Analytical solutions for irradiance of principal and subsidiary maxima in multiple-slit diffraction

Abstract

This paper presents analytical solutions for the irradiance of principal and subsidiary maxima in multiple-slit diffraction with arbitrary slit numbers. By analytically solving the equation for the principal and subsidiary maxima, the irradiance of the principal and subsidiary maxima can be obtained up to the twelfth order in $a/d$, where a is the width of each slit and d is the separation between two adjacent slits.

1. Introduction

Fraunhofer diffraction is an important and fundamental phenomenon in wave optics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. When a monochromatic light with a wavelength λ is illuminated on N-equidistant multiple slits with a slit width (a) and slit separation (d), the irradiance distribution of the diffracted light in the Fraunhofer regime is expressed as:

$$I=I_0{\left(\frac{\sin (\beta /2)}{\beta /2}\frac{\sin (N\varphi /2)}{N\sin (\varphi /2)}\right)}^2,$$ (1)

$$\beta =\frac{2\pi }{\lambda }a\sin \theta =\eta \varphi ,\varphi =\frac{2\pi }{\lambda }d\sin \theta ,$$ (2)

where $I_0$ is the irradiance of the central angular position, and θ is the angular position of the diffracted light. In Eq. (2), η is defined as

$$\eta \equiv a/d,$$

which is assumed to be small in this study. One can see that in Eqs. (1) and (2), the angular dependence is accumulated in the variable ϕ so in what follows, we consider the dependence on ϕ rather than on the other variables such as θ and β; for convenience, this definitive variable ϕ will be called phase.

The typical diffraction patterns calculated using Eq. (1) for $N=5$ are shown in Fig. 1. In this figure, the black curve denotes the diffraction pattern obtained for a negligible slit width ($\eta =0$), and the red curve represents the pattern for $\eta =1/10$. The green curve represents the single-slit diffraction pattern. When the slit width is negligible, the principal maxima (PM) exists at $\varphi =2\pi m$ (where m is an integer), and $N-2$ subsidiary maxima (SM) are observed between two adjacent PM. The positions of the minima, PM, and SM in the case of $\eta \to 0$ are reported in many textbooks on optics [1], [2], [3], [4], [5]. The phases and irradiances for the PM and SM at $\eta \ne 0$ are slightly different from those at $\eta =0$.

Figure 1.

Typical diffraction patterns calculated using Eq. (1) as functions of ϕ for N = 5. The diffraction patterns for η = 0 and η = 1/10 are represented as black and red curves, respectively. The green curve represents the envelope resulting from the single-slit diffraction pattern.

Recently, analytical solutions of the phases (ϕ) of the PM and SM were reported by the author [20], [21], [22]. In particular, the phases of the PM and SM for $\eta \ne 0$ were reported in Refs. [21] and [22], respectively. Although the phases of the PM and SM can be obtained by numerically solving Eq. (1), it would be useful if the analytical solutions are available. In particular, we can assess the difference between the PM and SM phases obtained at $\eta \ne 0$ and $\eta =0$. For the lowest order in η, the phases of the PM (${\varphi }_{\mathrm{PM}}$) and SM (${\varphi }_{\mathrm{SM}}$) vary as follows [21], [22]:

$${\varphi }_{\mathrm{PM}}\approx (1-\frac{{\eta }^2}{N^2-1}){\varphi }_0,{\varphi }_{\mathrm{SM}}\approx (1-\frac{{\eta }^2}{3(N^2-1)}){\psi }_0,$$ (3)

where ${\varphi }_0$ and ${\psi }_0$ are defined as ${\varphi }_{\mathrm{PM}}$ and ${\varphi }_{\mathrm{SM}}$ for $\eta =0$, respectively. To discern the SM phase for $\eta =0$ from the PM phase for $\eta =0$, a different notation (${\psi }_0$) is used for the SM. The analytical results of the phases shown in Eq. (3) provide useful information such as dependence of the phases on the slit number and η, and similarities and differences in the results for the PM and SM phases.

For the PM and SM irradiance, complicated expressions such as Eq. (9) in Ref. [21] and Eq. (22) in Ref. [22] have been reported. However, it is usually difficult to deduce the physical meaning from these complicated results. The analytical expression for the irradiance of the PM (SM) as a function of N and ${\varphi }_0$ (${\psi }_0$) in a power series of η reveals the behavior of the irradiances, and the differences and similarities in the PM and SM irradiances can be determined. Despite the significance of the analytical solutions, only the lowest order of η for the PM irradiance is reported in Ref. [21]. The present investigations were conducted as an extension of the previously reported studies [21], [22] to obtain the analytical solutions of PM and SM irradiances up to the twelfth order in η.

2. Theory

The phases for the PM and SM were previously reported in Ref. [21], [22]. Because the methods for calculating the PM and SM phases are similar, except for the phase at the $\eta \to 0$ limit, the method for calculating the PM phase is elucidated here in brief. The equation $dI/d\varphi =0$ can be solved for $\varphi ={\varphi }_{\mathrm{PM}}$ by inserting $\beta =\eta \varphi$ in Eq. (1) and by using the ϕ expanded in powers of η as follows:

$${\varphi }_{\mathrm{PM}}={\varphi }_0+\sum _{j=1}^{\infty }{a}_{2j}{\eta }^{2j}.$$ (4)

Because the derivative of Eq. (1) with respect to ϕ contains only even order terms of η, only the even order terms in η are considered in Eq. (4) [21], [22]. Further, the phase ${\varphi }_0=2\pi m$ (m: integers) and the values of the coefficients, $a_2$, $a_4$, ⋯, and $a_{12}$ are reported in a previous study [21].

The irradiance can be obtained analytically using the coefficients $a_j$, and the corresponding results are given by

$$I_{\mathrm{PM}}=I_{{\varphi }_0}+I_0\sum _{j=2}^{\infty }{b}_{2j}{\eta }^{2j},$$ (5)

where

$$I_{{\varphi }_0}=I_0\frac{{\sin }^2(\eta {\varphi }_0/2)}{{(\eta {\varphi }_0/2)}^2}.$$ (6)

The irradiance in Eq. (5) also contains only even order terms of η, because Eq. (1) contains even powers of β and ϕ. In Eq. (6), $I_{{\varphi }_0}$ is the irradiance of the envelope at $\varphi ={\varphi }_0=2\pi m$. The zeroth order value of I is defined here as $I_{{\varphi }_0}$ for convenience, although this definition does not apply in reality. By inserting the coefficients $a_{2j}$ presented in Ref. [21], the coefficients $b_{2j}$ in Eq. (5) for $j=2,3,\cdots ,8$ can be deduced as

$$b_4=\frac{{\varphi }_0^2}{12(N^2-1)},b_6=-\frac{{\varphi }_0^2}{12{(N^2-1)}^2}-\frac{{\varphi }_0^4}{240(N^2-1)},b_8=\frac{{\varphi }_0^2}{12{(N^2-1)}^3}+\frac{(2N^2-3){\varphi }_0^4}{720{(N^2-1)}^3}+\frac{{\varphi }_0^6}{11200(N^2-1)},b_{10}=-\frac{{\varphi }_0^2}{12{(N^2-1)}^4}+\frac{(N^2+1){\varphi }_0^4}{360{(N^2-1)}^4}-\frac{(4N^2-5){\varphi }_0^6}{43200{(N^2-1)}^3}-\frac{{\varphi }_0^8}{907200(N^2-1)},b_{12}=\frac{{\varphi }_0^2}{12{(N^2-1)}^5}-\frac{N^2{\varphi }_0^4}{72{(N^2-1)}^5}+\frac{(76N^4-113N^2+55){\varphi }_0^6}{302400{(N^2-1)}^5}+\frac{(2N^2-5){\varphi }_0^8}{4536000{(N^2-1)}^3}+\frac{{\varphi }_0^{10}}{111767040(N^2-1)}.$$

To complete the analytical solutions, $I_{{\varphi }_0}$ in Eq. (6) is expanded up to the twelfth order in η as follows:

$$\begin{gathered} I_{{\varphi }_0}\approx I_0(1-\frac{{\varphi }_0^2{\eta }^2}{12}+\frac{{\varphi }_0^4{\eta }^4}{360}-\frac{{\varphi }_0^6{\eta }^6}{20160}+\frac{{\varphi }_0^8{\eta }^8}{1814400} \\ -\frac{{\varphi }_0^{10}{\eta }^{10}}{239500800}+\frac{{\varphi }_0^{12}{\eta }^{12}}{43589145600}). \end{gathered}$$ (7)

Therefore, Eq. (7) may be included in Eq. (5) instead of $I_0\frac{{\sin }^2(\eta {\varphi }_0/2)}{{(\eta {\varphi }_0/2)}^2}$.

The phases of the SM are also expanded as follows:

$${\varphi }_{\mathrm{SM}}={\psi }_0+\sum _{j=1}^{\infty }{c}_{2j}{\eta }^{2j},$$ (8)

where ${\psi }_0$ denotes the phase ϕ of the SM in the $\eta \to 0$ limit, and the coefficients $c_2$, $c_4$, ⋯, and $c_{12}$ are reported in Ref. [22].

The SM irradiance ($I_{\mathrm{SM}}$) can be derived by inserting Eq. (8) into Eq. (1) as follows:

$$I_{\mathrm{SM}}=\tilde{I}(1+\sum _{j=1}^{\infty }{u}_{2j}{\eta }^{2j}),$$ (9)

where

$$\tilde{I}=\frac{2I_0}{N^2+1+(1-N^2)\cos {\psi }_0}$$

is $I_{\mathrm{SM}}$ at $\eta \to 0$. The coefficients $u_2$, $u_4$, ⋯, and $u_{12}$ in Eq. (9) are explicitly expressed by

$$u_2=-\frac{{\psi }_0^2}{12},u_4=\frac{{\psi }_0^2}{36(N^2-1)}+\frac{{\psi }_0^4}{360},u_6=-\frac{{\psi }_0^2}{108{(N^2-1)}^2}-\frac{\gamma {\psi }_0^3}{324{(N^2-1)}^2}-\frac{{\psi }_0^4}{720(N^2-1)}-\frac{{\psi }_0^6}{20160},u_8=\frac{{\psi }_0^2}{324{(N^2-1)}^3}+\frac{\gamma {\psi }_0^3}{324{(N^2-1)}^3}+\frac{(N^2-2+{\gamma }^2){\psi }_0^4}{3888{(N^2-1)}^3}+\frac{\gamma {\psi }_0^5}{9720{(N^2-1)}^2}+\frac{{\psi }_0^6}{33600(N^2-1)}+\frac{{\psi }_0^8}{1814400},u_{10}=-\frac{{\psi }_0^2}{972{(N^2-1)}^4}-\frac{\gamma {\psi }_0^3}{486{(N^2-1)}^4}+\frac{(2N^2+2-7{\gamma }^2){\psi }_0^4}{11664{(N^2-1)}^4}+\frac{(N^2+5)\gamma {\psi }_0^5}{58320{(N^2-1)}^4}-\frac{(11N^2-16+5{\gamma }^2){\psi }_0^6}{1166400{(N^2-1)}^3}-\frac{\gamma {\psi }_0^7}{510300{(N^2-1)}^2}-\frac{{\psi }_0^8}{2721600(N^2-1)}-\frac{{\psi }_0^{10}}{239500800},u_{12}=\frac{{\psi }_0^2}{2916{(N^2-1)}^5}+\frac{5\gamma {\psi }_0^3}{4374{(N^2-1)}^5}-\frac{(8N^2+2-25{\gamma }^2){\psi }_0^4}{34992{(N^2-1)}^5}-\frac{(33N^2+7-10{\gamma }^2)\gamma {\psi }_0^5}{174960{(N^2-1)}^5}+\frac{(117N^4-119N^2+72-5(39N^2+5){\gamma }^2-50{\gamma }^4){\psi }_0^6}{10497600{(N^2-1)}^5}+\frac{\gamma {\psi }_0^7}{340200{(N^2-1)}^3}+\frac{(N^2-6+5{\gamma }^2){\psi }_0^8}{40824000{(N^2-1)}^3}+\frac{\gamma {\psi }_0^9}{61236000{(N^2-1)}^2}+\frac{{\psi }_0^{10}}{335301120(N^2-1)}+\frac{{\psi }_0^{12}}{43589145600},$$

where

$$\gamma \equiv \cot \frac{{\psi }_0}{2}.$$

Equations (5) and (9) obtained for the PM and SM, respectively, are the main results of this study.

3. Discussion

Typical calculated diffraction patterns for $N=3,\eta =1/5$ and $N=4,\eta =1/4.5$ are presented in Figs. 2(a) and (b), respectively. In Figs. 2(a) and (b), the red dotted curves denote the envelopes solely represented by the single-slit diffraction pattern. Notably, an order is missing at $\varphi =10\pi$ in Fig. 2(a), where $1/\eta$ is an integer [1], whereas all the orders are present in Fig. 2(b), wherein $1/\eta$ is not an integer.

Figure 2.

Calculated diffraction patterns as functions of ϕ for (a) N = 3,η = 1/5 and (b) N = 4,η = 1/4.5. Comparison between the analytical (dots) and numerical (red lines) irradiances at ϕ = ϕ_PM for (c) N = 3,η = 1/5 and (d) N = 4,η = 1/4.5. The irradiances at ϕ₀ = 2π, 4π, 6π, and 8π as functions of the order are displayed as traces from the bottom to the top.

The calculated irradiances at $\varphi ={\varphi }_{\mathrm{PM}}$ for $N=3,\eta =1/5$ and $N=4,\eta =1/4.5$ are presented in Figs. 2(c) and (d), respectively. The results obtained at ${\varphi }_0=2\pi$, 4π, 6π, and 8π are presented in Figs. 2(c) and (d); the dots and red lines denote the analytical and numerical results, respectively. The abscissa, representing the orders, indicates the highest order in the calculation; e.g., the order of 10 indicates the analytical result obtained using Eq. (5), wherein the summation is performed up to the tenth order in η. Notably, the results at the zeroth order imply $I_{{\varphi }_0}$ in Eq. (6). As evident from Figs. 2(c) and (d), the analytical results approach the numerical ones with the increasing order. Although the analytical results up to the twelfth order in η are presented here, the results depicted in Figs. 2(c) and (d) include those of $b_{14}$, $b_{16}$, $b_{18}$, and $b_{20}$ as well. As reported in Ref. [21], the analytical solutions are valid only at $\varphi <2\pi /\eta$.

The typical diffraction patterns for the analysis of the SM are presented in Fig. 3(a), where $N=5$ and $\eta =1/5$. In Fig. 3(a), the red dotted curves, labeled as S₁, S₂, ⋯, and S₁₂, represent the SM enlarged by a factor of ten for clarity. The phases ${\psi }_0$ of S₁, S₂, and S₃ are ${\cos }^{-1}(-\frac{1}{4})$, π, and $2\pi -{\cos }^{-1}(-\frac{1}{4})$, respectively [20], [22]. Further, the phases of S${}_{3j+1}$, S${}_{3j+2}$, and S${}_{3(j+1)}$ are shifted by $2\pi j$ with respect to those of S₁, S₂, and S₃, respectively, for $j=1,2$, and 3.

Figure 3.

(a) Calculated diffraction pattern as a function of ϕ for N = 5 and η = 1/5; the diffraction patterns of the SM are enlarged by a factor of ten for clarity. (b) Calculated analytical and numerical irradiances for S₂, S₄, ⋯, and S₁₂ as functions of the order. The dots and red lines denote the analytical and numerical results, respectively.

The analytical and numerical results for S₂, S₄, S₆, S₈, S₁₀, and S₁₂ are shown in Fig. 3(b). Similar to Fig. 2, the dots and red lines denote the analytical and numerical results, respectively. Fig. 3(b) reveals that analytical results approach the numerical ones as the order increases. Similar to the case of PM, the results of $u_{14}$, $u_{16}$, $u_{18}$, and $u_{20}$ are also included to compare the analytical and numerical results.

4. Conclusions

This paper presents the analytical solutions of the irradiance of PM (SM) up to the twelfth order in η as functions of N and ${\varphi }_0$ (${\psi }_0$). Evidently, the analytical and numerical results are in excellent agreement with each other. Moreover, the analytical results up to approximately the tenth order in η are similar to the corresponding numerical results. Because the analytical solutions for the irradiance are functions of N, ${\varphi }_0$ (${\psi }_0$), and $a/d$, this method can be easily applied to deduce the phase and irradiance of the PM (SM) in a multiple-slit diffraction case. In addition to their practical applications, the analytical solutions of the PM and SM irradiances shown in Eqs. (5) and (9) exhibit different dependences on the zeroth order phase and η, as can be found in the case of the phases [21], [22].

Author contribution statement

Heung-Ryoul Noh: Conceived and designed the analysis; Analyzed and interpreted the data; Contributed analysis tools or data; Wrote the paper.

Funding statement

Professor Heung-Ryoul Noh was supported by National Research Foundation of Korea (NRF) [2020R1A2C1005499].

Declaration of Competing Interest

The authors declare no competing interests.

Data availability

No data was used for the research described in the article.