The proposed necessary and sufficient condition for spectral switches with concave reflectance of aluminum metal

Abstract

Spectral switches have been widely studied, and many related applications and phenomena were proposed or verified. However, the conditions under which spectral switches can be found are not completely specified. In the past, the zero-intensity components or phase singularities are used as the sufficient conditions for spectral switches. Some kind of oscillatory behavior in the modifier function is also proposed to be the necessary condition for it. In this work, we suggest that the necessary and sufficient condition can be reduced to a simple condition: a strictly concave modifier with a minimum. An example with this condition is given using the reflectance of aluminum in the visible to near-infrared range. This result clarifies the conditions for spectral switches and helps identifying it under different circumstances in the future. The related spectral shift amplification and polarization–controlled spectral shift effect are also presented.

Introduction

The problem of spectra invariance during propagation has been studied since 1980. It is found that contrary to previous thoughts, spectral changes occur unless the scaling law is satisfied, which is called the Wolf effect. ¹ This law requires that the complex degree of spectral coherence is a function of the product of wave number and relative position vector. After that, many studies investigate other mechanisms leading to spectral changes, including aperture diffractions,^2–4 spectral correlation, ⁵ material interactions, and scattering.^6–8 Also, many interesting phenomena or applications have been found, such as the singular optics, ⁹ spatial coherence spectroscopy, ⁵ lattice spectroscopy, ¹⁰ Talbot spectra, ¹¹ and the spectral switches (SSW).^3,12,13 The SSW are first proposed by Pu et al. ⁴ and are verified experimentally. ¹⁴ SSW are generally referred as a phenomenon in which the spectral shift of diffracted or modified polychromatic light experiences a discontinuous jump, like a switch, when one of the variables (e.g. the aperture size, the detected distance, the central wavelength, or the bandwidth of the light source) varies continuously. Many related interesting effects are found, such as the spectral shift amplification ¹⁵ polarization-controlled spectral shifts, ¹⁶ data transmission, ¹⁷ tunable SSW, ¹⁸ and surface plasmon effect. ¹⁹ It was first named by the observation that when polychromatic light passes a circular aperture, the detected spectral in the near field is modified, and the shift of the spectrum maximum goes through a discontinuous jump, just like a switch. Later, Foley and Wolf ²⁰ illustrated that those points are actually the positions of phase singularities, with features that the field amplitude at central wavelength is zero and the phase increases from 0 to 2π for an infinite small loop around the singular points. The light field is undetermined at those singular points. Consequently, the singular points in the modifier function (explained later) are used as the sufficient condition for the SSW. In 2009, Han ²¹ gave a SSW example without singular points and illustrated that the necessary condition may be some kind of oscillatory behavior in modifier. In this work, this issue is investigated further. We suggest that a simple but important condition is necessary and sufficient for the existence of SSW; that is, the modifier is a strictly concave function with a minimum in some range of variables, as long as the original spectrum distribution is a convex function with a maximum. An example is given to illustrate this condition with the reflectivity of aluminum in visible near-infrared (NIR) range. This article is organized as the follows. The first section introduces the background of the spectral changes by different mechanisms and the reviews of conditions for the appearance of SSW. In the second section, we present the definition of a strictly convex/concave function with a maximum/minimum. For the first time, SSW is successfully made with such a simple concave modifier with a minimum, utilizing the reflectance of aluminum metal as the illustration. The discussion of such a necessary and sufficient condition and some conclusions from this result are made in the final section.

Theory and calculations

The SSW have been studied satisfactorily within the frame of scalar wave optics. For a fully spatially coherent polychromatic light, the standard Gaussian form of the spectrum can be written as ³

$$S^{(i)}(\lambda )=I_0\exp [-{\left(\frac{\lambda -{\lambda }_c}{\mathrm{\Gamma }}\right)}^2]$$

(1)

where the superscript $(i)$ denotes the incident light wave, $I_0$ the intensity magnitude, ${\lambda }_c$ the central wavelength, and $\mathrm{\Gamma }$ the bandwidth. Its distribution is shown in Figure 1(a) with I₀ = 1.0, λc = 1.0 µm, and Γ = 0.5 µm. Obviously, this kind of function is strictly convex in the interval of the bandwidth with a maximum at ${\lambda }_c$ . A strictly convex function $F_1(x)$ with a maximum in the range of $(x_1,x_2)$ is illustrated in Figure 1(b) and is defined as ²²

$$\begin{array}{c}\forall x_1,x_2\in X,\forall t\in [0,1]:\hfill \\ F_1[t{x}_1+(1-t)x_2]<t{F}_1(x_1)+(1-t)F_1(x_2)\hfill \\ \multicolumn{1}{c}{∍x_3\in (x_1,x_2):{F\prime }_1\left(x_3\right)=0,{F″}_1\left(x_3\right)<0}\end{array}$$

(2)

where $F{\prime }_1(x_3),{F″}_1(x_3)$ are, respectively, the first and second derivatives of $F_1(x)$ at $x_3$ . The first condition in equation (2) is for convex property and the second assure the existence of the maximum (the black dot in the figure). To put it simply, it looks like a mountain shape with a single top. As shown in Figure 1(b), an easy way to check whether the function is convex is that by drawing a straight line from ( $x_1$ , $F_1(x_1)$ ) to ( $x_2$ , $F_1(x_2)$ ) and see whether all the corresponding function values in the interval [x₁, x₂] lie above the line. Similarly, if a function $F_2(x)$ is strictly concave with a minimum, it satisfies

$$\begin{array}{c}\forall x_1,x_2\in X,\forall t\in [0,1]:\hfill \\ F_2[t{x}_1+(1-t)x_2]>t{F}_2(x_1)+(1-t)F_2(x_2)\hfill \\ \multicolumn{1}{c}{∍x_3\in (x_1,x_2):{F\prime }_2\left(x_3\right)=0,{F″}_2\left(x_3\right)>0}\end{array}$$

(3)

Figure 1.

(a) The distribution of a Gaussian spectrum, (b) the plot of a strictly convex function with a maximum, and (c) the plot of a concave function with a minimum.

A strictly concave function with a minimum is shown in Figure 1(c), which has the function values lie below the straight line within the specific interval. It simply looks like a valley shape with a single bottom.

As mentioned in the “Introduction” section, the detected spectrum can be altered by different mechanisms, such as propagation, aperture diffraction, and interactions with matters. The resultant spectrum is usually written as ²⁰

$$S(\lambda )=M(\lambda )·S^{(i)}(\lambda )\mathrm{or}S(\omega )=M(\omega )·S^{(i)}(\omega )$$

(4)

where $M(\lambda )$ or $M(\omega )$ is called the modifier function (or simply the modifier), which indicates how the original spectrum is modified, and generally, it is a function of wavelength and/or detection position. Note that, for convenience, the variables $\omega$ (the angular frequency) and $\lambda$ (the wavelength) are used interchangeably, with the relation $\omega =2\pi \nu =2\pi c/\lambda$ , where $c$ is the speed of light in vacuum and $\nu$ is its frequency.

For example, the modifier on the axis for near-field circular aperture diffraction is $M_1(\omega )=4{\sin }^2[\omega a^2/4\mathrm{cz}]=4{\sin }^2[\pi a^2/2\lambda z]$ , where $a$ is the radius of the aperture and z is the distance from the center of the aperture. ²⁰ The configuration is plotted in Figure 2(a) and the modifier is shown in Figure 2(b) with ${\omega }_c{a}^2/2\pi \mathrm{cz}=8$ , ${\omega }_c$ being the central frequency. The subscripts 1, 2, 3, and so on of $M_1(\omega )$ are used to distinguish from other modifiers under different situations. Note that $M_1(\omega )$ is a periodic sinusoidal function and some spectral components disappear (the red circles). When ${\omega }_c{a}^2/2\pi \mathrm{cz}$ is an even integer, $M_1(\omega )$ at ${\omega }_c$ is zero (the blue circle in Figure 2(b)) and this zero point also makes $S^{(i)}(\omega )$ zero at $\omega ={\omega }_c$ , as seen from equation (4). The condition ${\omega }_c{a}^2/2\pi \mathrm{cz}=2n$ , n = 1, 2, 3, …, leads to the specific axial positions $z_m=a^2/(2n{\lambda }_c)$ , n = 1, 2, 3, … It is shown that these positions are also phase singularity points. ²⁰ The spectral switch is produced with the zero intensity at ${\omega }_c$ near those $z_m$ locations, as shown in Figures 5 and 6 in Foley and Wolf. ²⁰ Together with other examples, it is recognized that the zero intensity of modifier at $\omega ={\omega }_c$ and the phase singularities are suggested as sufficient conditions for SSW.

Figure 2.

(a) The configuration for light diffraction through a circular aperture and (b) the distribution of $M_1(\omega )$ . The circles are zero spectral intensities.

Later, the authors for the first time used a modifier without zero intensities and phase singularities to make SSW. ²¹ The geometry is shown in Figure 3(a) and (b), which is a right triangle aperture with an adjustable hypotenuse slope; the modifier is $M_2(\omega )=\{[\sin c^2(\mathrm{mb}\omega ·\sin (\theta )/2c)]+2[(\sin c(\mathrm{mb}\omega ·\sin (\theta )/2c))·\cos ((\mathrm{mb}\omega ·\sin (\theta )/2c)]+1\}$ , where $\theta$ is the angle between $\overline{o\prime p}$ and optical axis $\overline{o\prime o}$ for $p$ on the $y$ axis, $b$ is the length of the triangle base, and $m$ is the slope of the hypotenuse, as denoted in Figure 3. The behavior of $M_2(\omega )$ with $\mathrm{mb}{\omega }_0·\sin (\theta )/2c=5.0$ is shown in Figure 3(c), and obviously, there are no zero intensities or phase singularities, like the red or blue circles in Figure 2(b). Nevertheless, the SSW are still found at specific $\theta$ like Figure 3 in Han. ²¹ Although the modifier $M_2(\omega )$ is not a periodic oscillating function like $M_1(\omega )$ , it still has some kind of oscillatory property. An oscillatory function usually has more than one peak and one valley, as seen from Figures 2(b) and 3(c). Consequently, the author claim that this oscillating behavior may be the necessary condition for the existence of SSW because, for example, a monotonic increasing modifier without oscillations has been investigated and it only leads to pure blueshift without SSW like Figure 2 in Han. ²

Figure 3.

(a) Schematic diagram of the configuration and illustration of notation. An incoming light wave from left is incident on a right triangle aperture and diffracted toward the observation plane. (b) Dimension and structure of the triangle which has an adjustable hypotenuse’s slope m. (c) The distribution of $M_2(\omega )$ with $\mathrm{mb}{\omega }_0·\sin (\theta )/2c=5.0$ .

Then, Han and Tseng ²³ use the oscillatory behavior of spectral reflectance of silver metal to study the SSW and the spectral shift amplification effect. Figure 4 shows the reflectance of three metals: silver (Ag), gold (Au), and aluminum (Al). ²⁴ It is found that there is a sharp dip in the silver’s reflectance near 0.3 µm (indicated by the red oval) and its blowup is depicted on the left-hand side. From the blowup, we see that this dip actually contains two minima and again some kind of oscillatory behavior, similar to Figures 2(b) and 3(c), is observed. Han and Tseng ²³ successfully produce SSW with these two minima and oscillating property as shown in Figures 5 and 6. The dip of silver metal is caused by bound electrons resonant absorption and by free electron plasma, and it has higher reflectance at the dip than that of pure water by 10 times (the dielectrics containing only bound charges). Even so, the reflectance at the minimum is only 7.2%, which does not benefit the reflection spectrum. In this work, we want to show that the necessary and sufficient condition can be reduced from an oscillatory modifier to a simple strictly concave modifier with only one minimum. The former usually contains more than one dip and one peak, while the latter simply has one minimum. The example being given uses the reflectance of the aluminum in the NIR, as shown in the green oval in Figure 4 and its blowup inside the figure. It is easy to identify that it is a simple concave function with only one minimum and without oscillations in the interested wavelength range, consistent with the definition and the behavior in Figure 1(c). Another advantage of using Al metal is obvious by observing that the minimum of the reflectance is over 85% as shown in Figure 4, which is much larger than that of the Ag and benefits the reflection spectrum detection. To facilitate numerical calculations and illustrations, some parameters are set. The convex Gaussian spectrum in equation (1) with $\mathrm{\Gamma }=0.8\mu \mathrm{m}$ is used for the incident light $S^{(i)}(\lambda )$ , and we first consider the situation that it is normally incident on aluminum metal (Al) from the air, as shown in Figure 5.

Figure 4.

The reflectance of Ag, Au, and Al. The blowup on the left-hand side is the dip of the Ag at 0.32 µm and the other blowup inside the figure is the concave reflectance of Al in the range from 0.5 to 1.1 µm.

Figure 5.

The incident Gaussian spectrum is normally incident on the aluminum and the reflected spectrum is altered by the reflectance of Al.

The reflected spectrum $S^{(r)}(\lambda )$ from the Al surface is given as follows, with the superscription r denoting reflection

$$S^{(r)}(\lambda )=R(\lambda )·S^{(i)}(\lambda )=M(\lambda )·S^{(i)}(\lambda )$$

(5)

where $R(\lambda )$ is the reflectance of the air–Al interface and it is also the modifier function of $S^{(i)}(\lambda )$ . To obtain $R(\lambda )$ , the reflection coefficient $r(\lambda )$ by Fresnel equation ²⁵ is calculated as

$$r(\lambda )=\frac{N_{\mathrm{air}}-N_{\mathrm{Al}}(\lambda )}{N_{\mathrm{air}}+N_{\mathrm{Al}}(\lambda )}$$

(6)

where $N_{\mathrm{air}}$ is the refraction index of air and is set to 1.0 without loss of generality and $N_{\mathrm{Al}}(\lambda )=n(\lambda )-\mathrm{ik}(\lambda )$ is the complex refraction index of Al. The real refraction index $n(\lambda )$ and extinction coefficient $k(\lambda )$ of Al are plotted in Figure 6. The data marked with circle in Figure 6 are experimental values, ²⁶ and they are interpolated by the cubic spline to give the curves. Once $r(\lambda )$ is found, the reflectance $R(\lambda )=r(\lambda )\times r(\lambda {)}^{*}$ can be obtained, where $r(\lambda {)}^{*}$ is the complex conjugate of $r(\lambda )$ .

Figure 6.

(a) The refraction index n of Al versus wavelength and (b) the extinction coefficient k of Al versus wavelength. ²²

Spectral shift amplification

Figure 7(a)–(c) shows the normalized spectrum of $S^{(i)}(\lambda )$ (dotted line), $S^{(r)}(\lambda )$ (solid line), and $R(\lambda )$ (dashed line) for three incremental ${\lambda }_c$ ; the value of ${\lambda }_c$ is 0.78, 0.80, and 0.84 µm in Figure 7(a)–(c), respectively. In Figure 7(a), $S^{(r)}(\lambda )$ is modified into two peaks and the left one dominates; the peak of $S^{(r)}(\lambda )$ is blueshifted from $S^{(i)}(\lambda )$ , as indicated by the arrow on the top of Figure 7. In Figure 7(b), the two peaks reach the same height; while in Figure 7(c), the right peak becomes the major peak and the spectrum is red-shifted now. From Figure 7, it is observed that when ${\lambda }_c$ changes continuously, the spectral shift of the spectrum goes from a red-shift to a blueshift with a discontinuous jump, which is the spectral switch. Therefore, for the first time, we successfully show its existence with such a simple concave behavior in the aluminum reflectance, without resorting to the more complicate oscillatory property or singular points. The accompanied effect called spectral shift amplification can also be found in Figure 7(d), which is the recombination of Figure 7(a) and (c). It is found that the shift in $S^{(i)}(\lambda )$ (dashed arrow above the figure) is amplified into the shift of $S^{(r)}(\lambda )$ (solid arrow above the figure). The amplification factor is about 4.0 under the present situation.

Figure 7.

The normalized spectrum of $S^{(i)}(\lambda )$ (dotted line), $S^{(r)}(\lambda )$ (solid line), and $R(\lambda )$ (dashed line) for three ${\lambda }_c$ . (a) λ_c = 0.78 µm, (b) λ_c = 0.80 µm, and (c) λ_c = 0.84 µm. The shift of the spectrum maximum is indicated by the arrows on the top of the figure. (d) The spectral shift amplification effect with Al.

Polarization-controlled method

Another application for using the material reflectance as the modifier is its polarization dependence. It has been proposed by Han and Tseng ²³ and used as a data transmission tool through polarization control. Similar effects can also be found here. It is known that the reflectance is polarization dependent when the incident angle is not normal. For oblique incidence with incidence angle ${\theta }_1$ and refraction angle ${\theta }_2$ , the reflection coefficients ²⁵ are

$$\begin{array}{c}{r}_s(\lambda )=\frac{N_{\mathrm{air}}\cos ({\theta }_1)-N_{\mathrm{Al}}(\lambda )\cos ({\theta }_2)}{N_{\mathrm{air}}\cos ({\theta }_1)+N_{\mathrm{Al}}(\lambda )\cos ({\theta }_2)}\hfill \\ r_p(\lambda )=\frac{N_{\mathrm{air}}\cos ({\theta }_2)-N_{\mathrm{Al}}(\lambda )\cos ({\theta }_1)}{N_{\mathrm{air}}\cos ({\theta }_2)+N_{\mathrm{Al}}(\lambda )\cos ({\theta }_1)}\hfill \end{array}$$

(7)

where ${\theta }_2$ can be found with Snell’s law $N_{\mathrm{air}}\sin ({\theta }_1)=N_{\mathrm{Al}}(\lambda )\sin ({\theta }_2)$ and the subscripts s and p denote the TE (Transverse Electric) and TM (Transverse Magnetic) polarizations’ state, respectively. The reflected spectrum for the two polarizations is

$$S_{s,p}^{(r)}(\lambda )=S^{(i)}(\lambda )\times R_{s,p}(\lambda )$$

(8)

where $R_{s,p}(\lambda )=r_{s,p}(\lambda )\times r_{s,p}(\lambda {)}^{*}$ is the reflectance for the s and p polarizations as in equation (7), with $r_{s,p}(\lambda {)}^{*}$ being the complex conjugate of $r_{s,p}(\lambda )$ . It is known that the biggest difference between $R_s(\lambda )$ and $R_p(\lambda )$ exists at the Brewster angle ( ${\theta }_B$ ) for dielectrics or at the quasi-Brewster angle ${\theta }_{B\prime }$ (also called the principle angle of incidence ²⁵ ) for metals. The quasi-Brewster angle ${\theta }_{B\prime }$ is 80° at wavelength being 0.8 µm. Figure 8(a) shows the spectral reflectance for ${\theta }_1={\theta }_{B\prime }=80\circ$ , and it is found that the $R_s(\lambda )$ (dotted line) is greater than that of normal incidence (see Figure 4 for Al) and has a very shallow valley at 0.82 µm (the blue circle). However, $R_p(\lambda )$ is smaller than its normal incidence counterpart (see Figure 4 for Al), with a minimum being 55% also at 0.82 µm, and the valley is denoted by the red circle. Although both $R_s(\lambda )$ and $R_p(\lambda )$ have a minimum at λ = 0.82 µm, the very different distribution of the two modifiers still cause the polarization-dependent spectral shift variations. Figure 8(b) and (c) shows the spectrum for $S_s^{(r)}(\lambda )$ and $S_p^{(r)}(\lambda )$ , respectively, for the same $S^{(i)}(\lambda )$ with λ_c = 0.82 µm and the same $\mathrm{\Gamma }$ as in Figure 7. From Figure 8, it is seen that $S_s^{(r)}(\lambda )$ is blueshifted and $S_s^{(r)}(\lambda )$ is redshifted. This feature can be further employed to perform the digital data transmission at arbitrary angles with polarization control. The method is explained, and for further details, section 2.2 in Han and Tseng ²³ can be referred, where silver metal is used. The advantage of using Al over Ag is that again the former has much higher reflectance, benefiting the reflection detection.

Figure 8.

(a) The reflectance of $R_s(\lambda )$ and $R_p(\lambda )$ of Al at ${\theta }_1={\theta }_{B\prime }=80\circ$ . (b) Normalized spectrum for $S_s^{(r)}(\lambda )$ (solid line), $S^{(i)}(\lambda )$ (dotted line), and $R_s(\lambda )$ (dashed line). (c) Normalized spectrum for $S_p^{(r)}(\lambda )$ (solid line), $S^{(i)}(\lambda )$ (dotted line), and $R_p(\lambda )$ (dashed line). The shift of the spectrum maximum is blueshifted in $S_s^{(r)}(\lambda )$ and redshifted in $S_p^{(r)}(\lambda )$ , as indicated by the arrows on top of the figure.

Discussion and conclusion

In this study, we illustrate that the spectral switch can still be found for a simple strictly concave modifier with only one valley or minimum. An example using the aluminum’s reflectance is given to show that such a simple condition still leads to SSW; and as found from Figures 4 and 1(c), the behavior of the aluminum’s reflectance meets the strictly concave function. This condition is obviously simpler than the oscillatory modifiers or modifiers with zero points or phase singularities, as we compare Figure 4 for Al with Figures 2(b) and 3(c). The authors claim that this should be the necessary and sufficient condition for SSW for the following reasons. Referring to Figure 7, it shows that the most critical feature making a spectral switch is the splitting of the original spectrum into two peaks, which occurs when the minimum of the concave modifier locates approximately on the maximum of the convex original spectrum. Without this concave behavior in modifier, the splitting cannot be made (a necessary condition); with it, the splitting can always be made under proper choice of the central wavelength and bandwidth (a sufficient condition). For example, a monotonic increasing or decreasing modifier, like the form $M_3(\omega )={\omega }^2$ , was found and it leads to pure blueshift ² without SSW. Also, the property of a convex modifier function has been examined, and it gives smooth spectral shifts without jumping or switching as shown in Figure 3 in Han ² as expected. However, the oscillatory or singular behaviors of modifier in the previous works, like those shown in Figures 2(b) and 3(c), always contain many minima and valleys, without exceptions. No wonder they were used as the sufficient conditions in the past. In actual fact, they are more sufficient than required. This point can also be explained from mathematical point of view. From equation (4), the resultant spectrum is a product of two functions, the original spectrum and the modifier. As long as the former is a strictly convex function with a maximum and the latter is a strictly concave function with a minimum, the modifier always redistributes the original spectrum by weighting it to different extent. Most importantly, the original spectrum can be split into two parts because the concave modifier minimum weights the spectrum the least, and the other two sides of it weight more. Consequently, the modifier can always make a dip around the maximum of the original spectrum and splits it like the one in Figure 7(b).

Finally, in this work, we proposed that the simplest necessary and sufficient condition for SSW should be a strictly concave modifier with a minimum for a typical convex original spectrum with a maximum. The reflectance of aluminum metal with concave shape and only one minimum is used as the example to illustrate the proposition, with the extra advantage over silver that much higher reflectance can be utilized. Also, the spectral shift amplification effect and polarization-controlled spectral shift are presented. Although this proposition is not rigorously mathematically proved, it is still valuable to serve as a general guide to discover SSW. It also helps clarifying the conditions for appearance of a spectral switch and facilitates finding next SSW under other situations in the future.