Diffraction of partial temporal coherence beam from multi slits
E. KOUSHKI, S.A. ALAVI*
Department of Physics, Hakim Sabzevari University (HSU), Sabzevar, 96179- 76487, Iran Corresponding author: [email protected] *
Abstract:
We have generalized the notion of Franhoufer diffraction of temporal coherent light from a single slit to the case of arbitrary n-slits. A general numerical method has been developed for the simulation of the diffraction pattern of multi-slits which is also used to study the effects of coherence length on the diffraction pattern. The diffraction pattern is investigated for different values of recently introduced parameter “decoherence parameter”. It is shown that for multi-slits, the temporal decoherence effects appear for 1. The results of our study reproduce the previous studies for perfect temporal case, when coherence length tends to infinity.
Keywords: Multi-slits; Franhoufer diffraction; Partial temporal coherent beam.
1. Introduction
Diffractive multi-slits and Optical gratings are periodic systems which have very important applications in science and technology. Their potential in separating different frequencies of light marks them as the most useful devices in light spectroscopy. They are well-known instruments in optical spectroscopy of materials and light sources in both laboratory instruments and telescopes. Light diffraction from double and multi slits has been the subject of many
1 studies because due to its importance in the study of light properties, slits dimensions and optical measurements [1,2]. The technology of production and improvement of optical gratings is being extended using modern manufacturing potential of planar technologies such as lithography [3,4]. But the role of the light source coherence length has not been properly studied in the literature. The effects of temporal and spatial coherence on the diffraction from a few slits have been the subject of some researches [5-7]. Recently, the technology of measuring the temporal coherence properties of different sources has been extended. The electromagnetic degree of temporal coherence and the coherence time for quasi-monochromatic unpolarized light beams emitted by some different sources is studied in [8]. Also, the propagation of partially polarized and partially coherent beams in uniaxial crystals has been investigated [9]. Recently, we studied Fraunhofer diffraction from a single slit [10]. A general formula was derived to describe the intensity distribution of the Fraunhofer diffraction pattern of a slit aperture illuminated with partial temporal coherent light. The model was generalized to the case of circular aperture and the effects of the coherence length on the diffraction pattern were investigated. In the current work, the model is extended to the case of multi-slits.
2. Theory
In our previous work, the diffraction of a partial temporal coherent beam from a single-slit was studied. A general formula was derived for the intensity distribution from a single slit illuminated with partial temporal coherent light, at the far-field [10]. Theory was based on interference of waves from a source traveling along different paths. According to the fact that partial temporal coherence of light influences the interference intensity and considering a single slit as a continuous series of spot sources, we obtained an analytical relation for
2 the intensity distribution and then generalized the model to the circular apertures [10].
In this paper we generalize our previous studies to the case of multi slits. The superposition of N beams with the same intensity I and angular frequency ω at the far field point P is given by [10]:
N1 j I P NI 2I (N j)(1 )cos( j ) (1) j1 0 where τ and 0 are the constant time difference (time dilation) between two neighboring paths, and the coherence time, respectively. To apply this relation to an open slit, we consider the single slit as a continuous series of spot light sources. The details of calculations are provided in Appendix A.
Using calculations discussed in the Appendix A and by including the constant factor c 0 , the total intensity distribution from a single-slit is as follows: 2
2 yb c 0 b EL 2 EL 2 y y sin I P ( ) [ ( ) 2b( ) (1 )(1 )cos(kysin )dy] (2) 2 3 r r y0 b b where the dimensionless parameter which we name “slit decoherence
b k parameter” is defined as and l0 c 0 , c are the length of finite l0 wave train and the wave number respectively. Note that in the regime of Franhoufer diffraction theory, s i n is enough smaller than one so that the second parentheses in the integral is always positive.
By introducing: 1 sin , , k sin (3) b l0 the analytical solution of the integral in Eq.(2) is provided in Appendix B, so we arrive at the following expression for the intensity:
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c b2 E E 1 2 I () 0 [ ( L )2 2b( L )2 ( )sin(y) ( )ysin(y) ( )cos(y) P 2 3 r r 3 2
2 ( )y 2 sin(y) ( )y cos(y)] (4) 2
This is an exact expression for intensity distribution, which enable us to find exact results for the intensity of multi slits with given geometries.
Now, we generalize Eq.(2) to the case of n-slits which n may be even or odd. To do this we take the irradiating width of the slits as w na t , where a is the slits separation and t is the dark part between two neighboring slits ( t a b ). For even n we place the origin of the coordinates at the center of central slit, as shown in Fig.(1). The integral should be taken over all the width of slits and the
w w upper and lower bounds of the integral are 2 and 2 , respectively. For even number of slits, as shown in Fig.1, the lower bound of each slit is given by:
t (2 j 1) ( j 1)b , j 0 y(lower) 2 (5) j t (2 j 1) jb , j 0 2
n j n , j 0 where j is an integer and labels the consecutive slits ( 2 2 ), and t is the dark part between two neighboring slits ( ).
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y
J=2 3t/2+b
J=1 t/2 x
J=-1 -t/2-b
J=-2 -3t/2-2b
Fig.1. Geometry of multi slits diffraction with even number of slits.
For odd number of slits, the lower bound of each slit is as follows (Fig.2):
b y (lower) (2 j 1) jt , (n 1) j (n 1) (6) j 2 2 2
(upper) (lower) In both geometries the upper bound of each slit is given by y j y j b . It is worth mentioning that, for the case of single slit, in Eq.(2), 1/b and for the case of multi slits it is taken as 2/ w .
y
J=1 b/2+t x
J=0 -b/2
J=-1 -3b/2-t
Fig.2. Geometry of multi slits diffraction with odd number of slits.
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The last theoretical consideration is r i.e., the distance from the slits to a
r d typical point on the screen as shown in Fig (3), which is given by cos .
P r=d/cosθ
θ d
r d Fig.3. The distance of point P on the screen from the slits is cos .
The theoretical model discussed so far contains the effects of partial temporal coherence effects on the diffraction distribution pattern and could employ to study the intensity distribution at far field.
3. Results and discussion:
First, we check our general expression (4), for the case of single slit n=1, by comparing it to our previous work [10]. The results are shown in Fig.(4). The kbsin( ) intensities are plotted in terms of variable . As it is observed the 0 2 matching is excellent. In the case of perfect coherence ( →0), the model recover the well-known fact that the fist minimum is occurred at 0 rad (see e.g., [2]).
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Fig.4. comparison of two solutions for a single slit with b=50 micron illuminated by a He- Ne laser beam with wavelength of 633 nm at far-field r=250cm.
Now we study the effects of partial temporal decoherence on the multi-slits diffraction patterns through changing the coherence length l0 . In Figure (5) the diffraction pattern of two slits is plotted for different values of l0 and for
2(a ) 1 6 3 3 nm. For a double slits, we expect to have b interference peaks in the central diffraction peak [1,2]. As is shown in figure (5), our model satisfies this requirement. On the other hand, for values of b l0 more than 1, the effects of partial temporal coherence begin to appear, in other word the effects of partial temporal decoherence show themselves for 1 .
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Figure (5). Intensity distribution from a double-slits with a=90 microns, b=30 microns and different numbers of coherence length: (a) perfect temporal coherence ( 0 ). (b) 1 (c)
3 (d) 10 .
Figures (6) and (7) show diffraction pattern of a triple slits. It is observed that the different orders of diffraction become sharper when the number of slits increases, which is also expected in common models [1,2]. Also, again for values of b l0 more than 1, the effects of partial temporal coherence begin to appear. For other different values of n, the same results are obtained. It is an interesting result, because it shows that for multi slits, the effects of decoherence manifest themselves even for l0 which is in order of the slits dimension. In other words, the interaction of the coherence length and the slits begins when the coherence length is shorter that the size of slits. As an important note, increasing the decoherence parameter, influences the intensity distribution. In multi-slits, this effect seems to be destructive, as shown in figures (5-7). Destructive effects of partial coherence beam on the interference patterns have been approved and documented [11-12].
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Figure (6). Intensity distribution from triple-slits with a=90 microns, b=30 microns and different numbers of coherence length: (a) perfect temporal coherence ( 0 ). (b) 1 (c)
3 (d) 10 .
Figure (7). Intensity distribution from four slits with a=90 microns, b=30 microns and different numbers of coherence length: (a) perfect temporal coherence ( ). (b) (c)
3 (d) .
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4. Conclusion we have generalized the Fraunhofer diffraction from a single slit illuminated by a partial temporal coherent beam previously studied in our work [10] to the case of n-slits. To achieve this goal, theoretical and numerical methods have been developed for the simulation of the diffraction pattern of multi-slits which is also used to study the effects of coherence length on the pattern. The diffraction pattern is investigated for different values of the decoherence parameter . It is shown that for multi-slits, the temporal decoherence effects appear for 1. The results of our study reproduce the previous studies for perfect temporal case, when coherence length tends to infinity.
In conclusion our results may benefit optical engineering technologies in particular designing spectrometers. By adjusting the decoherence parameter one can adjust the brightness of the first and second orders of diffraction peaks which may create new features in spectroscopy.
5. Appendix A:
We assume that the slit width is b and contains N spot sources. We divide it into N equal segments. The quantity EL is defined as the amplitude per unit width of the slit at unit distance away, and is given by [1,2,10]:
1 EL lim ( 0 N) (A.1) b N where 0 is the electric field of each spot source. This definition avoids divergence at the irradiation of infinite number of sources. The electric field of j-th segment with width y j at distance r is given by
1 E j lim ( 0 N)y j . (A.2) br N
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So, the second term of Eq.(1) changes to the following form:
N1 1 2 j I s 2 lim lim ( ( 0 N)y j ) (N j)(1 )cos( j ) . (A.3) N N j1 br j 0
Using Eq.(A.1) it can be rewritten as;
N 1 EL 2 j 2 lim ( y j ) (N j)(1 )cos( j ) . (A.4) N j1 rj 0
bsin Using [10] it can be transformed into the form: Nc
N1 EL 2 y j jb sin jbsin 2 lim ( y j ) N(1 )(1 )cos( ) (A.5) N j1 rj b 0 Nc Nc
b jb j y j here, in the limit of N , we use y j , y j , , so: N N N b
N1 EL 2 b y j y j sin y j sin 2 lim ( ) y j y j (1 )(1 )cos( ) . N j1 rj y j b 0c c
Therefore Eq.(A.3) takes the form as follows:
E yb y y sin ysin I 2b( L )2 (1 )(1 )cos( )dy s y0 . (A.6) r b 0c c
In Eq.(1), the first term is the summation of N self-product of different segments at the point P. With the same method as employed for the second term, one can calculate the contribution of the first term in Eq.(1).
N 1 N 1 N 1 1 2 2 2 EL 2 2 I f lim I j lim ( lim 0 N y j ) lim ( ) (y j ) (A.7) N N 2 2 N N j 1 j1 b r r j1
Also we would have:
N 1 2 2 2 2 (y j ) (y1 y2 .... yN 1) j1 (A.8) 2 (y1 y2 .... yN 1) 2y1y2 2y1y3 ...
11 as we know, y1 y1 ... yN 1 . Thus:
N 1 2 2 2 2 (y j ) (y1 y2 .... yN 1) j1 2 2 2 2 (y1 y2 .... yN 1) 2(y1 y2 .... yN 2 ) (A.9) N 1 2 2 (b) 2((y j )) j1 This equation gives:
N 1 2 (y2 ) b j 3 j 1
So, we arrive at the following expression:
2 EL 2 b I f ( ) (A.10) r 3
Appendix B:
The analytical solution of the integral in Eq.(2), is as follows: (1y)(1 y)cos(y)dy 1 2 ( )sin(y) ( )ysin(y) ( )cos(y) 3 2 2 ( )y 2 sin(y) ( )y cos(y) 2
6. Acknowledgement We also acknowledge Dr. Asghar Moulavi Nafchi, from the department of English at HSU, for proofreading of the manuscript.
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