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A simple description of near-field and far-field diffraction
Accepted Manuscript A simple description of near-field and far-field diffraction Wipawee Temnuch, Sarayut Deachapunya, Pituk Panthong, Surasak Chiangga, Sorakrai Srisuphaphon
PII: DOI: Reference:
S0165-2125(18)30010-6 https://doi.org/10.1016/j.wavemoti.2018.01.002 WAMOT 2220
To appear in:
Wave Motion
Received date : 24 April 2017 Revised date : 28 December 2017 Accepted date : 8 January 2018 Please cite this article as: W. Temnuch, S. Deachapunya, P. Panthong, S. Chiangga, S. Srisuphaphon, A simple description of near-field and far-field diffraction, Wave Motion (2018), https://doi.org/10.1016/j.wavemoti.2018.01.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
*Research Highlights
Highlights 1. We present the simple method for explanation both the near-field and far-field diffractions with the same expressions. 2. Our method is based on the Huygens' principle alone. 3. Our method can identify the boundary between the near-field and far-field diffractions. 4. Our method is verified nicely by both the near-field and far-field, i.e. The Talbot effect, the single slit and the double-slit diffraction.
*Manuscript (Clear) Click here to view linked References
A simple description of near-field and far-field diffraction Wipawee Temnucha,c , Sarayut Deachapunyab,c , Pituk Panthonga , Surasak Chianggaa , Sorakrai Srisuphaphonb,∗ a
Department of Physics, Faculty of Science, Kasetsart University, Bangkok Province, 10900, Thailand b Department of Physics, Faculty of Science, Burapha University, Chonburi Province, 20131, Thailand c Thailand Center of Excellence in Physics, Commission on Higher Education, 328 Si Ayutthaya Road, Bangkok 10400, Thailand
Abstract We present an alternative explanation of near-field and far-field diffraction, i.e., the Talbot effect and single- and double-slit experiments, with the same expressions. Our method is based on the superposition of waves with different path lengths. A simple experiment is conducted as a demonstration. This simple experiment exhibits the transition from near-field to far-field diffraction. Keywords: Wave optics; Diffraction gratings; Wave propagation 1. Introduction Diffraction phenomena have been classified into two regimes, i.e., nearfield and far-field diffraction. Near-field effects rely on Fresnel diffraction, whereas far-field effects are based on Fraunhofer diffraction. One of the wellknown optical near-field effects is the Talbot effect, discovered by Henry Fox Talbot in 1836 [1]. When plane-parallel light falls onto a grating, it will generate a periodic interference pattern. This pattern will be similar to the grating if the observed screen is located at the Talbot length (LT ) [2, 3, 4, 5]. Since the Talbot effect was discovered, numerous theories have been introduced [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Those theories describe the effect in different ways. However, there is an alternative method that can be used to simplify ∗
Corresponding author Email address: [email protected] (Sorakrai Srisuphaphon)
Preprint submitted to Wave Motion
December 28, 2017
the Fresnel diffraction theory. It is based on wave propagation through the transmission function of a diffraction grating [5]. The Talbot effect has been widely used in many fields in both classical [12, 13, 14, 15, 16, 17, 18, 19] and quantum optics [20, 21, 22, 23, 24, 25]. Among them, the interesting notion of quantum carpets or carpets of light were explored [4]. Studies of the Talbot effect with completely controllable multiwave mixing signals were reported [26]. The angular Talbot effect has also been introduced [27]. The Talbot effect was explored with monolayer graphene sheet arrays [28]. Recently, a graphene sheet has been used as a diffraction grating for an electron Talbot effect experiment [29]. The Talbot effect can be exploited in many applications, for example, displacement sensors [15], surface profiling [16], X-ray phase imaging [17], vibration sensors [18], focal length measurement [30], wavefront sensors [31], digital holography [32], and lithography [33]. In telecommunications, such as in antenna design [34, 35], the near-field and far-field diffraction regimes are distinguished by the Fraunhofer distance and the near-field regime must be shorter than this distance limit. Therefore, it is important to study and simplify near-field and far-field diffraction for any application. In this paper, we introduce an alternative method that can explain both near-field and far-field diffraction. This method is based on the superposition of waves through the opening apertures. Simulations using our method show that it describes both near-field and far-field effects. We prove our method with a simple experimental setup. 2. Theory and numerical results We first consider the diffraction pattern of light behind a grating in both the far-field and near-field regimes. The intensity patterns in the intermediate area between the two regimes will be evaluated by considering the superposition of waves propagating through a static grating for which the time dependence of the field can be omitted. Here, a binary grating period d is placed along the x0 axis as shown in Fig. 1. By dividing the slit window into M points, a position labelled by j in the mth slit can be represented as xm,j = (|m| − 1)d +
jd d + . 4 2M
(1)
The parameter m is the order of the grating window, and d is the grating period. According to Huygens’ principle, let xm,j be a point source originating 2
Figure 1: A plane wave incident onto the grating along the x0 axis and propagating from the mth slit to the screen behind the grating with N slits.
√ a wave propagating a distance R = z 2 + (x − xm,j )2 to a position (x, z) on the screen. By using the Fresnel approximation, the distance R can be approximated as R ≃ z + (x − xm,j )2 /2z. Therefore, the wave function with wave number k = 2π/λ and wavelength λ at a position (x, z) is given by ψ(x, z) =
N/2 M ∑ ∑
m=−N/2 j=0
A ik(z+ (x−xm,j )2 ) 2z e , NM
(2)
where A represents the amplitude of the incident wave. For a symmetric grating, as shown in Fig. 1 with x = 0 at the middle position of the x axis, the wave function can be written as ψ(x, z) = ψ+ (x, z) + ψ− (x, z) N/2 M ik(x+xm,j )2 } A ikz ∑ ∑ { ik(x−xm,j )2 2z 2z e e . = +e NM m=1 j=0
(3)
The wave functions ψ± (x, z) correspond to a free wave that is generated from every point of xm,j on the upper and lower parts of the grating, respectively. Subsequently, the intensity distribution can be obtained as I = ψ ∗ ψ. The 3
obtained wave function given in Eq.(3) provides the diffraction patterns from near-field to far-field regimes with only one equation. Some simulations of carpets of light are presented in the following. The optical carpets exhibit variation of their interference patterns as a function of the distance behind the grating in the near-field range [5]. The far-field diffraction patterns are also presented in this paper. Nevertheless, the wave function ψ(x, z), which is in terms of a series, can be applied to estimate the intermediate zone between the two fields as mentioned. According to far-field diffraction, for a grating with an even number of N slits as shown in Fig. 1, the centre of the screen (x = 0) will be the position of the centre bright fringe at any distance z. However, the dark fringe can only occur in the near-field diffraction regime, as found in the Talbot effect. Therefore, one can find the zone by evaluating the shortest distance that gives the continuous bright fringe along the z axis at x = 0. The proposed distance, letting z = Z, may be used as the starting point of the far-field diffraction regime. Firstly, we consider the simplest case of N = 2 (double slits), in which case the wave function in Eq.(3) at x = 0 is reduced to M
A ia2z ∑ ib2z j ic2z2j2 e e M e M , M j=0
(4)
( ) d2 πLT =k z+ , b2z = c2z = , 32z 4z
(5)
ψ(x = 0, z) = where a2z
with the so-called Talbot length given by LT = d2 /λ.
(6)
The second exponential function in the series in Eq.(4) can be written in the series form ∞ ∑ ic2z j 2 1 ( ic2z j 2 )q e M2 = . (7) 2 q! M q=0 Therefore Eq. (4) becomes ∞
A ia2z ∑ 1 ψ(0, z) = e M q! q=0
( 4
ic2z M2
)q ∑ M j=0
j 2q e
ib2z j M
.
(8)
Figure 2: (a) Optical carpet in the xz plane for z = πLT /4 = 59.05 mm obtained from the modulus square of Eq.(3) for d = 200 µm, λ = 532 nm, M = 500, and N = 2. (b) Intensity distribution along the longitudinal direction (z = 6d ≃ 1.2 mm to z = 75 mm) at the centre of the grating. The vertical dashed line indicates the transition from near-field to far-field regimes. (c) and (d) show the optical carpet and the longitudinal intensity distribution (z = 4 to z = 17.5 mm) with the same parameters in the near-field diffraction regime.
5
To calculate the second series in Eq.(8), we apply the geometric series formula M ∑
e
iχj M
=
j=0
1−e
iχ(M +1) M iχ
1 − eM
,
(9)
with the series condition χ < 1. For a large number of waves (M ≫ 1), the above series can be approximated as M ∑ j=0
iχ(M )
e
iχj M
1−e M eiχ − 1 ≃ = . iχ iχ 1 − (1 + M ) M
(10)
Therefore, the second series in Eq.(8) with j 2q may be written in the form of a derivative as ) ( )2q 2q ( iχ )2q 2q M ( M ∑ ∑ iχj iχj M M d d e − 1 j 2q e M = eM ≃ . (11) iχ 2q 2q i dχ i dχ M j=0 j=0 Using the above formula with χ = b2z , we obtain the wave function (Eq.(8)) in the form of infinite series as ( ) ∞ ∑ 1 cq2z d2q 2ieib2z /2 sin(b2z /2) ia2z ψ(0, z) ≃ A e . (12) q! iq+1 db2q b2z 2z q=0 In the limit of large distance z, sin(b2z /2) ≃ b2z /2, and the derivative can be calculated as ( ) ( ) ( )2q d2q 2ieib2z /2 sin(b2z /2) d2q ieib2z /2 b2z i ib2z /2 ≃ 2q = ie . (13) 2q b2z b2z 2 db2z db2z The wave function, in the far-field regime, is then reduced to ( )q ∞ ∑ 1 ic2z ia2z ib2z /2 ψ(0, z) ≃ A e e = A eia2z +ib2z /2+ic2z /4 , q! 4 q=0
(14)
which corresponds to the constant intensity I = ψ ∗ ψ = A2 as assumed in Eq.(3). Nevertheless, according to the series condition χ = b2z < 1, the series does not diverge only if z>
πLT . 4 6
(15)
This suggests that the wave function in the near-field regime cannot be described by the series in Eq.(12). However, the wave function given by Eq.(3) can still be applied in the two regimes. Fig. 2(a) shows the optical carpet of I = ψ ∗ ψ from Eq. (3) with N = 2 for z = πLT /4; the intensity distribution along the z axis (z = 6d ≃ 1.2 mm to z = 75 mm) at x = 0 is shown in Fig. 2(b). It is clearly seen that the middle bright fringe obviously appears at the distance z = πLT /4, which may be used as the boundary distance Z between near-field and far-field diffraction. The near-field diffraction pattern can be found where z < πLT /4 (Figs. 2(c) and (d)). Therefore, we will employ this idea to find the distance Z for the other gratings, N > 2.
Figure 3: Optical carpets in the xz plane (d = 200 µm, λ = 532 nm, and M = 500) for Z = πLT (2N − 3)/4 obtained from the modulus square of Eq.(3) for N = 4 (a), 8 (b), 12 (c), and 20 (d). The dashed lines represent the triangle with a base of N d and the distance Z, which both enclose the diffraction patterns in the near-field regime.
7
According to Eq.(3) but N ̸= 2, the wave function at x = 0 is given by N/2 M A ∑ iaN z ∑ ibN z j icN z2j2 e M e M , ψ(x = 0, z) = e N M m=1 j=0
where aN z
) ( d2 πLT πLT 2 (4m − 3) , bN z = (4m − 3), cN z = . =k z+ 32z 4z 4z
(16)
(17)
Because N is a finite number equalling the number of slits of the grating, the summation over j needs to give a converged result. In analogy with the case of N = 2, we also have the condition bN z < 1. The maximum value of m is limited at N/2, thus the condition may be given by z>
πLT (2N − 3). 4
(18)
Therefore, the limit of near-field diffraction depends on the number of grating slits and also the wavelength of the source. Fig. 3 shows the theoretical simulations of the optical carpets for N = 4, 8, 12, and 20 in the domain involved in the above condition (Eq.(18)).
Figure 4: Optical carpet of the binary grating from 0.5LT to 1.5LT . The simulation was done with I = ψ ∗ ψ according to Eq. (3), and with d = 200 µm, M = 500, N = 50, and λ = 532 nm. The x axis is in units of the grating period. The z axis represents the distance between the grating and the detector in multiples of the Talbot length (Eq.(6)).
The near-field diffraction patterns are in the area that is approximately enclosed by the triangle with a base of N d and the distance Z, denoted by 8
the dashed lines. In the case of large N , for instance N = 20, it is easily seen that the self-imaging of the grating or the Talbot effect appears only in the triangle zone. Experimentally, the diameter of an incident beam can be substituted for the triangle base N d, and one can use the triangle zone to approximate the boundary of Fresnel diffraction behind the grating.
Figure 5: Simulation results according to Eq.(3) performed with the same parameters, i.e., d = 200 µm, λ = 532 nm, M = 500, and N = 2. The longitudinal distance (z) is plotted to 100 cm. The space between the two adjacent bright fringes is 2.7 mm, which corresponds to Fraunhofer diffraction in the double-slit experiment.
Fig. 4 exhibits the numerical result of the near-field optical carpet of the Talbot effect from 0.5LT to 1.5LT . The simulation was done with Eq. (3) and with d = 200 µm, N = 50, and λ = 532 nm. Then, the Talbot length (Eq.(6)) is ∼75.1 mm. This optical carpet is in good agreement with our previous work using the different approach [5]. To prove that our method can be applied for both near-field and farfield regimes, simulation of Young’s well-known double-slit experiment was conducted with the same equation, Eq.(3), and with the same parameters, i.e., d = 200 µm, N = 2, and λ = 532 nm. Fig. 5 shows the results in the longitudinal direction (z) of the far-field distance. The results correspond precisely to those of Fraunhofer diffraction. For a grating or slit with an odd number of N slits, the centre of the screen (x = 0) points at the centre of a grating window instead, and similar results can be obtained. 9
Figure 6: Results for single-slit diffraction with slit width a = 200 µm, λ = 532 nm, N = 1, m = 1, and M = 500, theoretically determined with Eqs. (19) and (20). The longitudinal distance (z) is plotted to 100 cm. The distance to the first dark fringe is 2.7 mm, which again corresponds to Fraunhofer diffraction.
In this case, we introduce the following modification: jd m = 1, 4M , xm,j = jd , m = 2, 3, 4, . . . , (|m| − 1) d2 + d4 + 2M
(19)
and
ψ(x, z) = ψ+ (x, z) + ψ− (x, z)
(N +1)/2 M ik(x+xm,j )2 } A ikz ∑ ∑ { ik(x−xm,j )2 2z 2z = e +e , e NM m=1 j=0
(20)
where N = 1, 3, 5, . . .. For this, we extend our study to cover single-slit diffraction (N = 1), the results of which are shown in Fig. 6. The parameters of slit width a = d2 = 200 µm, λ = 532 nm, N = 1, m = 1, and M = 500 were used in the simulation with Eqs. (19) and (20). The results also correspond to normal single-slit diffraction.
10
Figure 7: Comparison of (a) experimental results and (b) simulation results (from Eq.(3)) with M = 500 from double-slit diffraction with the parameters d = 648 µm, N = 2, and λ = 532 nm, monitored at the near-field distance of z = 9.87 cm < πLT /4. The interference pattern at the centre of the two slits (indicated by the dashed line) is the minimum intensity because the distance is in the near-field range.
3. Experimental comparison We confirm our study with experiments for the case of the double-slit experiment. The experiments were done with a laser diode (λ = 532 nm), and the double-slit for the first experiment had a slit spacing of d = 648 µm. The interference pattern was recorded at the distance of z = 9.87 cm < πLT /4. The intensity at the centre of the pattern, which is indicated by the dashed line in Fig. 7, is shown as the minimum. This is unusual for Young’s double-slit experiment because it should yield constructive interference. This unusual result occurs because the observed distance was in the near-field diffraction regime. This experimental result (Fig. 7(a)) is well verified by our simulation (Fig. 7(b)). We continued Young’s double-slit experiment with a laser diode (λ = 635 nm), using a slit spacing of d = 288 µm for the second experiment. The intensity at the centre of the two slits (x = 0) was measured with the longitudinal distance, z, as described previously with the simulation in Fig. 2(b). Fig. 8 confirms that our theory can explain the result of this experiment. The variation of intensity from near-field to far-field regimes can be seen clearly. 11
Figure 8: Intensity at the centre of the two slits (x = 0) as a function of the longitudinal distance z in steps of 0.5 mm in the double-slit experiment. The experiment was done with the parameters d = 288 µm, N = 2, and λ = 635 nm. The solid line is the calculation according to Eq.(3) with M = 500.
4. Conclusions We devise an alternative method for explaining the diffraction phenomena in both near-field and far-field diffraction regimes with the same expression. Our method is based on the superposition of waves through the opening apertures, namely the Huygens’ principle. Our recent experimental results are in good agreement with the simulations. The method can describe not only near-field diffraction but also far-field diffraction as well as the transition between them. The Talbot effect and the single- and double-slit diffraction were used for our demonstration. Acknowledgements We acknowledge the support from the Department of Physics, Faculty of Science, Burapha University, and the Department of Physics, Faculty of Science, Kasetsart University. SD gratefully acknowledges the support grant from the Thailand Center of Excellence in Physics (ThEP) under Contract No. ThEP-60-PET-BUU8.
12
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[14] B. Wang, X. Luo, T. Pfeifer, H. Mischo, Moire deflectometry based on Fourier-transform analysis, Measurement 25 (1999) 249-253. [15] G. Spagnolo, D. Ambrosini, D. Paoletti, Displacement measurement using the Talbot effect with a Ronchi grating, J. Opt. A 4(6) (2002) 376380. [16] S. Mirza, C. Shakher, Surface profiling using phase shifting Talbot interferometric technique, Opt. Eng. 44(1) (2004) 01360. [17] T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, E. Ziegler, X-ray phase imaging with a grating interferometer, Opt. Express 13 (2005) 6296-6304. [18] R. Rodriguez-Vera, K. Genovese, J.A. Rayas, F. Mendoza-Santoyo, Vibration analysis at micro-scable by Talbot fringe projection method, Strain 45 (2009) 249-258. [19] S. Deachapunya, S. Srisuphaphon, Sensitivity of transverse shift inside a double-grating Talbot interferometer, Measurement 58 (2014) 1-5. [20] M.S. Chapman, C.R. Ekstrom, T.D. Hammond, J. Schmiedmayer, B.E. Tannian, S. Wehinger, D.E. Pritchard, Near-field imaging of atom diffraction gratings: the atomic talbot effect, Phys. Rev. A 51 (1995) R14-R17. [21] B.J. McMorran, A.D. Cronin, An electron Talbot interferometer, New J. Phys. 11 (2009) 033021. [22] T. Juffmann, S. Truppe, P. Geyer, A.G. Major, S. Deachapunya, H. Ulbricht, M. Arndt, Wave and particle in molecular interference lithography, Phys. Rev. Lett. 103 (2009) 263601. [23] S. Deachapunya, S. Srisuphaphon, Accordion lattice based on the Talbot effect, Chin. Opt. Lett. 12 (2014) 031101. [24] H. Chen, X. Zhang, D. Zhu, C. Yang, T. Jiang, H. Zheng, and Y. Zhang, Dressed four-wave mixing second-order Talbot effect, Phys. Rev. A 90 (2014) 043846.
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[25] S. Deachapunya, S. Srisuphaphon, P. Panthong, T. Photia, K. Boonkham, S. Chiangga, Realization of the single photon Talbot effect with a spatial light modulator, Opt. Express 24 (2016) 20029-20035. [26] Y. Zhang, X. Yao, C. Yuan, P. Li, J. Yuan, W. Feng, S. Jia, Y. Zhang, Controllable multiwave mixing Talbot effect, IEEE Photonics Journal 4(6) (2012) 2057-2065. [27] J. Azaa, H.G. de Chatellus, Angular Talbot effect, Phys. Rev. Lett. 112 (2014) 213902. [28] Y. Fan, B. Wang, K. Wang, H. Long, P. Lu, Talbot effect in weakly coupled monolayer graphene sheet arrays, Opt. Lett. 39(12) (2014) 33713373. [29] J.A. Salas, K. Varga, J. Yan, K.H. Bevan, Electron Talbot effect on graphene, Phys. Rev. B 93 (2016) 104305. [30] J.C. Bhattacharya, A.K. Aggarwal, Measurement of the focal length of a collimating lens using the Talbot effect and the moir technique, Appl. Opt. 30 (1991) 4479-4480. [31] Ch. Siegel, F. Loewenthal, J.E. Balmer, A wavefront sensor based on the fractional Talbot effect, Opt. Commun. 194 (2001) 265-275. [32] L. Martnez-Len, M. Araiza-E, B. Javidi, P. Andrs, V. Climent, J. Lancis, E. Tajahuerce, Single-shot digital holography by use of the fractional Talbot effect, Opt. Express 17 (2009) 12900-12909. [33] L. Stuerzebecher, T. Harzendorf, U. Vogler, U. D. Zeitner, R. Voelkel, Advanced mask aligner lithography: Fabrication of periodic patterns using pinhole array mask and Talbot effect, Opt. Express 18 (2010) 19485-19494. [34] E. Budiarto, N-W Pu, S. Jeong and J. Bokor, Near-field propagation of terahertz pulses from a large-aperture antenna, Opt. Lett. 23 (1998) 213-215. [35] R.L. Olmon, P.M. Krenz, A.C. Jones, G.D. Boreman, M.B. Raschke, Near-field imaging of optical antenna modes in the mid-infrared, Opt. Express 16 (2008) 20295-20305. 15
PII: DOI: Reference:
S0165-2125(18)30010-6 https://doi.org/10.1016/j.wavemoti.2018.01.002 WAMOT 2220
To appear in:
Wave Motion
Received date : 24 April 2017 Revised date : 28 December 2017 Accepted date : 8 January 2018 Please cite this article as: W. Temnuch, S. Deachapunya, P. Panthong, S. Chiangga, S. Srisuphaphon, A simple description of near-field and far-field diffraction, Wave Motion (2018), https://doi.org/10.1016/j.wavemoti.2018.01.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
*Research Highlights
Highlights 1. We present the simple method for explanation both the near-field and far-field diffractions with the same expressions. 2. Our method is based on the Huygens' principle alone. 3. Our method can identify the boundary between the near-field and far-field diffractions. 4. Our method is verified nicely by both the near-field and far-field, i.e. The Talbot effect, the single slit and the double-slit diffraction.
*Manuscript (Clear) Click here to view linked References
A simple description of near-field and far-field diffraction Wipawee Temnucha,c , Sarayut Deachapunyab,c , Pituk Panthonga , Surasak Chianggaa , Sorakrai Srisuphaphonb,∗ a
Department of Physics, Faculty of Science, Kasetsart University, Bangkok Province, 10900, Thailand b Department of Physics, Faculty of Science, Burapha University, Chonburi Province, 20131, Thailand c Thailand Center of Excellence in Physics, Commission on Higher Education, 328 Si Ayutthaya Road, Bangkok 10400, Thailand
Abstract We present an alternative explanation of near-field and far-field diffraction, i.e., the Talbot effect and single- and double-slit experiments, with the same expressions. Our method is based on the superposition of waves with different path lengths. A simple experiment is conducted as a demonstration. This simple experiment exhibits the transition from near-field to far-field diffraction. Keywords: Wave optics; Diffraction gratings; Wave propagation 1. Introduction Diffraction phenomena have been classified into two regimes, i.e., nearfield and far-field diffraction. Near-field effects rely on Fresnel diffraction, whereas far-field effects are based on Fraunhofer diffraction. One of the wellknown optical near-field effects is the Talbot effect, discovered by Henry Fox Talbot in 1836 [1]. When plane-parallel light falls onto a grating, it will generate a periodic interference pattern. This pattern will be similar to the grating if the observed screen is located at the Talbot length (LT ) [2, 3, 4, 5]. Since the Talbot effect was discovered, numerous theories have been introduced [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Those theories describe the effect in different ways. However, there is an alternative method that can be used to simplify ∗
Corresponding author Email address: [email protected] (Sorakrai Srisuphaphon)
Preprint submitted to Wave Motion
December 28, 2017
the Fresnel diffraction theory. It is based on wave propagation through the transmission function of a diffraction grating [5]. The Talbot effect has been widely used in many fields in both classical [12, 13, 14, 15, 16, 17, 18, 19] and quantum optics [20, 21, 22, 23, 24, 25]. Among them, the interesting notion of quantum carpets or carpets of light were explored [4]. Studies of the Talbot effect with completely controllable multiwave mixing signals were reported [26]. The angular Talbot effect has also been introduced [27]. The Talbot effect was explored with monolayer graphene sheet arrays [28]. Recently, a graphene sheet has been used as a diffraction grating for an electron Talbot effect experiment [29]. The Talbot effect can be exploited in many applications, for example, displacement sensors [15], surface profiling [16], X-ray phase imaging [17], vibration sensors [18], focal length measurement [30], wavefront sensors [31], digital holography [32], and lithography [33]. In telecommunications, such as in antenna design [34, 35], the near-field and far-field diffraction regimes are distinguished by the Fraunhofer distance and the near-field regime must be shorter than this distance limit. Therefore, it is important to study and simplify near-field and far-field diffraction for any application. In this paper, we introduce an alternative method that can explain both near-field and far-field diffraction. This method is based on the superposition of waves through the opening apertures. Simulations using our method show that it describes both near-field and far-field effects. We prove our method with a simple experimental setup. 2. Theory and numerical results We first consider the diffraction pattern of light behind a grating in both the far-field and near-field regimes. The intensity patterns in the intermediate area between the two regimes will be evaluated by considering the superposition of waves propagating through a static grating for which the time dependence of the field can be omitted. Here, a binary grating period d is placed along the x0 axis as shown in Fig. 1. By dividing the slit window into M points, a position labelled by j in the mth slit can be represented as xm,j = (|m| − 1)d +
jd d + . 4 2M
(1)
The parameter m is the order of the grating window, and d is the grating period. According to Huygens’ principle, let xm,j be a point source originating 2
Figure 1: A plane wave incident onto the grating along the x0 axis and propagating from the mth slit to the screen behind the grating with N slits.
√ a wave propagating a distance R = z 2 + (x − xm,j )2 to a position (x, z) on the screen. By using the Fresnel approximation, the distance R can be approximated as R ≃ z + (x − xm,j )2 /2z. Therefore, the wave function with wave number k = 2π/λ and wavelength λ at a position (x, z) is given by ψ(x, z) =
N/2 M ∑ ∑
m=−N/2 j=0
A ik(z+ (x−xm,j )2 ) 2z e , NM
(2)
where A represents the amplitude of the incident wave. For a symmetric grating, as shown in Fig. 1 with x = 0 at the middle position of the x axis, the wave function can be written as ψ(x, z) = ψ+ (x, z) + ψ− (x, z) N/2 M ik(x+xm,j )2 } A ikz ∑ ∑ { ik(x−xm,j )2 2z 2z e e . = +e NM m=1 j=0
(3)
The wave functions ψ± (x, z) correspond to a free wave that is generated from every point of xm,j on the upper and lower parts of the grating, respectively. Subsequently, the intensity distribution can be obtained as I = ψ ∗ ψ. The 3
obtained wave function given in Eq.(3) provides the diffraction patterns from near-field to far-field regimes with only one equation. Some simulations of carpets of light are presented in the following. The optical carpets exhibit variation of their interference patterns as a function of the distance behind the grating in the near-field range [5]. The far-field diffraction patterns are also presented in this paper. Nevertheless, the wave function ψ(x, z), which is in terms of a series, can be applied to estimate the intermediate zone between the two fields as mentioned. According to far-field diffraction, for a grating with an even number of N slits as shown in Fig. 1, the centre of the screen (x = 0) will be the position of the centre bright fringe at any distance z. However, the dark fringe can only occur in the near-field diffraction regime, as found in the Talbot effect. Therefore, one can find the zone by evaluating the shortest distance that gives the continuous bright fringe along the z axis at x = 0. The proposed distance, letting z = Z, may be used as the starting point of the far-field diffraction regime. Firstly, we consider the simplest case of N = 2 (double slits), in which case the wave function in Eq.(3) at x = 0 is reduced to M
A ia2z ∑ ib2z j ic2z2j2 e e M e M , M j=0
(4)
( ) d2 πLT =k z+ , b2z = c2z = , 32z 4z
(5)
ψ(x = 0, z) = where a2z
with the so-called Talbot length given by LT = d2 /λ.
(6)
The second exponential function in the series in Eq.(4) can be written in the series form ∞ ∑ ic2z j 2 1 ( ic2z j 2 )q e M2 = . (7) 2 q! M q=0 Therefore Eq. (4) becomes ∞
A ia2z ∑ 1 ψ(0, z) = e M q! q=0
( 4
ic2z M2
)q ∑ M j=0
j 2q e
ib2z j M
.
(8)
Figure 2: (a) Optical carpet in the xz plane for z = πLT /4 = 59.05 mm obtained from the modulus square of Eq.(3) for d = 200 µm, λ = 532 nm, M = 500, and N = 2. (b) Intensity distribution along the longitudinal direction (z = 6d ≃ 1.2 mm to z = 75 mm) at the centre of the grating. The vertical dashed line indicates the transition from near-field to far-field regimes. (c) and (d) show the optical carpet and the longitudinal intensity distribution (z = 4 to z = 17.5 mm) with the same parameters in the near-field diffraction regime.
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To calculate the second series in Eq.(8), we apply the geometric series formula M ∑
e
iχj M
=
j=0
1−e
iχ(M +1) M iχ
1 − eM
,
(9)
with the series condition χ < 1. For a large number of waves (M ≫ 1), the above series can be approximated as M ∑ j=0
iχ(M )
e
iχj M
1−e M eiχ − 1 ≃ = . iχ iχ 1 − (1 + M ) M
(10)
Therefore, the second series in Eq.(8) with j 2q may be written in the form of a derivative as ) ( )2q 2q ( iχ )2q 2q M ( M ∑ ∑ iχj iχj M M d d e − 1 j 2q e M = eM ≃ . (11) iχ 2q 2q i dχ i dχ M j=0 j=0 Using the above formula with χ = b2z , we obtain the wave function (Eq.(8)) in the form of infinite series as ( ) ∞ ∑ 1 cq2z d2q 2ieib2z /2 sin(b2z /2) ia2z ψ(0, z) ≃ A e . (12) q! iq+1 db2q b2z 2z q=0 In the limit of large distance z, sin(b2z /2) ≃ b2z /2, and the derivative can be calculated as ( ) ( ) ( )2q d2q 2ieib2z /2 sin(b2z /2) d2q ieib2z /2 b2z i ib2z /2 ≃ 2q = ie . (13) 2q b2z b2z 2 db2z db2z The wave function, in the far-field regime, is then reduced to ( )q ∞ ∑ 1 ic2z ia2z ib2z /2 ψ(0, z) ≃ A e e = A eia2z +ib2z /2+ic2z /4 , q! 4 q=0
(14)
which corresponds to the constant intensity I = ψ ∗ ψ = A2 as assumed in Eq.(3). Nevertheless, according to the series condition χ = b2z < 1, the series does not diverge only if z>
πLT . 4 6
(15)
This suggests that the wave function in the near-field regime cannot be described by the series in Eq.(12). However, the wave function given by Eq.(3) can still be applied in the two regimes. Fig. 2(a) shows the optical carpet of I = ψ ∗ ψ from Eq. (3) with N = 2 for z = πLT /4; the intensity distribution along the z axis (z = 6d ≃ 1.2 mm to z = 75 mm) at x = 0 is shown in Fig. 2(b). It is clearly seen that the middle bright fringe obviously appears at the distance z = πLT /4, which may be used as the boundary distance Z between near-field and far-field diffraction. The near-field diffraction pattern can be found where z < πLT /4 (Figs. 2(c) and (d)). Therefore, we will employ this idea to find the distance Z for the other gratings, N > 2.
Figure 3: Optical carpets in the xz plane (d = 200 µm, λ = 532 nm, and M = 500) for Z = πLT (2N − 3)/4 obtained from the modulus square of Eq.(3) for N = 4 (a), 8 (b), 12 (c), and 20 (d). The dashed lines represent the triangle with a base of N d and the distance Z, which both enclose the diffraction patterns in the near-field regime.
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According to Eq.(3) but N ̸= 2, the wave function at x = 0 is given by N/2 M A ∑ iaN z ∑ ibN z j icN z2j2 e M e M , ψ(x = 0, z) = e N M m=1 j=0
where aN z
) ( d2 πLT πLT 2 (4m − 3) , bN z = (4m − 3), cN z = . =k z+ 32z 4z 4z
(16)
(17)
Because N is a finite number equalling the number of slits of the grating, the summation over j needs to give a converged result. In analogy with the case of N = 2, we also have the condition bN z < 1. The maximum value of m is limited at N/2, thus the condition may be given by z>
πLT (2N − 3). 4
(18)
Therefore, the limit of near-field diffraction depends on the number of grating slits and also the wavelength of the source. Fig. 3 shows the theoretical simulations of the optical carpets for N = 4, 8, 12, and 20 in the domain involved in the above condition (Eq.(18)).
Figure 4: Optical carpet of the binary grating from 0.5LT to 1.5LT . The simulation was done with I = ψ ∗ ψ according to Eq. (3), and with d = 200 µm, M = 500, N = 50, and λ = 532 nm. The x axis is in units of the grating period. The z axis represents the distance between the grating and the detector in multiples of the Talbot length (Eq.(6)).
The near-field diffraction patterns are in the area that is approximately enclosed by the triangle with a base of N d and the distance Z, denoted by 8
the dashed lines. In the case of large N , for instance N = 20, it is easily seen that the self-imaging of the grating or the Talbot effect appears only in the triangle zone. Experimentally, the diameter of an incident beam can be substituted for the triangle base N d, and one can use the triangle zone to approximate the boundary of Fresnel diffraction behind the grating.
Figure 5: Simulation results according to Eq.(3) performed with the same parameters, i.e., d = 200 µm, λ = 532 nm, M = 500, and N = 2. The longitudinal distance (z) is plotted to 100 cm. The space between the two adjacent bright fringes is 2.7 mm, which corresponds to Fraunhofer diffraction in the double-slit experiment.
Fig. 4 exhibits the numerical result of the near-field optical carpet of the Talbot effect from 0.5LT to 1.5LT . The simulation was done with Eq. (3) and with d = 200 µm, N = 50, and λ = 532 nm. Then, the Talbot length (Eq.(6)) is ∼75.1 mm. This optical carpet is in good agreement with our previous work using the different approach [5]. To prove that our method can be applied for both near-field and farfield regimes, simulation of Young’s well-known double-slit experiment was conducted with the same equation, Eq.(3), and with the same parameters, i.e., d = 200 µm, N = 2, and λ = 532 nm. Fig. 5 shows the results in the longitudinal direction (z) of the far-field distance. The results correspond precisely to those of Fraunhofer diffraction. For a grating or slit with an odd number of N slits, the centre of the screen (x = 0) points at the centre of a grating window instead, and similar results can be obtained. 9
Figure 6: Results for single-slit diffraction with slit width a = 200 µm, λ = 532 nm, N = 1, m = 1, and M = 500, theoretically determined with Eqs. (19) and (20). The longitudinal distance (z) is plotted to 100 cm. The distance to the first dark fringe is 2.7 mm, which again corresponds to Fraunhofer diffraction.
In this case, we introduce the following modification: jd m = 1, 4M , xm,j = jd , m = 2, 3, 4, . . . , (|m| − 1) d2 + d4 + 2M
(19)
and
ψ(x, z) = ψ+ (x, z) + ψ− (x, z)
(N +1)/2 M ik(x+xm,j )2 } A ikz ∑ ∑ { ik(x−xm,j )2 2z 2z = e +e , e NM m=1 j=0
(20)
where N = 1, 3, 5, . . .. For this, we extend our study to cover single-slit diffraction (N = 1), the results of which are shown in Fig. 6. The parameters of slit width a = d2 = 200 µm, λ = 532 nm, N = 1, m = 1, and M = 500 were used in the simulation with Eqs. (19) and (20). The results also correspond to normal single-slit diffraction.
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Figure 7: Comparison of (a) experimental results and (b) simulation results (from Eq.(3)) with M = 500 from double-slit diffraction with the parameters d = 648 µm, N = 2, and λ = 532 nm, monitored at the near-field distance of z = 9.87 cm < πLT /4. The interference pattern at the centre of the two slits (indicated by the dashed line) is the minimum intensity because the distance is in the near-field range.
3. Experimental comparison We confirm our study with experiments for the case of the double-slit experiment. The experiments were done with a laser diode (λ = 532 nm), and the double-slit for the first experiment had a slit spacing of d = 648 µm. The interference pattern was recorded at the distance of z = 9.87 cm < πLT /4. The intensity at the centre of the pattern, which is indicated by the dashed line in Fig. 7, is shown as the minimum. This is unusual for Young’s double-slit experiment because it should yield constructive interference. This unusual result occurs because the observed distance was in the near-field diffraction regime. This experimental result (Fig. 7(a)) is well verified by our simulation (Fig. 7(b)). We continued Young’s double-slit experiment with a laser diode (λ = 635 nm), using a slit spacing of d = 288 µm for the second experiment. The intensity at the centre of the two slits (x = 0) was measured with the longitudinal distance, z, as described previously with the simulation in Fig. 2(b). Fig. 8 confirms that our theory can explain the result of this experiment. The variation of intensity from near-field to far-field regimes can be seen clearly. 11
Figure 8: Intensity at the centre of the two slits (x = 0) as a function of the longitudinal distance z in steps of 0.5 mm in the double-slit experiment. The experiment was done with the parameters d = 288 µm, N = 2, and λ = 635 nm. The solid line is the calculation according to Eq.(3) with M = 500.
4. Conclusions We devise an alternative method for explaining the diffraction phenomena in both near-field and far-field diffraction regimes with the same expression. Our method is based on the superposition of waves through the opening apertures, namely the Huygens’ principle. Our recent experimental results are in good agreement with the simulations. The method can describe not only near-field diffraction but also far-field diffraction as well as the transition between them. The Talbot effect and the single- and double-slit diffraction were used for our demonstration. Acknowledgements We acknowledge the support from the Department of Physics, Faculty of Science, Burapha University, and the Department of Physics, Faculty of Science, Kasetsart University. SD gratefully acknowledges the support grant from the Thailand Center of Excellence in Physics (ThEP) under Contract No. ThEP-60-PET-BUU8.
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