WANG Xue-fei, LU Zhen-wu, WANG Tai-sheng , YU Wei-xing

(1.Changchun Institute of Optics,Fine Mechanics & Physics,Chinese Academy of Sciences,Changchun 130033，China；2.University of the Chinese Academy of Sciences,Beijing 100049,China;3.Xi′an Institute of Optics and Precision Mechanics,Chinese Academy of Sciences,Xi′an 710119,China)

1 Introduction

引 言

Surface Plasmon Polariton wave(SPP) is an electron density wave with collective oscillation generated by the interaction between photons and free electron on the metal surface. It can gather electromagnetic field energy in a small space[1], which has shown great potential in the field of nanophotonics and has become a hotspot in the research of nanophotonics[1]. SPP has been hailed as the most promising information carrier for integrated nano-photonic devices and has been widely used in many fields such as nano-lithography, new energy sources, highly sensitive biochemical sensing, solar cells and high-efficiency photonic components[2-5]; SPP has also an important application in the area of super-resolution imaging. Yang Jingzhong, use light soure with wavelength of 7 μm to illuminate a graphene nanocavity on metasurface to excite the SPP wave, and generate a standing wave with wavelength of 52 nm by interference. When this standing wave is applied to a microscope illumination source[6], the resolution of the microscope can reach 26 nm, which is nearly 100 times of the conventional fluorescent microscope.

表面等離激元激化波(Surface Plasmonic Polariton Wave,SPP)是光子和金屬表面的自由電子相互作用所產(chǎn)生的集體振蕩的電子疏密波，它能夠?qū)㈦姶艌龅哪芰烤奂谝粋€很小的空間范圍內(nèi)[1]，在納米光子學領(lǐng)域顯示出了巨大的應(yīng)用潛力，并成為當前納米光子學研究的熱點。SPP被譽為當前最有希望的納米集成光子器件的信息載體，并已被廣泛地應(yīng)用在納米光刻、新型能源、高靈敏生物化學傳感、太陽能電池以及高效光子元器件等諸多領(lǐng)域中[2-5]。SPP在超分辨成像中也有重要應(yīng)用，楊靖忠利用波長7 μm的光源入射到超表面的石墨烯納米腔結(jié)構(gòu)激發(fā)SPP波，干涉產(chǎn)生了波長為52 nm的駐波。將該駐波應(yīng)用到顯微鏡的照明光源時，顯微鏡分辨率可達到26 nm,比傳統(tǒng)熒光顯微鏡提高了近100倍左右[6]。

As a device to excite SPP, the metal grating has a simple structure and is easy to control, so it is widely used to generate SPP waves[7].

金屬光柵作為激發(fā)SPP的器件，結(jié)構(gòu)簡單且易于控制，被廣泛應(yīng)用于SPP波的產(chǎn)生[7]。

The SPP wave is an electromagnetic wave and is essentially the same as the spatial light wave thus possess characteristics of interference and diffraction. A number of nanodevices, such as super-lenses, have been developed by using the diffractive properties of SPP waves. In recent years, the research on the diffraction of SPP waves has been increasing. R. Ziaetal. studied the effects of the width of the metallic waveguide on the diffraction of the SPP wave at the waveguide end when they conducted Young′s double-slit experiment of SPP waves and found that the SPP wave is not diffracted at the waveguide end face if the waveguide width is smaller than a critic value[8]. Feng Liangetal. focused SPP wave on the metal surface by using its near-field diffractive property, and found that the electric field strength of the focusing point was increased by three times[9].

SPP波在本質(zhì)上與空間光相同，為電磁波，因此具有干涉及衍射特性。利用SPP波的衍射特性已經(jīng)制作出了很多納米器件，例如超透鏡等。近年來對SPP波衍射特性的研究不斷增多。R.Zia等人在做SPP波的楊氏雙縫實驗時，研究了金屬波導(dǎo)的寬度對SPP波在波導(dǎo)端部衍射現(xiàn)象的影響，并發(fā)現(xiàn)存在一個極限值，當波導(dǎo)寬度小于這個值時，SPP波在波導(dǎo)端面不發(fā)生衍射[8]。馮亮等人在金屬表面上利用SPP波的近場衍射現(xiàn)象對其進行聚焦，使聚焦點的電場強度增大了3倍之多[9]。

The near-field diffraction phenomenon of SPP waves on the metal surface is mainly affected by the structural parameters such as the period and filling factor of the diffraction grating. However, only few work have been reported in this area. In this paper, we use the metallic grating couple method to excite SPP waves, and studied the influence of diffraction grating on the near-field diffraction phenomenon of SPP wave. We found that of SPP wave has an obvious diffractive effect in near-field. The spatial light is transformed into SPP wave, and the spectral signal can be separated at the micro-nanoscale by optimizing the structural parameters of the diffraction grating.

SPP波在金屬表面的近場衍射現(xiàn)象主要受衍射光柵的周期、占空比等結(jié)構(gòu)參數(shù)的影響，然而鮮見關(guān)于這方面的公開報道。本文利用金屬光柵耦合方式激發(fā)SPP波，研究了衍射光柵對SPP波近場衍射現(xiàn)象的影響，并發(fā)現(xiàn)SPP波的近場衍射有分光作用。將空間光轉(zhuǎn)變成SPP波，通過優(yōu)化衍射光柵的結(jié)構(gòu)參數(shù)，可以實現(xiàn)在微納尺度上對光譜信號的分離。

2 Surface plasmon resonance theory and three characteristic lengths of SPP waves

表面等離子體共振理論及SPP波的3個特征長度

SPP wave is a kind of electromagnetic wave propagating along the metal/dielectric interface formed by the coupling of the surface charge group′s oscillation and the electromagnetic field on the metal/dielectric interface[10]. The SPP field intensity component is maximum at the metal/dielectric interface and decays exponentially on both sides of the interface. In the visible and near-infrared wave bands, the real part of the dielectric constant of most metals is negative, so the sign of dielectric constant of the metal and its neighboring dielectric medium is opposite, and only P-polarized light(TM) can excite SPPs. As shown in Fig.1, when light(including P-polarized light) is incident on the metallic grating, diffraction occurs on the grating surface, and different diffraction angles correspond to different diffraction orders. According to the grating equation, it can be seen that the component of the wave vector of them-th(m=±1,±2,±3,…，±n) order diffracted light in the direction parallel to the interface is

Fig.1 Cross section of rectangular metal gratings in one-dimension 圖1 一維矩形金屬光柵橫截面

SPP是金屬/介質(zhì)界面上由表面電荷的集體震蕩與電磁場耦合所形成的沿著金屬/介質(zhì)界面?zhèn)鞑サ囊环N電磁波[10]。SPP的場分量在金屬/介質(zhì)界面上取得最大值，在金屬兩側(cè)的介質(zhì)中場分量呈e指數(shù)衰減。在可見光及近紅外波段內(nèi)，絕大多數(shù)金屬介電常數(shù)的實部為負數(shù)，因此金屬的介電常數(shù)與其相鄰介質(zhì)的介電常數(shù)異號，只有P偏振光(TM)才能激發(fā)出SPP。如圖1所示，當光波(含P偏振光)入射到金屬光柵時，在光柵表面將發(fā)生衍射現(xiàn)象，不同的衍射角度對應(yīng)于不同的衍射級次。根據(jù)光柵方程可知第m(m=±1,±2,±3,…，±n)級衍射光的波矢在平行于界面方向上的分量如式(1)。

(1)

Wherek0is the light wave vector in free space,θis the incident angle of the light wave,ε2is the dielectric constant of the medium, andΛis the grating period. It can be seen from equation (1) that the wave vector of the diffracted light can be increased due to the diffraction of the grating so that the wave vector of them-th order diffracted light parallel to the interface can be equal to the wave vector of the SPP wave at the interface, that is

式中,k0是自由空間光波波矢，θ為光波的入射角，ε2為介質(zhì)的介電常數(shù)，Λ為光柵周期。從式(1)可知，光柵的衍射作用可以使衍射光的波矢得到增大，從而可以使平行于界面的第m級衍射光波矢分量與界面上SPP波的波矢相等，即有式(2)。

(2)

As can be seen from the formula (2) ，the surface plasmon wave wavelength is as follows：

由公式(2)可知，表面等離激元波的波長為：

(3)

where，λ0is incident wavelength.

式中，λ0為入射光波長。

SPP波有3個特征長度，分析如下。

Three characteristic lengths of SPP wave are analyzed as follows.

(4)

(5)

(6)

Where,k0is the incident light wave vector[12]. From equation (4) to (6), the distance of light in 550-700 nm waveband propagating on the silver surface is in the range of 12.5-60 μm, the penetration depth of SPP in air and silver is in the range of 279-500 nm and 22.7-29.5 nm, respectively.

式中，k0為入射光的波矢量[12]。由式(4)～(6)可得，550～700 nm波長的光波在金屬銀表面的傳播距離為12.5～60 μm時，SPP在空氣和銀中的穿透深度分別為279～500 nm和22.7～29.5 nm。

3 Determination of optimal structure of metallic grating coupler

最優(yōu)激發(fā)金屬光柵結(jié)構(gòu)的確定

Fig.2 shows the model for studying the near-field diffraction of SPP waves.The left metal grating is used to as grating coupler to excite the SPP wave, the right metal grating serves as the diffraction grating. When light is incident on the metal grating structure, an SPP wave is generated on the metal surface and propagates to both left and right along the metal surface. Near-field diffraction occurs when a diffraction grating is encountered.

圖2是研究SPP波超表面近場衍射的模型。左側(cè)是激發(fā)SPP波的金屬光柵結(jié)構(gòu)，右側(cè)是衍射光柵。當光入射到金屬光柵結(jié)構(gòu)上時，會在金屬表面產(chǎn)生SPP波，產(chǎn)生的SPP波將沿著金屬表面向右傳播，遇到衍射光柵時便會發(fā)生近場衍射現(xiàn)象。

In this paper, the near-field diffraction of SPP waves excited by incident light with wavelength in the range of 550-700 nm and the central wavelength of 625 nm is studied. The coupling efficiency of incident light to SPP wave is mainly related to the structure parameters of metal grating coupler such as cycle, filling factor and modulation depth. The material of the metal grating coupler is set as silver, and it can be seen from equation (2) that for the incident light with a wavelength of 625 nm, the coupling efficiency of the incident light is the highest when the grating periodΛis 607 nm. The grating filling factorDis set to 0.5 and the grating depthhis 200 nm. By emplaying the rigorous coupled wave theory, the reflectivity of the incident light against the wavelength ranging from 550-700 nm, and the incident angle ranging from 10-15° was calculated by using the SPR angle scanning method[13]. Fig.3(a) shows the three-dimensional map of the reflectivity and against the wavelength and the angle of incidence. Fig.3(b) illustrates the relationship of the reflectivity and wavelength for different incident angles.

本文主要研究中心波長為625 nm，波段為550～700 nm的入射光激發(fā)的SPP波近場衍射現(xiàn)象。而入射光耦合激發(fā)SPP波的效率主要與金屬光柵的面型參數(shù)如周期、占空比及調(diào)制深度等有關(guān)。設(shè)定耦合金屬光柵的材料為銀，通過式(2)可知，對于波長為625 nm的入射光，光柵周期Λ為607 nm時，入射光的耦合率最高。光柵占空比D設(shè)為0.5，光柵厚度h設(shè)為200 nm。根據(jù)嚴格耦合波理論，利用SPR角度掃描方法[13]，計算入射光波長為550～700 nm、入射角在10°～ 15°范圍內(nèi)入射光的反射率，得到反射率與波長及入射角度的三維關(guān)系圖，如圖3(a)所示。圖3(b)為反射率與波長及入射角的關(guān)系曲線圖。

It can be seen from Fig.3(a) that for the above mentioned metal grating structure, the reflectivity is lower for 625 nm wavelength when incident angle falls on the range of 12.5°-15°. It can be further deduced from Fig.3(b) that when the incident angle is about 14°, the reflectivity is the mimimum, that means the coupling efficiency is the highest, about 35%. At this incident angle, the coupling ratio is 25% and 10% for 550 nm and 700 nm respectively.

由圖3(a)可以看出，在上述金屬光柵結(jié)構(gòu)下，波長為625 nm的光以12.5°～15°入射時，反射率較低。由圖3(b)進一步可得，當入射角約等于14°時，反射率最小，即耦合成SPP波的效率最高，大約在35%左右。在此入射角下，550 nm波長的入射光的耦合率為25%，700 nm波長的入射光的耦合率在10%左右。

4 Diffraction results and analysis

衍射結(jié)果與分析

When the period of the metal silver grating is 607 nm, the wavelength of SPP waves excited by the incident light with a wavelength ranging from 550-700 nm is in the range of 525 to 683 nm. As shown in Fig.2, the metal grating coupler has a period of 607 nm, filling factor of 0.5 and a groove depth of 200 nm. The right diffractive metal grating located on the metal surface has a height of 600 nm and a thickness of 200 nm in theXdirection. When studying the diffractive behavior of SPP waves through a metal grating, two cases are considered:one is the case where the grating height and filling factor keep no change but the grating period changes; the other case is that the grating height and period keep no change but the filling factor changes. At the same time, the excitation wavelength and the incident angle are fixed at 625 nm and 14°.

當金屬銀的光柵周期為607 nm時，波長為550～700nm的入射光激發(fā)出的SPP波波長在525～683 nm之間。設(shè)定圖2所示的近場衍射模型中激發(fā)SPP波的金屬光柵周期為607 nm，占空比為0.5，凹槽深度為200 nm。右方衍射金屬光柵位于金屬表面上，高度為600 nm，在X方向上的厚度為200 nm。當研究SPP波經(jīng)過其的衍射行為時，對以下兩種情況分別研究：一種是光柵高度和占空比不變，而光柵周期發(fā)生變化時的情況；另一種是光柵高度和周期不變，占空比發(fā)生改變的情況。同時激發(fā)光波長和入射角度固定為625 nm和14°。

4.1 Diffraction situation by fixed diffraction grating filling factor but different period

衍射光柵占空比固定,周期不同時的衍射情況

Figs.4(a)-4(e) show the near-field diffraction of the SPP wave excited by the incident light with the wavelength of 625 nm when the filling factor of the diffraction grating is constant(0.5) and the period is different. In the figures,XYplane is the metal film surface, and the diffraction grating is located on theY-axis.

圖4(a)～4(e)是波長為625 nm的入射光所激發(fā)的SPP波在衍射光柵占空比一定(0.5)而周期不同時的近場衍射現(xiàn)象。圖中XY平面為金屬薄膜表面，衍射光柵位于Y軸上。

Fig.4 Diffraction phenomenon of SPP wave of incidence light with wavelength of 625 nm when diffraction grating with different period but the same filling factor(It should be noted that the white dashed line represents the location of the diffraction grating) 圖4 波長λ=625 nm的入射光激發(fā)的SPP波在周期不同，占空比一定的條件下的衍射現(xiàn)象(圖中白色斷線表示金屬衍射光柵的位置)

It can be seen that there is only the 0th diffraction order at this time. The transmitted light intensity is quite weak, below 0.2, and the transmission distance is only about 2.3 μm after passing through the diffraction grating; Fig.4(b) is the case when the period of the diffraction grating isλspp. In this case, the diffraction phenomenon can be observed and one can see that besides the 0th order, ±1 storders are also appear. The light intensity at this time also increases significantly with the maximum reaches to about 1, while the transmission distance also increases to about 4 μm; Figs.4(c)-(e) depict the diffraction of the SPP wave when the diffraction grating period is 1.5λspp, 4λsppand 6λspp, respectively. It can be seen that the diffraction order increases correspondingly as the period of the diffraction grating increases. When the diffraction grating period is 1.5 times of the SPP wavelength, the highest order of the diffraction is ±2nd and the distinction of diffraction fringes is relatively clear. When the grating period is 4-6 times of the SPP wavelength, the distribution of diffraction fringes becomes more disordered due to even higher diffraction orders appear.

圖4(a)為當衍射光柵周期為0.5λspp時SPP波的衍射情況。可以看到此時只有零級衍射，且透射光強較弱，在0.2 μm以下，入射光透過衍射光柵后傳播距離在2.3 μm左右；圖4(b)為衍射光柵周期為λspp的情形，此時已有明顯的衍射現(xiàn)象，可以看到0級和±1級衍射，并且此時光強有明顯的增強，最大達到了1左右，同時透過衍射光柵后SPP光的傳播距離在4 μm左右；圖4(c)～4(e)為衍射光柵周期分別為1.5λspp、4λspp和6λspp時SPP波的衍射情形?？梢钥吹?，隨著衍射光柵的周期不斷增大，衍射級次相應(yīng)地增多。當衍射光柵周期為SPP波長的1.5倍時，最高衍射級次為±2級，并且衍射條紋的分布比較清晰；而當光柵周期為SPP波長的4～6倍時，由于更高衍射級次的出現(xiàn)使得衍射條紋的分布變得較為紊亂。

4.2 Diffraction behavior of diffraction grating with fixed period but different filling factor

衍射光柵周期固定,占空比不同時的衍射情況

Figs.5(a)-5(e) show the near-field diffraction of the SPP wave at the metaface with a constant period of the diffraction grating of 910 nm, but the filling factor changes. In the Fig.5, theXYplane is the metal film surface, and the diffraction grating is located on theY-axis. Fig.5(f) illustates the electric field intensity distribution corresponding to the Fig.5(a)-5(d) near field diffraction on the surface of the metal thin film atx=0.

Fig.5 Diffraction phenomenon of SPP wave for an incidence light with wavelength of 625 nm for diffraction grating with different duty ratios and the fixed grating period. (a)Duty radio is 0.1; (b)Duty radio is 0.3； (c)Duty radio is 0.5; (d)Duty radio is 0.7; (e)Duty radio is 0.9; (f)Electric field intensity distribution of the diffraction patterns along y axis under different duty radio at x=0 圖5 入射光波長λ=625 nm,衍射光柵周期為910 nm，占空比不同時的衍射現(xiàn)象，其中白色虛線表示金屬光柵的位置。(a)占空比為0.1；(b)占空比為0.3；(c)占空比為0.5；(d)占空比為0.7；(e)占空比為0.9；(f)衍射光柵占空比不同時，金屬薄膜表面上x=0直線上的近場衍射電場強度曲線圖

圖5(a)～5(e)為衍射光柵周期固定為910 nm不變，占空比變化時SPP波在超表面的近場衍射情形。圖中XY平面為金屬薄膜表面，且衍射光柵位于Y軸上。 圖5(f)為對應(yīng)于5(a)～5(d)金屬薄膜表面直線x=0上近場衍射的電場強度分布圖。

As can be seen from the graphs in Figs.5(a)-(e), the SPP wave finally converges on the metal surface after passing through the diffraction grating, and the diffraction order does not change substantially, but the intensity changes as the filling factor of the diffraction grating changes. When the filling factor of the grating changes from 0.1 to 0.7, the diameter of the converging light beam gradually becomes smaller. When the grating filling factor is 0.9, the light transmittance has become very low, so that the maximum electric field intensity is only about 0.3. It can be seen from Fig.5(f) that the diffraction angle the SPP wave in of near-field also varies when the filling factor of grating is different. The diffraction angle of the ±1 storderis in the range of 39.79° to 42.67° for different filling factors. When the filling factor is 0.1, the diffraction angle is the smallest, i.e. 39.79°. The diffraction angle is the largest at 42.67° when the filling factor is 0.5. The diffraction angle is very close, i.e. about 41°, when the filling factor ranges from 0.3 to 0.7. The light field intensity of ±1 diffraction order varies with different filling factor, which is related to the different intensity of the SPP wave passing through the diffraction grating.

從圖5(a)～5(e)可以看到，SPP波透過衍射光柵后,在金屬表面上最終都匯聚到一起,衍射級次基本不發(fā)生變化，但強度隨著衍射光柵占空比的變化而變化。當光柵占空比從0.1變化到0.7時,匯聚光束的直徑逐漸變小。當光柵占空比為0.9時,光的透過率已經(jīng)變得非常低，電場強度最大只有0.3左右。從圖5(f)中可以看出，光柵占空比不同時，SPP波的近場衍射角也有所變化，幾種不同占空比的衍射光柵對應(yīng)的±1級衍射角在39.79°～42.67°內(nèi),其中:當占空比為0.1時,衍射角最小,為39.79°;占空比為0.5時,衍射角最大,為42.67°;占空比為0.3或0.7時,衍射角接近,在41°左右。占空比不同,±1級的光場強度也不同,這也與SPP波透過衍射光柵的光強不同有關(guān)。

4.3 Comparison of diffraction behaviors of metal gratings in free space and on meta-surface

自由空間中與超表面上金屬光柵衍射行為比較分析

To further illustrate the difference between the grating diffraction behavior of SPP waves on the meta-surface and that in free-space light, we also calculated the diffraction effect of metal gratings in free space and compared them with those on the meta-surface. Fig.6(a) shows the free space grating diffraction model. Here for the convenience of comparison, let the incident light wavelength and SPP wavelength to be 607 nm as well, and let the thickness of the metal grating also equal to 200 nm. However, the difference is that the height of the metal grating on the meta-surface is limited to 600 nm in the direction perpendicular to the period, while the size of the metal grating in the free space in the direction perpendicular to the period is infinitely large.

Fig.6 Comparison of near field diffraction of SPP wave and free space light: (a)theoretical model for near field diffraction of free space light; (b)near field diffraction of SPP on meta-surface; (c)near field diffraction in free space light; (d)electric field distribution in X direction at 0.95 μm from diffraction grating corresponding to (b); (e)electric field distribution in Z direction at 0.95 μm corresponding to (c); (f)electric field intensity at the surface of the metal film corresponding to (d), (e) 圖6 自由空間光與超表面上光柵近場衍射行為對比結(jié)果圖：(a)空間光的近場衍射裝置圖；(b)SPP波超表面上的光柵近場衍射圖；(c)自由空間光的光柵近場衍射圖；(d)對應(yīng)(b)在距衍射光柵0.95 μm處X方向上的電場分布圖；(e)對應(yīng)(c)在距衍射光柵0.95 μm處Z方向上的電場分布圖；(f)對應(yīng)(d)、(e)在金屬薄膜表面處的電場強度曲線

為進一步說明超表面上SPP波的光柵衍射行為和自由空間光的光柵衍射行為的區(qū)別。本文對自由空間中金屬光柵進行了計算，并和超表面上的情況進行比較。圖6(a)所示為自由空間光柵的衍射模型圖。這里為方便比較，使入射光波長和SPP波波長相等，都為607 nm，并且金屬光柵厚度相等，皆為200 nm，但不同之處在于超表面上金屬光柵在和周期垂直方向上高度有限制，為600 nm，而自由空間中的金屬光柵在和周期垂直方向上的尺寸取無限大。

Fig.6(b) shows the grating near-field diffraction of the SPP wave for an incident wavelength of 625 nm. The wavelength of the SPP wave excited at this time is 607 nm and the period of the diffraction grating is 910 nm, which is about 1.5λspp, the filling factor is 0.5. Fig.6(c) shows the near-field diffraction of the space light with an incident wavelength of 607 nm. In order to compare the diffraction phenomenon on the metasurface to that in free space, the period, filling factor and thickness of the diffraction grating is set to 910 nm、 0.5 and 200 nm respectively.

圖6(b)是入射光波長為625 nm時激發(fā)的SPP波在超表面上的光柵近場衍射情形，此時激發(fā)的SPP波波長為607 nm，衍射光柵周期設(shè)為910 nm，約為1.5λspp，占空比為0.5。圖6(c)為入射光波長為607 nm的空間光的近場衍射情形。為了對比分析空間光與SPP波在超表面的衍射現(xiàn)象，設(shè)衍射光柵周期為910 nm，占空比為0.5，金屬光柵厚度為200 nm。

Comparing Fig.6(b) and 6(c), one can see that the diffractive behavior of SPP wave is similar to the phenomenon when space light is diffracted in the near field. Fig.6(d) and 6(e) correspond to the electric field intensity plots forZ>0(i.e SPP waves in air) atX=0.95 μm in Fig.6(b) and 6(c). The SPP wave exponentially decays in the medium and in the metal, so that on the metal film surface, the electric field intensity reaches its maximum value and decays rapidly in theZdirection. Where as space light propagates uniformly in the medium[14], the intensity of the electric field in theZ-direction is uniformly distributed. Fig.6(c) is light intensity distribution curve on the metal surface corresponds to Fig.6(d) and 6(e). As can be seen from the figure, in addition to the difference in field intensity, the position of 0th and ±1th diffraction orders are basically the same.

比較圖6(b)和6(c)可以看出，SPP波與空間光在近場衍射現(xiàn)象類似。圖6(d)和6(e)為對應(yīng)圖6(b)、6(c)中X=0.95 μm處，Z>0部分(即SPP波在空氣中)的電場強度圖，SPP波在介質(zhì)和金屬中光強呈指數(shù)衰減，因此在金屬薄膜表面上，電場強度取得最大值，在Z軸方向迅速衰減。而空間光在介質(zhì)中是均勻傳播的[14]，所以在Z軸方向電場強度是均勻分布的。圖6(c)對應(yīng)圖6(d)、6(e)在金屬表面上光強分布曲線圖，從圖中可以看出，二者除了在強度上有所區(qū)別外，0、±1衍射級的位置基本一致。

In addition, it is clear that the near-field diffraction angles of the SPP waves excited by different incident wavelengths on the meta-surface are also different after diffraction by metal grating on metasurface. It is known from equation (3), when the incident light wavelengths is 550 nm, 581 nm, 616 nm, 655 nm and 700 nm respectively, the wavelength of SPP waves excited by the device in Fig.2 is 524.7 nm, 557.9 nm, 594.8 nm, 635.8 nm and 682.9 nm, and the ±1 order near-field diffraction angles after metal grating diffraction on the metasurface is 35.737°, 36.966°, 38.157°, 39.311° and 41.507°, respectively. For diffraction of light in free space, when light with a wavelengths of 524.7 nm, 557.9 nm, 594.8 nm, 635.8 nm and 682.9 nm respectively is incident on the diffraction grating with the same structural parameters as shown in Fig.6(a), the resulting far-field diffraction angle can be determined by:

此外，很顯然超表面上不同入射波長所激發(fā)的SPP波經(jīng)光柵衍射后的近場衍射角也不同，由式(3)可知，當入射光波長分別為550、581、616、655及700 nm時，利用圖2的裝置激發(fā)出的SPP波波長依次為524.7、557.9、594.8、635.8及682.9 nm，且經(jīng)過超表面上金屬光柵衍射后的±1級的近場衍射角分別為35.737°、36.966°、38.157°、39.311°及41.507°。而對于自由空間中光的衍射，將波長分別為524.7、557.9、594.8、635.8及682.9 nm的光入射到圖6(a)所示具有相同結(jié)構(gòu)參數(shù)的衍射光柵上時產(chǎn)生的遠場衍射角可由下面式(7)決定：

dsinθ=kλ.

(7)

By using this formula, the corresponding diffraction angles are determined to be 31.653°, 33.916°, 36.503°, 39.485° and 43.071° respectively. In comparison with the case on meta-surface, it can be found that the near-field diffraction angle is almost the same, and the maximum error is only about 5°. This means that the diffraction angle of the SPP wave after transmitted through the metal grating on the meta-surface can be roughly estimated by using the conventional grating diffraction formula in free-space-of, but more accurate results have to be obtained by rigorous numerical calculations.

利用該公式可計算得出所對應(yīng)的衍射角分別為31.653°、33.916°、36.503°、39.485°及43.071°。兩相對比，可發(fā)現(xiàn)其與SPP波透過衍射光柵后的近場衍射角相差不大，最大誤差僅約5°。這說明，采用常規(guī)的用于自由空間光光柵衍射公式，可以對超表面上光柵的衍射角度進行粗略估計，但更準確的結(jié)果必須通過嚴格數(shù)值計算來獲得。

4.4 Comparison of diffraction of two kinds of diffraction gratings on metasurface

超表面上兩種衍射光柵衍射情況的對比

Figs.7(a) and 7(b) show two kinds of diffraction gratings on the meta-surface, Fig.7(a) shows the case where the diffraction grating protrudes on the metal surface; and Fig.7(b) shows the case where the diffraction grating is recessed on the metal surface. In Fig.7(a), in order to avoid the influence of stray light on the diffraction of near-field SPP wave, the height of the diffraction grating is set to be 600 nm, which is equal to the penetration depth of the SPP wave in the air and the width is 200 nm. In order to compare the near-field diffraction of the two kinds of diffraction gratings, the depth of the diffraction grating grooves in Fig.7(b) is set to be 600 nm and the width is set to be 200 nm which is the same as that in Fig.7(a). The two kinds of diffraction gratings are set to have the same period and filling factor.

圖7(a)、7(b)為超表面上兩種衍射光柵示意圖，圖7(a)為衍射光柵凸起于金屬表面的情形；圖7(b)為衍射光柵凹進金屬表面的情形。圖7(a)中，為了避免雜散光對SPP波近場衍射的影響，設(shè)衍射光柵的高度為600 nm，與SPP波在空氣中的穿透深度相等，寬度為200nm。為對比兩種衍射光柵的近場衍射情況，取圖7(b)中衍射光柵凹槽深度為600 nm，寬度為200 nm，與圖7(a)相同，且設(shè)定兩種衍射光柵的周期和占空比均相同。

Fig.7 Schematic diagrams of two kinds of diffraction gratings 圖7 兩種衍射光柵的示意圖

The grating period, the filling factor and depth of grating coupler of excitation SPP wave on the left of the above two structures is set to be 607 nm, 0.5, 200 nm respectively, the incident light wavelength is in the range of 550-700 nm and the incident angle is 14°. Figs.8(a) and 8(b) show the near-field diffraction of SPP waves under the above mentioned two grating structures, respectively. In Fig.8,XYplane is the metal film surface, the diffraction grating is located on theY-axis.

設(shè)定上述兩種結(jié)構(gòu)左側(cè)激發(fā)SPP波的光柵周期為607 nm，占空比為0.5，光柵深度為200 nm，入射光波長為550～700 nm，入射角度為14°。圖8(a)、8(b)分別為在上述兩種光柵結(jié)構(gòu)下，SPP波的近場衍射情況。圖中XY平面為金屬薄膜表面，衍射光柵位于Y軸上。

Fig.8 (a)Near field diffraction of the incident light at 625 nm Corresponding to Fig.7(a); (b)near field diffraction of the incident light at 625 nm Corresponding to Fig.7(b); (c)curve of near field diffraction of the incident light at 550-700 nm corresponding to Fig.7(b). White dashed line in the figure(ieft sode) represents the grating location 圖8 (a)對應(yīng)圖7(a)所示結(jié)構(gòu)，入射波長為625 nm的近場衍射；(b)對應(yīng)圖7(b)所示結(jié)構(gòu)，波長625 nm時近場衍射；(c)對應(yīng)圖7(b)所示結(jié)構(gòu)，入射光波長為550～700 nm的近場衍射曲線圖。圖中左側(cè)白色虛線表示衍射光柵所在位置

Fig.8(c) shows the near-field diffraction curve with an incident light wavelength of 550-700 nm corresponding to the structure shown in Fig.7(b). As can be seen from the figure, the zero-th diffraction order of each wavelength center is located aty=0, but for diffraction orders of ±1, ±2, there have been obvious diffractive effect. The reason is that the intensity of the electric field diffracted by the SPP wave is different that for the fixed metal grating coupling structure there is only one optimal incident wavelength, and at this wavelength, the light is coupled into the SPP wave most efficiently[15].

圖8(c)為對應(yīng)圖7(b)所示結(jié)構(gòu)，入射光波長為550～700 nm的近場衍射曲線圖。從圖中可以看出，各波長的零級衍射的中心均在y=0上，但在±1、±2衍射級時已經(jīng)有明顯的分光現(xiàn)象。不同波長的入射光所激發(fā)的SPP波衍射后的電場強度不同，原因是，固定的金屬光柵耦合結(jié)構(gòu)只對應(yīng)一個最佳的入射波長，這個波長的光耦合成SPP波的效率最高[15]。

In Fig.8(d), the incident light of 550-581 nm is separated by 1.23°and the incident light of 581-616 nm is separated by 1.20°on the metal surface. The incident light of 616-655 nm is separated by 1.15° and the incident light at 655-700 nm is separated by 2.20° on the metal surface due to the near-field diffraction of the SPP wave. Since the excitation and propagation of SPP waves in near-field diffraction are on the order of micrometers[15], the signal of incident light can be separated by the near-field diffraction of SPP wave to realize micron-scale spectrometer.

圖8(d)中由于SPP波的近場衍射， 550～581 nm的入射光在金屬表面上分開1.23°，581～ 616 nm的入射光在金屬表面上分開1.20°，616～655 nm的入射光在金屬表面上分開1.15°，655～700 nm的入射光在金屬表面上分開2.20°。在近場衍射中由于SPP波的激發(fā)和傳播都在微米量級[15]，可以利用SPP波的近場衍射現(xiàn)象對入射光的信號進行分離,制備微米量級的光譜儀器。

5 Conclusions

結(jié) 論

In this paper, the near-field diffraction of SPP waves excited by visible light incident on metal grating structures in the wavelength range of 550-700 nm is studied by a method of metal grating coupling. First of all, according to the rigorous coupling wave theory, the SPR angle scanning method is used to find the optimal metal grating structure with the highest coupling efficiency for the incident light with the central wavelength of 625 nm. Then the effect of metal grating structure parameters on the near-field diffraction of SPP wave is studied. The results show that the diffraction effect of SPP is most pronounced when the period of the diffraction grating is 1.5 times the wavelength of SPP and the filling factor is 0.5. Finally, using the phenomenon of SPP wave near-field diffraction, the incident light of 550-558 nm can be separated by 1.23° on the metal surface, the incident light of 581-616 nm is separated by 1.20° on the metal surface and the incident light of 616-655 nm can be separated by 1.15° on the metal surface, the 655-700 nm incident light can be separated by 2.20° on the metal surface. Finally, the diffraction of SPP wave and the diffraction of free space light are compared. We discover that the diffraction angle calculated from the free-space grating diffraction formula is similar to that of the SPP wave. The formula can be used to make a rough estimate of the diffraction angle of a grating on a meta-surface.

本文利用金屬光柵耦合方式,對波長在550～700 nm范圍內(nèi)的可見光入射到金屬光柵結(jié)構(gòu)上激發(fā)的SPP波的近場衍射現(xiàn)象進行了研究。首先根據(jù)嚴格耦合波理論，利用SPR角度掃描方法，找出中心波長625 nm的入射光耦合效率最高的金屬光柵結(jié)構(gòu)。然后研究了金屬光柵結(jié)構(gòu)參數(shù)對SPP波近場衍射的影響。結(jié)果表明，當衍射光柵的周期為SPP波波長的1.5倍、占空比為0.5時，SPP波的衍射效果最明顯。最后，利用SPP波的近場衍射現(xiàn)象，可將550～581 nm的入射光在金屬表面上分開1.23°，581～616 nm 的入射光在金屬表面上分開1.20°，616～655 nm的入射光在金屬表面上分開1.15°，可將655～700 nm的入射光在金屬表面上分開2.20°。最后將SPP波的衍射情況和自由空間光的衍射情況進行比較，發(fā)現(xiàn)根據(jù)自由空間光柵衍射公式計算出的衍射角度和SPP波的情況相差不大，可以用該公式對超表面上光柵的衍射角度進行粗略估算。

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