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. 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 the m-th(m=±1, ±2, ±3, …，±n) order diffracted light in the direction parallel to the interface is
SPP是金属/介质界面上由表面电荷的集体震荡与电磁场耦合所形成的沿着金属/介质界面传播的一种电磁波。SPP的场分量在金属/介质界面上取得最大值，在金属两侧的介质中场分量呈e指数衰减。在可见光及近红外波段内，绝大多数金属介电常数的实部为负数，因此金属的介电常数与其相邻介质的介电常数异号，只有P偏振光(TM)才能激发出SPP。如图 1所示，当光波(含P偏振光)入射到金属光栅时，在光栅表面将发生衍射现象，不同的衍射角度对应于不同的衍射级次。根据光栅方程可知第m(m=±1, ±2, ±3, …，±n)级衍射光的波矢在平行于界面方向上的分量如式(1)。
Where k0 is the light wave vector in free space, θ is the incident angle of the light wave, ε2 is the dielectric constant of the medium, and Λ is 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 the m-th order diffracted light parallel to the interface can be equal to the wave vector of the SPP wave at the interface, that is
is the wave vector of the SPP wave at the metal/dielectric interface and ε1 is the dielectric constant of the metal. When the equation (2) is satisfied, the incident light is coupled to the SPP wave so that it is excited, and the surface plasmon resonance(SPR) phenomenon occurs.
As can be seen from the formula (2)，the surface plasmon wave wavelength is as follows：
where，λ0 is incident wavelength.
Three characteristic lengths of SPP wave are analyzed as follows.
The SPP wave propagation distance at the metal/dielectric interface is defined as the distance traveled of the SPP wave when the electric field decreases to 1/e of the initial value, denoted as δspp, which is mainly determined by the imaginary part k″spp of the wave vector and
Where εm′ and εm″ are respectively the real and imaginary parts of the complex dielectric constant of the metal respectively, λ0 is the wavelength of the incident light, and the penetration depth of the SPP wave in the medium and the metal is respectively
Where, k0 is the incident light wave vector. 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为入射光的波矢量。由式(4)~(6)可得，550~700 nm波长的光波在金属银表面的传播距离为12.5~60 μm时，SPP在空气和银中的穿透深度分别为279~500 nm和22.7~29.5 nm。
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 the X direction. 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°.
当金属银的光栅周期为607 nm时，波长为550~700nm的入射光激发出的SPP波波长在525~683 nm之间。设定图 2所示的近场衍射模型中激发SPP波的金属光栅周期为607 nm，占空比为0.5，凹槽深度为200 nm。右方衍射金属光栅位于金属表面上，高度为600 nm，在X方向上的厚度为200 nm。当研究SPP波经过其的衍射行为时，对以下两种情况分别研究：一种是光栅高度和占空比不变，而光栅周期发生变化时的情况；另一种是光栅高度和周期不变，占空比发生改变的情况。同时激发光波长和入射角度固定为625 nm和14°。
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, XY plane is the metal film surface, and the diffraction grating is located on the Y-axis.
图 4 波长λ=625 nm的入射光激发的SPP波在周期不同，占空比一定的条件下的衍射现象(图中白色断线表示金属衍射光栅的位置)
Figure 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)
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 is λspp. 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, 4λspp and 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.
图 4(a)为当衍射光栅周期为0.5λspp时SPP波的衍射情况。可以看到此时只有零级衍射，且透射光强较弱，在0.2 μm以下，入射光透过衍射光栅后传播距离在2.3 μm左右；图 4(b)为衍射光栅周期为λspp的情形，此时已有明显的衍射现象，可以看到0级和±1级衍射，并且此时光强有明显的增强，最大达到了1左右，同时透过衍射光栅后SPP光的传播距离在4 μm左右；图 4(c)~4(e)为衍射光栅周期分别为1.5λspp、4λspp和6λspp时SPP波的衍射情形。可以看到，随着衍射光栅的周期不断增大，衍射级次相应地增多。当衍射光栅周期为SPP波长的1.5倍时，最高衍射级次为±2级，并且衍射条纹的分布比较清晰；而当光栅周期为SPP波长的4~6倍时，由于更高衍射级次的出现使得衍射条纹的分布变得较为紊乱。
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, the XY plane is the metal film surface, and the diffraction grating is located on the Y-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 at x=0.
图 5 入射光波长λ=625 nm, 衍射光栅周期为910 nm，占空比不同时的衍射现象，其中白色虚线表示金属光栅的位置。(a)占空比为0.1；(b)占空比为0.3；(c)占空比为0.5；(d)占空比为0.7；(e)占空比为0.9；(f)衍射光栅占空比不同时，金属薄膜表面上x=0直线上的近场衍射电场强度曲线图
Figure 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
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.
从图 5(a)~5(e)可以看到，SPP波透过衍射光栅后, 在金属表面上最终都汇聚到一起, 衍射级次基本不发生变化，但强度随着衍射光栅占空比的变化而变化。当光栅占空比从0.1变化到0.7时, 汇聚光束的直径逐渐变小。当光栅占空比为0.9时, 光的透过率已经变得非常低，电场强度最大只有0.3左右。从图 5(f)中可以看出，光栅占空比不同时，SPP波的近场衍射角也有所变化，几种不同占空比的衍射光栅对应的±1级衍射角在39.79°~42.67°内, 其中:当占空比为0.1时, 衍射角最小, 为39.79°; 占空比为0.5时, 衍射角最大, 为42.67°; 占空比为0.3或0.7时, 衍射角接近, 在41°左右。占空比不同, ±1级的光场强度也不同, 这也与SPP波透过衍射光栅的光强不同有关。
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.
图 6 自由空间光与超表面上光栅近场衍射行为对比结果图：(a)空间光的近场衍射装置图；(b)SPP波超表面上的光栅近场衍射图；(c)自由空间光的光栅近场衍射图；(d)对应(b)在距衍射光栅0.95 μm处X方向上的电场分布图；(e)对应(c)在距衍射光栅0.95 μm处Z方向上的电场分布图；(f)对应(d)、(e)在金属薄膜表面处的电场强度曲线
Figure 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)
为进一步说明超表面上SPP波的光栅衍射行为和自由空间光的光栅衍射行为的区别。本文对自由空间中金属光栅进行了计算，并和超表面上的情况进行比较。图 6(a)所示为自由空间光栅的衍射模型图。这里为方便比较，使入射光波长和SPP波波长相等，都为607 nm，并且金属光栅厚度相等，皆为200 nm，但不同之处在于超表面上金属光栅在和周期垂直方向上高度有限制，为600 nm，而自由空间中的金属光栅在和周期垂直方向上的尺寸取无限大。
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.
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 for Z > 0(i.e SPP waves in air) at X=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 the Z direction. Where as space light propagates uniformly in the medium, the intensity of the electric field in the Z-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.
比较图 6(b)和6(c)可以看出，SPP波与空间光在近场衍射现象类似。图 6(d)和6(e)为对应图 6(b)、6(c)中X=0.95 μm处，Z＞0部分(即SPP波在空气中)的电场强度图，SPP波在介质和金属中光强呈指数衰减，因此在金属薄膜表面上，电场强度取得最大值，在Z轴方向迅速衰减。而空间光在介质中是均匀传播的，所以在Z轴方向电场强度是均匀分布的。图 6(c)对应图 6(d)、6(e)在金属表面上光强分布曲线图，从图中可以看出，二者除了在强度上有所区别外，0、±1衍射级的位置基本一致。
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，很显然超表面上不同入射波长所激发的SPP波经光栅衍射后的近场衍射角也不同，由式(3)可知，当入射光波长分别为550、581、616、655及700 nm时，利用图 2的装置激发出的SPP波波长依次为524.7、557.9、594.8、635.8及682.9 nm，且经过超表面上金属光栅衍射后的±1级的近场衍射角分别为35.737°、36.966°、38.157°、39.311°及41.507°。而对于自由空间中光的衍射，将波长分别为524.7、557.9、594.8、635.8及682.9 nm的光入射到图 6(a)所示具有相同结构参数的衍射光栅上时产生的远场衍射角可由下面式(7)决定：
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.
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.
图 7(a)、7(b)为超表面上两种衍射光栅示意图，图 7(a)为衍射光栅凸起于金属表面的情形；图 7(b)为衍射光栅凹进金属表面的情形。图 7(a)中，为了避免杂散光对SPP波近场衍射的影响，设衍射光栅的高度为600 nm，与SPP波在空气中的穿透深度相等，宽度为200nm。为对比两种衍射光栅的近场衍射情况，取图 7(b)中衍射光栅凹槽深度为600 nm，宽度为200 nm，与图 7(a)相同，且设定两种衍射光栅的周期和占空比均相同。
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, XY plane is the metal film surface, the diffraction grating is located on the Y-axis.
Figure 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
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 at y=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.
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, the signal of incident light can be separated by the near-field diffraction of SPP wave to realize micron-scale spectrometer.
图 8(d)中由于SPP波的近场衍射，550~581 nm的入射光在金属表面上分开1.23°，581~ 616 nm的入射光在金属表面上分开1.20°，616~655 nm的入射光在金属表面上分开1.15°，655~700 nm的入射光在金属表面上分开2.20°。在近场衍射中由于SPP波的激发和传播都在微米量级，可以利用SPP波的近场衍射现象对入射光的信号进行分离, 制备微米量级的光谱仪器。