The wave vector of a one-dimensional simple grating, *k*_{grating}, has a magnitude given by 2*π*/*P*, where *P* is the period, and a direction that is perpendicular to the axis of the grating and in-plane with the grating. For any wavelength of light *λ* the input light from a microscope has a wave vector in air, *k*_{in}, whose magnitude is given by 2*π*/*λ*, while the direction is determined by the light path through the optical components. Thus, the shortest blue wavelengths of light (~400 nm) have the largest wave vectors, while the longest red wavelengths (~700 nm) have the smallest wave vectors. When light is reflected by a perfect reflector, the wave vector of the reflected light retains its original magnitude, but the vertical or *z*-component of the vector reverses its sign.

Wave vector diagrams can be used to explain the diffraction of the input light, as shown in Figure 1G and H, for the case of a one-dimensional grating. The grating wave vector, incident and output wave vectors, and incident and collection cones are as indicated. (The right half of the incident cone is omitted for clarity.) We consider the case where the grating vector *k*_{grating} has a larger magnitude than the wave vectors of the input light, as the period of the disk array is smaller than the wavelength of violet light, which is 380 nm. The incident cone for a 50×/0.8 NA objective has incident angles between ~53° and 73°. The input and output wave vectors lie on a circle because of conservation of energy. To find the direction of the output wave vector for a particular input angle, in-plane phase matching along the *x*-direction is used (the magnitudes of the vectors *k*_{grating}, *k*_{in,x}, and *k*_{in,y} are not in bold):

$${k}_{\text{out},x}={k}_{\text{in},x}-m\hspace{0.17em}{k}_{\text{grating}},$$(1)

$$\frac{2\pi}{\lambda}\mathrm{sin}{\theta}_{\text{out}}=\frac{2\pi}{\lambda}\mathrm{sin}{\theta}_{\text{in}}-m\left(\frac{2\pi}{P}\right),$$(2)

$$\mathrm{sin}{\theta}_{\text{out}}=\mathrm{sin}{\theta}_{in}-m\left(\frac{\lambda}{P}\right).$$(3)

The diffraction order, *m*, is taken as positive, as negative values would give a diffraction angle that lies outside the collection cone. *θ*_{in} and *θ*_{out} are measured with respect to the normal to the surface and are limited to the upper half-plane (*z*>0). By definition, if the output wave vector lies in the same quadrant as the input vector, then *θ*_{out} is negative, whereas *θ*_{out} is positive if the output and input wave vectors lie in different quadrants. From Eq. (3), *θ*_{out} is negative when the magnitude of *k*_{in,x} is smaller than that of *m* *k*_{grating}. With small periodicities, no light is collected as *k*_{grating} is too large. However, with increasing periodicity *k*_{grating} decreases, and we start to observe some light.

In Figure 1G we consider the specific condition where the input light is blue and has an incident angle equal to the smallest angle of the incident light cone (*θ*_{in1}). This illumination with wave vector *k*_{in,blue1} is diffracted and deflected in the opposite direction such that the output light *k*_{out,blue1} just fits into the collection cone (|*θ*_{out}|<*θ*_{collection}/2). For larger incident angles, e.g. for *k*_{in,blue2}, the input *x*-component increases, so the output light *k*_{out,blue2} has a smaller *x*-component and easily fits into the collection cone. Thus, all the first-order diffracted blue light, regardless of incident angle, is collected by the objective. The threshold (maximum) wavelength at a particular incident angle for which light will be first-order diffracted into the collection cone of the objective occurs when *θ*_{out} is equal to −sin^{−1} NA, or sin *θ*_{out}=−NA. *θ*_{out} has a negative sign as it lies in the same quadrant as *θ*_{in}. Thus, the following equations apply, where *θ*_{in1}<*θ*_{in2} and *λ*_{threshold1}<*λ*_{threshold2}:

$$m\hspace{0.17em}{\lambda}_{\text{threshold}1}=P\mathrm{(}NA+\mathrm{sin}{\theta}_{\text{in1}}\mathrm{)},$$(4)

$$m\hspace{0.17em}{\lambda}_{\text{threshold}2}=P\mathrm{(}NA+\mathrm{sin}{\theta}_{\text{in2}}\mathrm{)}.$$(5)

Figure 1H shows the wave vector diagram for red light. Since red light has a longer wavelength than blue light, its wave vector is shorter and the *x*-component of the diffracted light is larger, producing a larger output angle that falls outside of the collection cone, e.g. for *k*_{in,red1}. This condition applies for all the incident angles of red light smaller than *θ*_{in2}, which is the largest angle of the incident light cone. Therefore, in contrast to the case for blue light, almost no red light is collected by the objective. The resultant spectrum collected by the objective will mostly consist of short blue wavelengths of light, and thus the image will appear blue.

Since the disk arrays are two-dimensional, the diffraction grating vector is not fixed along one axis and can point in various directions. In fact, there are two grating vectors, *k*_{grating,x} and *k*_{grating,y}, and two boundary conditions to be satisfied. For the disk arrays in this section, the *x*-period *P*_{x} and *y*-period *P*_{y} are the same (*P*_{x}=*P*_{y}=*P*), so the magnitudes of *k*_{grating,x} and *k*_{grating,y} are the same. In general,

$${k}_{\text{out},x}={k}_{\text{in},x}-m\hspace{0.17em}{k}_{\text{grating},x}={k}_{\text{in},x}-m\left(\frac{2\pi}{{P}_{x}}\right)$$(6)

$${k}_{\text{out},y}={k}_{\text{in},y}-n\hspace{0.17em}{k}_{\text{grating},y}={k}_{\text{in},y}-n\left(\frac{2\pi}{{P}_{y}}\right)$$(7)

$$\begin{array}{l}{k}_{\text{out},z}=\sqrt{{k}_{\text{in}}{}^{2}-{k}_{\text{out},x}{}^{2}-{k}_{\text{out},y}{}^{2}}=\\ \text{\hspace{1em}}\sqrt{{k}_{\text{in}}{}^{2}-{\left({k}_{\text{in},x}-m\left(\frac{2\pi}{{P}_{x}}\right)\right)}^{2}-{\left({k}_{\text{in},y}-n\left(\frac{2\pi}{{P}_{y}}\right)\right)}^{2}}\end{array}$$(8)

When the array period is small, only the first-order diffracted light in either the *x*- or *y*-direction can be collected, so the disk array can be analyzed in terms of a one-dimensional (1D) grating.

Brightfield and darkfield images of the Al nanodisk arrays are taken using a Nikon Eclipse LV100ND microscope and shown in Figure 1C and D. A 50×/0.8 NA objective is used, which corresponds to a half-acceptance angle of 53°. In Figure 1D, saturated blue and cyan colors are observed for arrays with large periods greater than 250 nm. These colors form bands of similar color along the diagonal, as the array periods are constant along each diagonal line from the upper left to the lower right. These colors are caused by diffraction, as explained in the previous section. For periods around 250 nm, only short wavelengths of light are diffracted by the array and collected by the objective, so the arrays appear blue. When the period increases, longer green wavelengths are also collected, so the arrays appear cyan.

For arrays with smaller disk diameters (<240 nm), the central part of the array appears black but the square borders are colorful due to enhanced plasmon scattering by the individual disks, spanning blue, green, yellow, orange, and red. These colors are size-dependent and remain the same throughout each column of arrays and had been demonstrated previously using gold nanoparticles separated from a gold film by a sub-nanometer spacer [26].

The reflectance spectra for arrays of 280 nm wide disks were measured with a CRAIC 508 PV spectrophotometer and show a clear peak-and-side-lobe profile (Figure 1E). The inter-disk gap *g* is varied from 20 nm to 130 nm; thus, the periods *P* are 300 nm to 410 nm. The peaks in the spectra occur around 400 to 450 nm and increase in amplitude when the period increases. The reflectance gradually decreases to nearly 0 at a characteristic threshold wavelength. This threshold wavelength is the maximum wavelength at which the diffracted light from the array falls into the collection cone of the objective. Since the incident light cone has a spread of polar angles, there are two threshold wavelengths, *λ*_{threshold1} and *λ*_{threshold2}, corresponding to the minimum and maximum polar angles, respectively. Both threshold wavelengths redshift when the inter-disk gap increases. The threshold wavelengths are also compared to the theoretical prediction in Figure 1F and show good agreement. Here the theoretical relationship between the threshold wavelength and the array period is obtained from the diffraction grating equation, where the diffraction order *m* is +1:

$${\lambda}_{\text{threshold}}=P\mathrm{(}NA+\mathrm{sin}{\theta}_{\text{in}}\mathrm{)}.$$(9)

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