XI, edited by Selim M. Shahriar, Jacob Scheuer, Proc. of SPIE Vol. 10548, 105480N · © 2018 SPIE · CCC code: 0277-786X/18/$18 · doi: 10.1117/12.2299172 Proc. of SPIE Vol. 10548 105480N-1

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Objective Lens

Source

Drain

Sample

Graphene monolayer

Gratings Laser beam

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gate

Figure 1. The diagram of the microscopy scheme. The bottom gate can control the Fermi level of the monolayer graphene. GPs can be generated by the dielectric gratings.

2. SUB-WAVELENGTH MICROSCOPY WITH TUNABLE PLASMONS Our model is schematically shown in Fig. 1. The monolayer graphene is placed at z = 0 between two dielectric slabs with permittivities ε1 and ε2 , respectively. The imaging sample is located on slab ε2 and has a surrounding permittivity ε3 . The objective lens can collect the photons emitted by the sample for imaging. A laser beam illuminates the gratings to excite the GPs and consequently generates a periodic field pattern. In the high doping and long-wavelength limit, the in-plane conductivity of monolayer graphene can be described as σ(ωp ) = ie2 EF /π~2 (ω +iτ −1 ).8 Here, EF is the Fermi level and e is the electron charge. τ = µEF /evF2 is the relaxation time with µ to be the mobility of the graphene charge carriers and vF to be the Fermi velocity. √ Near the Dirac point, the magnitude of Fermi level can be expressed as EF = ~vF πn with n to be the carrier density. In addition, the carrier density depends on the voltage of the top gate as VGD = ne/Cg + EF /e. Here, VGD denotes the voltage difference between the bottom gate and the drain; Cg refers to the charge capability of the system, which depends on the structure size and the properties of the dielectric ε3 . Here, we choose Cg to be 0.025F/m2 .9 As shown in Fig. 1, we denote the interface between dielectric ε1 and ε2 as interface A, while the interface between dielectric ε2 and ε3 as interface B. The height of dielectric ε2 is d. Considering the continuity of the electric field on the interfaces, we assume the electric field parallel to the interface A as EA and the electric field component parallel to the interface B as EB . The components up- and down-prorogation in dielectric 2 are Eu and Ed , respectively. They have the relations

Here ki =

−Eu + Ed = EA ;

(1)

−Eu e−iβ2 d + Ed eiβ2 d = EB .

(2)

√

εi k0 (iq= 1, 2, 3) are the wave numbers in dielectric i, where k0 = ωp /c and c is the vacuum light velocity. βi = ki2 − kg2 are the wave number components perpendicular to the graphene. The magnetic √ √ √ field above interface A is HA1 = EA k1 ε1 ε0 / µ0 β1 , while the magnetic field below interface A is HA2 = √ √ √ (Eu + Ed )k2 ε2 ε0 / µ0 β2 . Here, ε0 and µ0 are the vacuum permittivity and permeability. The magnetic fields have the relation HA1 − HA2 = σEA , (3) which means

√ √ √ √ (Eu + Ed )k2 ε2 /β2 − EA k1 ε1 /β1 = −σEA µ0 / ε0 .

Similarly, the magnetic fields on interface B have the relation √ √ (Eu e−iβ2 d + Ed eiβ2 d )k2 ε2 /β2 = EB k3 ε3 /β3 .

(4)

(5)

Combining Eqs. (1-5), we can obtain p p 2iβ2 d 1 − α21 α23 e = 0,

Proc. of SPIE Vol. 10548 105480N-2 Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 2/26/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use

(6)

z φ଴

(a)

a

x b

(b) T.

2

,.'

1

(c)

0 2

. 0

-0.10

1

ll

-0.05

V 0.05

0.10

Figure 2. (a) Two perpendicular located dielectric gratings couple two TM-polarized laser beams to the GPs. The period of the gratings is a = 720nm, the width of the silicon rectangular is b = 36nm, and the relative permittivity of the silicon is 12. Additionally, the distance between the gratings is 6µm and the height of the silicon rectangular is 150nm. The COMSOL simulations of the field intensity on the sample for (b) VGD = 1.73 with φ0 = 180 and (c) 0.56V with φ0 = 00 . Dielectric 1, 2 and 3 have permittivities ε = 2.0.

Here p α21 =

ε1 β2 − ε2 β1 + σβ1 β2 /ε0 ωp ε1 β2 + ε2 β1 + σβ1 β2 /ε0 ωp

(7)

ε 3 β2 − ε 2 β3 . ε 3 β2 + ε 2 β3

(8)

and p α23 =

p The physical meaning of α21 is the reflection coefficient of a TM-polarized plane wave incident from dielectric ε2 p to ε1 with a monolayer graphene between them, while α23 is the reflection coefficient of a TM-polarized plane wave incident from dielectric ε2 to ε3 . If we have assumed ε1 = ε2 = ε3 , the plasmonic wave number can be expressed by the relation ε1 ωp Re(kg (ωp )) ≈ k0 . (9) 2α ωF

where α = e2 /4π~ε0 c ≈ 1/137 is the fine structure constant and ωF = EF /~. As a consequence, the plasmonic wave number can be two orders larger than that in vacuum. For a GP along the x direction, the electric field near the graphene is proportional to (kg ez + β3 ex )eikg x . Due to the huge value of the plasmonic wave number, the electric fields along x direction and z directions have almost the same amplitude but a phase difference π/2. Two counter-propagation GPs along the x have a z component proportional to eikg x ez + e−ikg x ez = 2 cos(kg x)ez and a x component proportional to eikg x ex − e−ikg x ex = 2 sin(kg x)ex . As a consequence, the field pattern constructed by the two counter-propagating plasmons along x direction has a constant intensity. However, due to the interference between incident wave and the plasmons, the total field always shows periodic pattern rather than a constant. For instance as shown in Fig. 2(a), we assume the gratings are located parallel to each other and homogeneous along y direction. A TM-polarized


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Figure 3. (a) The diagram of the spatial frequency. The blue and orange circles at the origin contribute to the conventional microscopy with imaging frequency ωp and ωcd respectively. The circles with the center at ±kg1/2 correspond to the linear response of the illumination pattern with period 2π/kg1/2 . The period of illumination plasmons can be tuned by the gate voltage. (b) The energy structure of the sample molecule.

wave incidents to the gratings with an incident angle φ0 . The electric field of the plane wave can be expressed as (cos φ0 ex + sin φ0 ez )eik3 sin φ0 x eiωp t with amplitude set to be 1. The plasmonic electric field distribution is E0 (cos(kg x + θ)ex + sin(kg x + θ)ez )eiωp t . Here, E0 is the amplitude multiplied by the relative phase difference between the incident wave and the plasmonic pattern. θ denotes the position of the pattern and always dependent on the arrival time difference of the plane wave to the two gratings. If we set φ0 = π/4, he total field intensity can be obtained as9 |E0 cos(kg x + θ) + cos φ0 eik3 sin φ0 x |2 + |E0 sin(kg x + θ) + sin φ0 eik3 sin φ0 x |2 2

= 1 + |E0 | + 2Re(E0 e

√ i 2k3 x/2

(10)

) cos(kg x + θ − π/4).

√

Due to that kg 2k3 /2. The above expression describes a standing wave pattern with an effective period of π/kg and a background 1 + |E0 |2 . And, more remarkable, the phase shift θ − π/4 in the above equation is dependent on φ0 . If we change the angle φ0 , we can change the phase shift in a large range. The means that we can obtain sinusoidal as well as cosine pattern, which plays indispensable role in the imaging. In Fig. 2, we give some COMSOL simulations of the field intensity and the results show that periodic field patterns with tiny periods can be realized. The pattern also can be shifted by controlling the incident angle φ0 . As we know, the spatial density of the sample atoms can be decomposed into its spatial Fourier components7, 8 ZZ F (x, y) = f (kx , ky )eikx x+iky y dkx dky . (11) As in incoherent fluorescence microscopy, the measured image M (x, y) can be described by a multiplication of the local excitation intensity I(x, y) by the local fluorescence concentration F (x, y), followed by a convolution with the point spread function (PSF) T (x, y) of the incoherent imaging system for the emitted field. ZZ 0 0 0 0 0 0 0 0 M (x, y) = (F (x , y )I(x , y ))T (x − x , y − y ) dx dy , (12) In the spatial frequency domain, the image reads m(kx , ky ) = t(kx , ky )φ(kx , ky ), where m(kx , ky ) and optical transfer function (OTF) t(kx , ky ) are the corresponding two-dimensional Fourier transformation of M (x, y) and T (x, y). Here φ(kx , ky ) is the spatial frequency spectrum. q In conventional fluorescence microscopy, only the Fourier components within the passband kx2 + ky2 = k|| 6 2N Ak0 = ν can be observed as the OTF is nonzero only in this region. As shown in Fig. 3(a), only the spatial


(a)

(b)

Figure 4. (a) The atom distribution. (b) The image simulation with EFmin = 0.2eV and ~ωp = 0.2eV . Here, Re(kg ) = 68.5k0 .

frequencies inside the circle with center at origin contribute to the image. The linear SIM uses a structured illumination pattern and utilizes the so-called ”Moire´ e effect” to couple some of the high spatial frequency information from outside of the circle into the circle to improve the resolution. If the illumination field intensity pattern is sinusoidal with period 2π/kg , the spatial frequency spectrum has the following form: φ(kx , ky ) = 2f (kx , ky ) + f (kx − kg , ky )ei∆x +f (kx + kg , ky )e−i∆x .

(13)

where ∆x is the shift of the pattern. Since the plasmon frequency ωp falls in the infrared region, a relatively small ν implies that we need to image the sample a large number of times, which results in a slow imaging process. The process can be expedited by using an additional illumination such as a laser with frequency ωL in the visible region. As shown in Fig. 3(b), the molecules in the sample are first excited from the ground state, |gi to the energy level |ei by using a laser of frequency ωL . The plasmonic pattern then excites the molecules to an additional level, |ci. Utilizing the spontaneous decay of the molecules from |ci, we image the sample with photons of frequency ωcd = ωc − ωd . Here, ωc and ωd are the respective frequencies of energy levels |ci and |di. As a consequence of the preceding discussion, the resulting passband in the spatial frequency is now bounded by (kx2 + ky2 )1/2 = 2kcd = 2ωcd /c = κ, which is the larger circle illustrated in Fig. 3(a). Additionally, we have proved that we can tune the kg by the gate voltage conveniently. Meanwhile, in order to excite the GPs with a high efficiency, the wave vector matching condition k1 sin φ0 + mΛ ≈ mΛ = ε1 ωp k0 /2αωF must be satisfied, which means that we only can tune the plasmonic pattern period with separated values. However, if 2π/a < κ, all the spatial frequency can be resolved by rotating the sample or the plasmonic structure. Since κ is much larger than k0 , high resolution can be realized by imaging the sample only a few times. Knowing all the Fourier information, the image of the sample can be reconstructed. In Fig. 4, we demonstrate the image simulations. Here, we assume N A = 1.8 We can see a 68.5 times resolution increase. In the preceding discussion, we did not consider the influence of the loss of the GPs on the results, which can destroy the regularity of the field pattern and subsequently errors if we still use Eq. (13) to solve the spatial frequencies. However, if the imaging size is limited, we can do Fourier expansion on the field intensity and only several Fourier components have values big enough to overcome the noise as shown in Fig. 5. As a consequence, we still can solve the spatial frequencies just by imaging the sample several more times. Due to the loss, the propagation distances of the GPs are limited, in order to image a sample with large size, we can use laser ωL to control the imaging area and in this area the field pattern is approximately periodical. After we image this area, we move the laser to another position and image the new area again. The whole sample can be imaged after scanning the sample by laser ωL . In addition, in order to avoid the rotation of the sample or the plasmonic structure, a circular grating can be used to excite the GPs, where we just need to modulate the polarization of the incident lasers to rotate the sample equivalently.


(a)

(b)

1.6 3

|f(kx)|

I(x)

1.2 0.8

1

0.4 0.0

2

-4

-2

0

x(a)

2

4

0

-20

-10

0

kx(Λ)

10

20

Figure 5. (a) The field intensity distribution with the loss. (b) The Fourier expansion components of the illumination field. Here, Λ = 2π/L and L is the total size of the field pattern.

3. CONCLUSIONS In conclusion, we have proposed a scheme for sub-wavelength microscopy by using the GPs. The gratings can couple the incident laser to the GPs and consequently generate field pattern with extremely small period. Additionally, due to the tunability of the graphene, we can obtain all the spatial frequencies of the sample by tuning the gate voltage. Due to our scheme works in the linear regime, nanometer scale resolution can be obtained under very weak light intensity, which is important in the imaging of the biological systems. ACKNOWLEDGMENTS This research is supported by a grant from King Abdulaziz City for Science and Technology (KACST).

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