- Email: [email protected]
Analytical solutions for the principal maxima in multiple slit diffraction: Effects of slit width
Journal Pre-proof Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Heung-Ryoul Noh
PII:
S0030-4026(19)31961-8
DOI:
https://doi.org/10.1016/j.ijleo.2019.164062
Reference:
IJLEO 164062
To appear in:
Optik
Received Date:
9 November 2019
Accepted Date:
11 December 2019
Please cite this article as: Heung-Ryoul Noh, Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width, (2019), doi: https://doi.org/10.1016/j.ijleo.2019.164062
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Elsevier Editorial System(tm) for Optik – International Journal for Light and Electron Optics Manuscript Draft Manuscript Number: IJLEO-D-19-04333 Title: Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Article Type: Full length article Section/Category: All other topics Keywords: principal maxima; multiple slits; diffraction
ro of
Corresponding Author: Professor Heung-Ryoul Noh, Corresponding Author's Institution: Chonnam National First Author: Heung-Ryoul Noh Order of Authors: Heung-Ryoul Noh
University
re
-p
Abstract: We present analytical solutions for the principal maxima in multiple slit diffraction. By expanding the angular position of the principal maxima in orders of $a/d$ where $d$ is the distance between two adjacent slits and $a$ is the width of each slit, we obtain the angular position of the principal maxima up to the twelfth order in $a/d$.
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Suggested Reviewers: Xinye Xu Professor, Physics Department, East China Norman University [email protected]
na
Charles Adams Professor, Physics Department, Durham University [email protected] Testuo Kishimoto Professor, Physics Department, University of Electrocommunications [email protected]
ur
Young-Tak Chough Professor, Physics Department, Gwangju University [email protected]
Jo
Matt Jones Professor, Physics Department, Durham University [email protected] Opposed Reviewers:
Author Agreement
Title: Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Author: Heung-Ryoul Noh
Author Agreement
I certify that all authors have seen and approved the final version of the manuscript being submitted. I warrant that the article is the authors' original work, hasn't received prior publication and isn't under
Jo
ur
na
lP
re
-p
ro of
consideration for publication elsewhere.
*Impact Statement
Impact Statement Dear editor, This is Heung-Ryoul Noh of Chonnam National University, Gwangju, Korea. I am now submitting a paper entitled “Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width’’. This work includes an analytical study on the effect of the
ro of
finite width of slits on the angular position of the principal maxima in the intensity distribution pattern diffracted from multiple slits with arbitrary slit numbers. Although the diffraction phenomena from multiple slits have been well known, the accurate positions of the principal maxima depending on the width of the slit have not been obtained to the best of our knowledge.
I calculated accurate position of the principal maxima analytically up to the 12th order in the slit
-p
width. I think the submitted paper will be helpful for those who are working on optics and laser
re
spectroscopy.
Jo
ur
na
Heung-Ryoul Noh
lP
Sincerely yours,
*Cover Letter
Nov. 9, 2019
Dear editor, This is Heung-Ryoul Noh of Chonnam National University, Gwangju, Korea. I am now submitting a paper entitled “Analytical solutions for the principal maxima in multiple slit
ro of
diffraction: effects of slit width’’. This work includes an analytical study on the effect of the finite width of slits on the angular position of the principal maxima in the intensity distribution
pattern diffracted from multiple slits with arbitrary slit numbers. I think the submitted paper will
-p
be helpful for those who are working on optics and laser spectroscopy.
lP
My full information is as follows.
re
Sincerely yours,
Prof. Heung-Ryoul Noh
na
Department of Physics,
Chonnam National University Gwangju 61186, Korea
ur
Tel: +82-62-530-3366
Fax: +82-62-530-3369
Jo
Email: [email protected]
Sincerely yours, Heung-Ryoul Noh
*Manuscript
Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Heung-Ryoul Noh∗
of
Department of Physics, Chonnam National University, Gwangju 61186, Korea
ro
Abstract
We present analytical solutions for the principal maxima in multiple slit
-p
diffraction. By expanding the angular position of the principal maxima in orders of a/d where d is the distance between two adjacent slits and a is the
up to the twelfth order in a/d.
lP
Keywords:
re
width of each slit, we obtain the angular position of the principal maxima
Principal Maxima, Multiple Slits, Diffraction
ur na
PACS: 42.25.Fx, 42.25.Hz, 01.55.+b 1. Introduction
When a monochromatic light is diffracted from multiple slits with an
equidistant slit separation (d) and finite width (a), the intensity distribution of the diffracted light in the Fraunhofer regime is given by
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
∗
I = I0
sin2 (β/2) sin2 (Nφ/2) , (β/2)2 N 2 sin2 (φ/2)
(1)
Corresponding author Email address: [email protected] (Heung-Ryoul Noh)
Preprint submitted to Optik
November 9, 2019
where N is the number of slits and I0 is the intensity at the central angular position. Here, β =
2π a sin θ λ
and φ =
2π d sin θ, λ
where λ is the wavelength
of light, and θ is the angular position of the diffracted beam. The intensity minima, principal maxima (PM), and subsidiary maxima in the diffraction pattern in Eq. (1) are well-known in textbooks on optics [1, 2, 3, 4]. Recently,
of
accurate analytical solutions for the subsidiary maxima in the multiple slit
interference with a negligible slit width were reported [5]. In the limit of
ro
η(≡ a/d) → 0, the PM occurs at φ = 2πm where m = 0, 1, 2, · · ·. This results in the well-known formula for the PM, d sin θ = mλ. However, when
-p
η 6= 0, the phases of the PM differ from φ = 2πm. In this paper, we present
re
an analytical study of the phases φ for the PM as functions of N and η. 2. Calculation
dI dφ
lP
To calculate the angular positions of the PM, we solve the equation, = 0, for φ = φPM , where φPM is the phase for the PM, as given by the
ur na
following expansion:
φPM = φ0 +
X
a2j η 2j .
(2)
j=1
We note that the derivative of Eq. (2) with respect to φ contains only even order terms of η: thus, the expansion in Eq. (2) is justified. Because the central PM at φ = 0 is unaltered, in Eq. (2), φ0 = 2π, 4π, 6π, · · ·. By solving
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
each order in η, we obtain the coefficients in Eq. (2) as functions of N and
φ0 .
From the equations of order η 2 , we obtain the following: a2 = − 2
φ0 . −1
N2
(3)
Calculating similarly for higher orders, we obtain the coefficients, a4 , a6 , · · ·, a12 as follows:
(5)
(6)
(7)
ur na
a12 =
lP
re
a10 =
(4)
of
a8 =
ro
a6 =
φ30 φ0 − , (N 2 − 1)2 60 (N 2 − 1) φ0 (5N 2 − 3) φ30 − + (N 2 − 1)3 60 (N 2 − 1)3 φ50 , − 2520 (N 2 − 1) (7N 2 − 3) φ30 φ0 − (N 2 − 1)4 30 (N 2 − 1)4 φ70 (17N 2 − 10) φ50 − + , 100 800 (N 2 − 1) 4200 (N 2 − 1)3 (3N 2 − 1) φ30 φ0 + − (N 2 − 1)5 6 (N 2 − 1)5 (268N 4 − 341N 2 + 100) φ50 − 12 600 (N 2 − 1)5 (251N 2 − 149) φ70 φ90 + − , 1 512 000 (N 2 − 1)3 3 991 680 (N 2 − 1) (11N 2 − 3) φ30 φ0 − (N 2 − 1)6 12 (N 2 − 1)6 (985N 4 − 1031N 2 + 250) φ50 + 12 600 (N 2 − 1)6 (2055N 4 − 2528N 2 + 743) φ70 − 1 512 000 (N 2 − 1)5 (21 587N 2 − 12 963) φ90 + 3 492 720 000 (N 2 − 1)3 691φ11 0 . − 108 972 864 000 (N 2 − 1)
-p
a4 =
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(8)
When 1/η is an integer, the PM at φ0 = 2π/η is absent and is called a missing
order [1]. Because the highest order term in coefficient a2j is proportional to φ02j−1, the PM at φ0 = 2π/η−2π converges, whereas the PM at φ0 = 2π/η+2π 3
does not converge as η becomes smaller. Therefore, our results are valid for the PMs at φ < 2π/η. The intensity at PM, calculated by inserting Eq. (2) into Eq. (1), is given by sin2 (ηφPM /2) sin2 (NφPM /2) , (ηφPM /2)2 N 2 sin2 (φPM /2)
Iφ 0 = I0
sin2 (ηφ0 /2) . (ηφ0 /2)2
ro
and the intensity at φ = φ0 is given by
(9)
of
IPM = I0
(10)
is given by up to the fourth order in η
re
3. Discussion
φ20 η 4 + O(η 6 ). 12(N 2 − 1)
lP
IPM − Iφ0 =
-p
Then, it is worth mentioning that the difference between Eqs. (9) and (10)
Figure 1(a) shows typical calculated diffraction patterns for N = 3 and
ur na
η = 1/5. Because η = 1/5, the PM at φ0 = 10π vanishes. The calculated φPM for these conditions are shown in Fig. 1(b). In Fig. 1(b), the results for φ0 = 2π, 4π, 6π, and 8π are presented from the bottom to the top panels. The red lines denote the numerically calculated results and the dots represent the analytical results obtained from Eq. (2) using the coefficients presented previously. Here, the orders 2, 4, · · ·, 12 imply the highest order considered in
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
each calculated result. As can be expected, the analytical results approach the numerical results as higher-order terms are included. It can also be noticed that that the results for smaller φ approach the numerical results faster than in the case of larger φ. 4
The dependence of φPM on N is shown in Fig. 1(c). Fig. 1(c) shows the results for φ0 = 2π and 8π when η = 1/5. The insets show the differences in the numerical (φnum ) and analytical (φana ) results of φPM . As can be readily expected from Eqs. (3)–(8), we find that φPM approaches φ0 fast as N increases, as shown in Fig. 1(c). The deviations of the analytical results
of
from the numerical results are smaller for the smaller φ0 values.
ro
4. Conclusions
-p
In summary, we presented an analytical study on the effect of the finite width of slits on the angular position of the PM in the intensity distribution pattern diffracted from multiple slits. Equation (2) and the coefficients
re
given in Eqs. (3)–(8) are our main results. We obtained the analytical results up to the twelfth order in η and found excellent agreement between the
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numerical and analytical results. Because our results are valid even for arbitrary slit numbers, these analytical results can be used in easy expectation
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of diffraction patterns and for pedagogical purposes.
Funding
This research was supported by Basic Science Research Program through
the National Research Foundation of Korea(NRF) funded by the Ministry of
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Science, ICT and future Planning(2017R1A2B4003483). References [1] E. Hecht, Optics, Addison-Wesley, Reading, 2002. 5
[2] M. Born, E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. [3] F.L. Pedrotti, L.S. Pedrotti, L.M. Pedrotti, Introduction to Optics, Prentice-Hall, Englewood Cliffs, 2007.
of
[4] B.D. Guenther, Modern Optics, Oxford University Press, New York,
ro
2015.
[5] H.R. Noh, Analytical solutions for the subsidiary maxima in multiple
-p
slit interference, Opt. Rev. 2019. https://doi.org/10.1007/s10043-019-
re
00526-2.
lP
FIGURES
• FIG. 1: (a) Typical calculated diffraction pattern using Eq. (1) as a function of φ for N = 3 and η = 1/5. (b) Typical analytical and
ur na
numerical results for φPM when N = 3 and η = 1/5, depending on the
order of η. (c) Dependence of φPM on N when η = 1/5 and φ0 = 2π (bottom panel) and 8π (top panel).
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
6
4
6
f/p
8
f 0=8 p X10
-3
h =1/5
0
7.8
2.00
2
4
6
f 0=2 p
1.99
X10
-11
8
ur na
-4
1.97
2
2
4
4
6
6
8
f 0=6 p
N=3 h =1/5
5.96
3.980
f 0=4 p
10
f 0=2 p
1.9898
10
8
3.978
1.9900
( f ana- f num)/ p
0
1.98
5.97
lP
f PM/ p
2
7.92
10
Numerical Analytical
7.9
ro
of 2
-p
8.0
0
Analytical Numerical
f 0=8 p
re
(c)
N=3 h =1/5
0.5 0.0
(b) 7.96
f 0=2 p f 0=4 p f 0=6 p f 0=8 p
f PM/ p
I/I0
(a)1.0
2
10
4
6
8
Order
N
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
7
10
12
Highlights
Title: Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Author: Heung-Ryoul Noh
Highlights
Investigating analytical solutions for the principal maxima in multiple slit diffraction.
•
Studying the effect of slit width on the angular position of the principal maxima.
•
Obtaining the angular position of the principal maxima up to the twelfth order in slit width.
Jo
ur
na
lP
re
-p
ro of
•
*Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Jo
ur na
lP
re
-p
ro
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
PII:
S0030-4026(19)31961-8
DOI:
https://doi.org/10.1016/j.ijleo.2019.164062
Reference:
IJLEO 164062
To appear in:
Optik
Received Date:
9 November 2019
Accepted Date:
11 December 2019
Please cite this article as: Heung-Ryoul Noh, Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width, (2019), doi: https://doi.org/10.1016/j.ijleo.2019.164062
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Elsevier Editorial System(tm) for Optik – International Journal for Light and Electron Optics Manuscript Draft Manuscript Number: IJLEO-D-19-04333 Title: Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Article Type: Full length article Section/Category: All other topics Keywords: principal maxima; multiple slits; diffraction
ro of
Corresponding Author: Professor Heung-Ryoul Noh, Corresponding Author's Institution: Chonnam National First Author: Heung-Ryoul Noh Order of Authors: Heung-Ryoul Noh
University
re
-p
Abstract: We present analytical solutions for the principal maxima in multiple slit diffraction. By expanding the angular position of the principal maxima in orders of $a/d$ where $d$ is the distance between two adjacent slits and $a$ is the width of each slit, we obtain the angular position of the principal maxima up to the twelfth order in $a/d$.
lP
Suggested Reviewers: Xinye Xu Professor, Physics Department, East China Norman University [email protected]
na
Charles Adams Professor, Physics Department, Durham University [email protected] Testuo Kishimoto Professor, Physics Department, University of Electrocommunications [email protected]
ur
Young-Tak Chough Professor, Physics Department, Gwangju University [email protected]
Jo
Matt Jones Professor, Physics Department, Durham University [email protected] Opposed Reviewers:
Author Agreement
Title: Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Author: Heung-Ryoul Noh
Author Agreement
I certify that all authors have seen and approved the final version of the manuscript being submitted. I warrant that the article is the authors' original work, hasn't received prior publication and isn't under
Jo
ur
na
lP
re
-p
ro of
consideration for publication elsewhere.
*Impact Statement
Impact Statement Dear editor, This is Heung-Ryoul Noh of Chonnam National University, Gwangju, Korea. I am now submitting a paper entitled “Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width’’. This work includes an analytical study on the effect of the
ro of
finite width of slits on the angular position of the principal maxima in the intensity distribution pattern diffracted from multiple slits with arbitrary slit numbers. Although the diffraction phenomena from multiple slits have been well known, the accurate positions of the principal maxima depending on the width of the slit have not been obtained to the best of our knowledge.
I calculated accurate position of the principal maxima analytically up to the 12th order in the slit
-p
width. I think the submitted paper will be helpful for those who are working on optics and laser
re
spectroscopy.
Jo
ur
na
Heung-Ryoul Noh
lP
Sincerely yours,
*Cover Letter
Nov. 9, 2019
Dear editor, This is Heung-Ryoul Noh of Chonnam National University, Gwangju, Korea. I am now submitting a paper entitled “Analytical solutions for the principal maxima in multiple slit
ro of
diffraction: effects of slit width’’. This work includes an analytical study on the effect of the finite width of slits on the angular position of the principal maxima in the intensity distribution
pattern diffracted from multiple slits with arbitrary slit numbers. I think the submitted paper will
-p
be helpful for those who are working on optics and laser spectroscopy.
lP
My full information is as follows.
re
Sincerely yours,
Prof. Heung-Ryoul Noh
na
Department of Physics,
Chonnam National University Gwangju 61186, Korea
ur
Tel: +82-62-530-3366
Fax: +82-62-530-3369
Jo
Email: [email protected]
Sincerely yours, Heung-Ryoul Noh
*Manuscript
Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Heung-Ryoul Noh∗
of
Department of Physics, Chonnam National University, Gwangju 61186, Korea
ro
Abstract
We present analytical solutions for the principal maxima in multiple slit
-p
diffraction. By expanding the angular position of the principal maxima in orders of a/d where d is the distance between two adjacent slits and a is the
up to the twelfth order in a/d.
lP
Keywords:
re
width of each slit, we obtain the angular position of the principal maxima
Principal Maxima, Multiple Slits, Diffraction
ur na
PACS: 42.25.Fx, 42.25.Hz, 01.55.+b 1. Introduction
When a monochromatic light is diffracted from multiple slits with an
equidistant slit separation (d) and finite width (a), the intensity distribution of the diffracted light in the Fraunhofer regime is given by
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
∗
I = I0
sin2 (β/2) sin2 (Nφ/2) , (β/2)2 N 2 sin2 (φ/2)
(1)
Corresponding author Email address: [email protected] (Heung-Ryoul Noh)
Preprint submitted to Optik
November 9, 2019
where N is the number of slits and I0 is the intensity at the central angular position. Here, β =
2π a sin θ λ
and φ =
2π d sin θ, λ
where λ is the wavelength
of light, and θ is the angular position of the diffracted beam. The intensity minima, principal maxima (PM), and subsidiary maxima in the diffraction pattern in Eq. (1) are well-known in textbooks on optics [1, 2, 3, 4]. Recently,
of
accurate analytical solutions for the subsidiary maxima in the multiple slit
interference with a negligible slit width were reported [5]. In the limit of
ro
η(≡ a/d) → 0, the PM occurs at φ = 2πm where m = 0, 1, 2, · · ·. This results in the well-known formula for the PM, d sin θ = mλ. However, when
-p
η 6= 0, the phases of the PM differ from φ = 2πm. In this paper, we present
re
an analytical study of the phases φ for the PM as functions of N and η. 2. Calculation
dI dφ
lP
To calculate the angular positions of the PM, we solve the equation, = 0, for φ = φPM , where φPM is the phase for the PM, as given by the
ur na
following expansion:
φPM = φ0 +
X
a2j η 2j .
(2)
j=1
We note that the derivative of Eq. (2) with respect to φ contains only even order terms of η: thus, the expansion in Eq. (2) is justified. Because the central PM at φ = 0 is unaltered, in Eq. (2), φ0 = 2π, 4π, 6π, · · ·. By solving
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
each order in η, we obtain the coefficients in Eq. (2) as functions of N and
φ0 .
From the equations of order η 2 , we obtain the following: a2 = − 2
φ0 . −1
N2
(3)
Calculating similarly for higher orders, we obtain the coefficients, a4 , a6 , · · ·, a12 as follows:
(5)
(6)
(7)
ur na
a12 =
lP
re
a10 =
(4)
of
a8 =
ro
a6 =
φ30 φ0 − , (N 2 − 1)2 60 (N 2 − 1) φ0 (5N 2 − 3) φ30 − + (N 2 − 1)3 60 (N 2 − 1)3 φ50 , − 2520 (N 2 − 1) (7N 2 − 3) φ30 φ0 − (N 2 − 1)4 30 (N 2 − 1)4 φ70 (17N 2 − 10) φ50 − + , 100 800 (N 2 − 1) 4200 (N 2 − 1)3 (3N 2 − 1) φ30 φ0 + − (N 2 − 1)5 6 (N 2 − 1)5 (268N 4 − 341N 2 + 100) φ50 − 12 600 (N 2 − 1)5 (251N 2 − 149) φ70 φ90 + − , 1 512 000 (N 2 − 1)3 3 991 680 (N 2 − 1) (11N 2 − 3) φ30 φ0 − (N 2 − 1)6 12 (N 2 − 1)6 (985N 4 − 1031N 2 + 250) φ50 + 12 600 (N 2 − 1)6 (2055N 4 − 2528N 2 + 743) φ70 − 1 512 000 (N 2 − 1)5 (21 587N 2 − 12 963) φ90 + 3 492 720 000 (N 2 − 1)3 691φ11 0 . − 108 972 864 000 (N 2 − 1)
-p
a4 =
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
(8)
When 1/η is an integer, the PM at φ0 = 2π/η is absent and is called a missing
order [1]. Because the highest order term in coefficient a2j is proportional to φ02j−1, the PM at φ0 = 2π/η−2π converges, whereas the PM at φ0 = 2π/η+2π 3
does not converge as η becomes smaller. Therefore, our results are valid for the PMs at φ < 2π/η. The intensity at PM, calculated by inserting Eq. (2) into Eq. (1), is given by sin2 (ηφPM /2) sin2 (NφPM /2) , (ηφPM /2)2 N 2 sin2 (φPM /2)
Iφ 0 = I0
sin2 (ηφ0 /2) . (ηφ0 /2)2
ro
and the intensity at φ = φ0 is given by
(9)
of
IPM = I0
(10)
is given by up to the fourth order in η
re
3. Discussion
φ20 η 4 + O(η 6 ). 12(N 2 − 1)
lP
IPM − Iφ0 =
-p
Then, it is worth mentioning that the difference between Eqs. (9) and (10)
Figure 1(a) shows typical calculated diffraction patterns for N = 3 and
ur na
η = 1/5. Because η = 1/5, the PM at φ0 = 10π vanishes. The calculated φPM for these conditions are shown in Fig. 1(b). In Fig. 1(b), the results for φ0 = 2π, 4π, 6π, and 8π are presented from the bottom to the top panels. The red lines denote the numerically calculated results and the dots represent the analytical results obtained from Eq. (2) using the coefficients presented previously. Here, the orders 2, 4, · · ·, 12 imply the highest order considered in
Jo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
each calculated result. As can be expected, the analytical results approach the numerical results as higher-order terms are included. It can also be noticed that that the results for smaller φ approach the numerical results faster than in the case of larger φ. 4
The dependence of φPM on N is shown in Fig. 1(c). Fig. 1(c) shows the results for φ0 = 2π and 8π when η = 1/5. The insets show the differences in the numerical (φnum ) and analytical (φana ) results of φPM . As can be readily expected from Eqs. (3)–(8), we find that φPM approaches φ0 fast as N increases, as shown in Fig. 1(c). The deviations of the analytical results
of
from the numerical results are smaller for the smaller φ0 values.
ro
4. Conclusions
-p
In summary, we presented an analytical study on the effect of the finite width of slits on the angular position of the PM in the intensity distribution pattern diffracted from multiple slits. Equation (2) and the coefficients
re
given in Eqs. (3)–(8) are our main results. We obtained the analytical results up to the twelfth order in η and found excellent agreement between the
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numerical and analytical results. Because our results are valid even for arbitrary slit numbers, these analytical results can be used in easy expectation
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of diffraction patterns and for pedagogical purposes.
Funding
This research was supported by Basic Science Research Program through
the National Research Foundation of Korea(NRF) funded by the Ministry of
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Science, ICT and future Planning(2017R1A2B4003483). References [1] E. Hecht, Optics, Addison-Wesley, Reading, 2002. 5
[2] M. Born, E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999. [3] F.L. Pedrotti, L.S. Pedrotti, L.M. Pedrotti, Introduction to Optics, Prentice-Hall, Englewood Cliffs, 2007.
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[4] B.D. Guenther, Modern Optics, Oxford University Press, New York,
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2015.
[5] H.R. Noh, Analytical solutions for the subsidiary maxima in multiple
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slit interference, Opt. Rev. 2019. https://doi.org/10.1007/s10043-019-
re
00526-2.
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FIGURES
• FIG. 1: (a) Typical calculated diffraction pattern using Eq. (1) as a function of φ for N = 3 and η = 1/5. (b) Typical analytical and
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numerical results for φPM when N = 3 and η = 1/5, depending on the
order of η. (c) Dependence of φPM on N when η = 1/5 and φ0 = 2π (bottom panel) and 8π (top panel).
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6
4
6
f/p
8
f 0=8 p X10
-3
h =1/5
0
7.8
2.00
2
4
6
f 0=2 p
1.99
X10
-11
8
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-4
1.97
2
2
4
4
6
6
8
f 0=6 p
N=3 h =1/5
5.96
3.980
f 0=4 p
10
f 0=2 p
1.9898
10
8
3.978
1.9900
( f ana- f num)/ p
0
1.98
5.97
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f PM/ p
2
7.92
10
Numerical Analytical
7.9
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of 2
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8.0
0
Analytical Numerical
f 0=8 p
re
(c)
N=3 h =1/5
0.5 0.0
(b) 7.96
f 0=2 p f 0=4 p f 0=6 p f 0=8 p
f PM/ p
I/I0
(a)1.0
2
10
4
6
8
Order
N
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7
10
12
Highlights
Title: Analytical solutions for the principal maxima in multiple slit diffraction: effects of slit width Author: Heung-Ryoul Noh
Highlights
Investigating analytical solutions for the principal maxima in multiple slit diffraction.
•
Studying the effect of slit width on the angular position of the principal maxima.
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Obtaining the angular position of the principal maxima up to the twelfth order in slit width.
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*Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: