Arbitrary laser beam propagation in free space

Optics Communications 282 (2009) 3216–3222

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Arbitrary laser beam propagation in free space Çag˘lar Arpali a,1, Yahya Baykal a,*, Cem Nakibog˘lu b,2 a b

Çankaya University, Electronic and Communication Engineering Department, Ög˘retmenler Cad. No: 14, Yüzüncüyıl, 06530 Balgat, Ankara, Turkey Gazi University, Electric and Electronic Engineering Department, Eti Mah., Yükselisß sok. No: 5, 06570 Maltepe, Ankara, Turkey

a r t i c l e

i n f o

Article history: Received 3 January 2009 Received in revised form 6 May 2009 Accepted 14 May 2009

Keywords: 200.2605 050.1940 230.6080 240.5770

a b s t r a c t The propagation of arbitrary laser beams in free space is examined. For this purpose, starting with an incident field of arbitrary field distribution, the intensity at the receiver plane is formulated via Huygens Fresnel diffraction integral. Arbitrary source field profile is produced by decomposing the source into incremental areas (pixels). The received field through the propagation in free space is found by superposing the contributions from all source incremental areas. The proposed method enables us to evaluate the received intensity originating from any type of source field. Using the arbitrary beam excitation, intensity of various laser beams such as cos-Gaussian, cosh-Gaussian, general type beams are checked to be consistent with the already existing results in literature, and the received intensity distributions are obtained for some original arbitrary beam field profiles. Our received intensity formulation for the arbitrary source field profiles presented in this paper can find application in optics communication links, reflection from rough surfaces, optical cryptography and optical imaging systems. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction For various types of optical incidences, propagation properties and especially the intensity distributions at the receiver plane are examined in free space [1–14]. Studies on the atmospheric propagation of different beam types as applied to atmospheric optical communications have been reported [15–21]. Applications such as optical cryptography [22,23] and inverse problems [24,25] are also reported. These studies are restricted to irradiance distribution of known beam shape types. In some applications, discretionary beam shapes can be of interest. Source excitation and beam reshaping for arbitrary fields have been studied [26–28]. Also studies involving propagation of beams with any spectral, coherence and polarization properties [29] and with arbitrary spatial and temporal coherence in turbulent atmosphere exist in literature [30]. Banakh and Falits proposed a method for the simulation of arbitrary laser beam propagation with the help of the laser radiation scattered by particles [31]. Basics of our arbitrary beam formulation as applied to turbulent atmosphere are provided in our earlier work [32–34]. In this paper, we have introduced optical beams having arbitrary field excitation at the source plane and by employing the Huygens Fresnel principle, we have formulated the intensity distribution at the receiver plane after such beams propagate in free space. Arbitrary source field profile is produced * Corresponding author. Tel.: +90 312 284 4500/132; fax: +90 312 284 8043. E-mail addresses: [email protected] (Ç. Arpali), [email protected] (Y. Baykal), [email protected] (C. Nakibog˘lu). 1 Tel.: +90 312 284 4500/318; fax: +90 312 284 8043. 2 Tel.: +90 312 231 7400/2342; fax: +90 312 230 8434. 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.05.034

by decomposing the source into pixels and the received field is formulated as the superposition of all the fields at the receiver plane originating from all the source pixels. Using the arbitrary beam excitation, intensities of various laser beams such as cos-Gaussian, cosh-Gaussian, general type beams are checked to match the existing results in literature. Generating some original arbitrary beams, their received intensity profiles are found at the receiver plane. 2. Theoretical model In our formulation, propagation geometry is composed of a transversal source and receiver planes with coordinates [s = (sx, sy),z = 0] and [p = (px, py),z = L], respectively [32]. Both planes are perpendicular to the propagation axis, z, and L is the link length. Arbitrary source field profile is constructed by decomposing the source into pixels and the total field arriving at the receiver plane is found by superposing the contributions of the fields originating from all the source pixels. Configuration of the arbitrary source field distribution is given in Fig. 1. Source plane is split into n  m pixels. At the source plane, central coordinate of each pixel is given as snm = [(snm)x, (snm)y]. The i h points nhcoordinates of the corner Ds of pixels can be found to be ðsnm Þx  D2sx ; ðsnm Þy  2y ; ðsnm Þx þ ih i h io Ds Ds Ds Dsx ; ðsnm Þy  2y ; ðsnm Þx þ D2sx ; ðsnm Þy þ 2y ; ðsnm Þx  D2sx ; ðsnm Þy þ 2y . 2 The field of the arbitrary optical beam at the source plane can be expressed as

uðs; z ¼ 0Þ ¼

N2 M2 X X n¼N 1 m¼M1

unm ðs; z ¼ 0Þ:

ð1Þ

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Applying Huygens Fresnel principle for a horizontal link, the field at the receiver point p = (px, py) emanating from each source pixel is found as

unm ðp; LÞ expðikLÞ ¼ ikL

Z

Z

ðsnm Þx þD2sx

ðsnm Þy þ

Dsy 2

D sy 2

ðsnm Þx D2sx

2



 ik jp  sj2 ; 2L

d sunm ðs; z ¼ 0Þ exp

ðsnm Þy 

ð5Þ where k is the wavelength, k = 2p/k is the wavenumber. Total field at the receiver field is obtained by adding the received field contributions from each pixel, i.e.,

uðp; LÞ ¼ Fig. 1. Source composed of incremental areas (pixels).

unm ðp; LÞ

ð6Þ

n¼N1 m¼M 1

Excluding the pixel located at the origin, N1 and N2 denote the number of pixels at the left and right side of the sy axis, respectively. Similarly, excluding the pixel located at the origin, M1 and M2 denote the number of pixels below and above sx axis, respectively. Thus, there are N1 + N2 + 1 pixels in sx direction and M1 + M2 + 1 pixels in sy direction so total number of pixels is (N1 + N2 + 1) (M1 + M2 + 1). The field at each pixel is defined by

unm ðs; z ¼ 0Þ ¼ unm ½ðsnm Þx ; ðsnm Þy ; z ¼ 0 ¼ Anm expðiunm Þ;

N2 M2 X X

Inserting Eq. (5) into Eq. (6), total field at the receiver plane becomes

uðp; LÞ ¼

N2 M2 X X expðikLÞ ikL n¼N m¼M 1



Z

1

ðsnm Þx þD2sx

ðsnm Þx D2sx

Z

ðsnm Þy þ

Dsy 2

D sy 2

2

d sunm ðs; z ¼ 0Þ exp

ðsnm Þy 



ik jp  sj2 2L



ð2Þ

ð7Þ

where i = (1) , Anm and unm are the amplitude and phase at the h i pixel whose central coordinates are snm ¼ ðsnm Þx ; ðsnm Þy . Intensity at the source plane is found from

The integrand in Eq. (5) does not vary with s if the pixel size is very small, i.e., if Dsx ? 0 and Dsy ? 0. Employing Eq. (2), the integrand in Eq. (7) can be approximated by

1/2

Is ðs; z ¼ 0Þ ¼ uðs; z ¼ 0Þu ðs; z ¼ 0Þ;

ð3Þ

where  denotes the complex conjugate. Substituting Eq. (1) together with Eq. (2) into Eq. (3), we obtain the intensity of the arbitrary beam on the source plane as

Is ðs; z ¼ 0Þ ¼

N2 M2 X X n¼N1 m¼M 1

Anm exp ðiunm Þ

N2 M2 X X n0 ¼N

1

m0 ¼M

An0 m0 exp ðiun0 m0 Þ: 1

ð4Þ

 ik jp  sj2 2L   ik ffi unm ðsnm ; z ¼ 0Þ exp jp  snm j2 2L   ik ffi Anm expðiunm Þ exp jp  snm j2 ¼ constant; 2L

unm ðs; z ¼ 0Þ exp



ð8Þ

Substituting Eq. (8) into Eq. (7), total field at the receiver plane is determined as

Fig. 2. Progress of cos-Gaussian beam (Vx = Vy = 90 m1) along the propagation axis.

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uðp; LÞ ¼

¼

Ds Dsx Z Z N2 M2 ðsnm Þy þ 2y X X expðikLÞ ðsnm Þx þ 2 2 d sAnm Dsy ikL ðsnm Þx D2sx ðsnm Þy  2 n¼N 1 m¼M 1   ik  exp ðiunm Þ exp jp  snm j2 2L

N2 M2 X X expðikLÞ Anm expðiunm Þ ikL n¼N 1 m¼M 1  n o ik  exp ½px  ðsnm Þx 2 þ ½py  ðsnm Þy 2 2L Z ðsnm Þx þDsx Z ðsnm Þy þDsy 2 2  dsx dsy ; Ds ðsnm Þx D2sx

ðsnm Þy 

y 2

ðsnm Þx ¼ nDsx ;

ðsnm Þy ¼ mDsy ;

ðsn0 m0 Þx ¼ n0 Dsx ; ðsn0 m0 Þy ¼ m0 Dsy ; ð12Þ

Substituting Eq. (12) into Eq. (11), the intensity of an arbitrary beam at the receiver plane becomes

Ir ðp; LÞ ¼

ð9Þ

N2 M2 X ðDsx Dsy Þ2 X

Anm expðiunm Þ ðkLÞ2 n¼N1 m¼M1  n N2 M2 o X X ik  exp An0 m0 ½px  nDsx 2 þ ½py  mDsy 2  2L 0 0 n ¼N 1 m ¼M 1  o ik n 2 0  expðiun0 m0 Þ exp  ½px  n Dsx  þ ½py  m0 Dsy 2 : 2L ð13Þ

Evaluating the integral in Eq. (9) and rearranging N2 M2 X expðikLÞ X uðp; LÞ ¼ Dsx Dsy Anm expðiunm Þ ikL n¼N1 m¼M 1  n o ik  exp ½px  ðsnm Þx 2 þ ½py  ðsnm Þy 2 ; 2L

Eq. (13) implies that by using the amplitude and phase information at each pixel of an arbitrary source field, the corresponding intensity distribution can be constructed at the receiver plane in free space.

ð10Þ 3. Results and discussions

From Eq. (10), the intensity at the receiver plane is found as

The intensity profiles at the source and receiver planes are produced using Eqs. (4) and (13), respectively. Source plane intensity is normalized as

Ir ðp; LÞ ¼ uðp; LÞu ðp; LÞ ¼

N2 M2 X expðikLÞ X Dsx Dsy unm Anm expðiunm Þ ikL n¼N 1 m¼M1  n o ik  exp ½px  ðsnm Þx 2 þ ½py  ðsnm Þy 2 2L N2 M2 X expðikLÞ X Dsx Dsy An0 m0 expðiun0 m0 Þ ðikLÞ n0 ¼N m0 ¼M 1 1  o ik n  exp  ½px  ðsn0 m0 Þx 2 þ ½py  ðsn0 m0 Þy 2 ; 2L

IsN ðs; z ¼ 0Þ ¼ Is ðs; z ¼ 0Þ=Max½Is ðs; z ¼ 0Þ;

IrN ðp; z ¼ LÞ ¼ Ir ðp; z ¼ LÞ=Max½Is ðs; z ¼ 0Þ:



where x and y components of the source point snm are

ð14Þ

where Max operator denotes the maximum value of Is(s, z = 0). Also, we apply the following normalization at the receiver plane

ð11Þ

ð15Þ

Receiver intensity profiles of arbitrary beams are checked to match the intensity profiles of many individual beam types. As check cases, we have applied our formulation to cos-Gaussian, coshGaussian and general type beams, compared the arbitrary beam intensities with their existing intensity patterns in the literature for selected propagation lengths. Then our formulation is applied

Fig. 3. Progress of cosh-Gaussian beam (Vx = Vy = 55i m1) along the propagation axis.

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Fig. 4. Progress of general type beam which is composed of seven different Gaussian beams along the propagation axis with the settings of h = [0 0 p p 0 p p] rad, A = [0.5 0.5 1 0.5 0.5 1 0.5], n = [1 1 1 0 2 1 0], m = [0 0 1 0 2 1 1], Vx = Vy = [10i 10i 10i 10i 10i 10i 10i] m1.

Fig. 5. Intensity distribution of a chess shaped beam at (a) the source plane with 71  71 pixels and the receiver planes with (b) 71  71 pixels, (c) 141  141 pixels, (d) 355  355 pixels for L = 1.5 km.

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to arbitrary profiles. As one of the check cases, we applied our formulation to the known sinusoidal-Gaussian beam wave fields which has the form at the source plane (z = 0) [15]

where A and u are, respectively, the amplitude and phase of the field at the source plane, asx and asy are the source sizes of Gaussian beam in sx and sy directions, Vx = Vxr + iVxi,Yx = Yxr + iYxi and Vy = Vyr + iVyi,Yy = Yyr + iYyi are the complex displacement parameters along the sx and sy directions, Vxr, Vxi symbolize the real and imaginary components of Vx and similarly Vyr, Vyi indicate the real and imaginary parts of Vy, Yxr, Yxi are the real and imaginary components of Yx and Yyr, Yyi are the y counterparts of Yxr, Yxi. Focal lengths along sx and sy directions are taken as infinite, i.e., collimated beams are considered. Cos-Gaussian laser beam can be produced by assigning Vx = Yx = Vxr and Vy = Yy = Vyr. Likewise, cosh-Gaussian laser beam can be established by setting Vx = Yx = iVxi and Vy = Yy = iVyi where Vxr, Vxi, Vyr and Vyi are assigned positive values. It is noted that our results in this paper are obtained for the symmetric cases of cos and cosh-Gaussian beams, however, our formulations can

also be applied to astigmatic beams. To construct the arbitrary source plane field for cos-Gaussian and cosh-Gaussian beams, we take the required phase and amplitude information by using Eq. (16). Taking the phase and amplitude information for each source pixel which has the center point snm = [(snm)x, (snm)y] and using Eq. (13), received intensity distribution of cos and cosh-Gaussian beams after propagating in free space can be obtained. In general, the necessary source and medium parameters, i.e. the source sizes asx and asy, displacement parameters Vx, Vy, wavelength, k, and the link length L, are indicated in our figures and their captions. In all of the Figs. 2–8, the total numbers of pixels taken at the source planes are 71  71 = 5041. In Figs. 2–4, i.e., for cos, cosh and general type beams, the same number of pixels as in the source planes, i.e., 71  71 = 5041 pixels are also taken at the receiver planes. Fig. 2 displays the progress of the intensity distribution of cos-Gaussian beam along the propagation axis, the plot at the upper left corner showing the intensity distribution at the source plane. Fig. 3 shows the progress of the intensity distribution of cosh-Gaussian beam at the source and receiver planes. It is verified that the intensity profiles of cos-Gaussian and cosh-Gaussian beams agree with the results of the general beam simulator in Ref. [15]. Again we apply the same discretionary procedure mentioned above to construct

Fig. 6. Intensity distribution of a square shaped beam at (a) the source plane with 71  71 pixels and (b) the receiver plane with 355  355 pixels for L = 1.5 km.

Fig. 7. Intensity distribution of an arbitrary beam that consists of letters, Ç.Ü. at (a) the source plane with 71  71 pixels and (b) the receiver plane with 141  141 pixels for L = 1.2 km.

h i us ðs; z ¼ 0Þ ¼ 0:5A expði/Þ exp 0:5ðs2x =a2sx þ s2y =a2sy Þ    exp½iðV x sx þ V y sy Þ þ exp½iðY x sx þ Y y sy Þ ;

ð16Þ

Ç. Arpali et al. / Optics Communications 282 (2009) 3216–3222

the source plane excitation of a general type beam which is composed of seven different Gaussian beams. These Gaussian beams are derived by using Eq. (1) of Ref. [15] taking different values of the parameters, namely phase h, amplitude factor A, Hermite mode orders n, m, displacement parameters Vx, Vy, and the source sizes asx, asy Using Eq. (13), received intensity distribution of this beam is obtained. In Fig. 4, selecting the source and medium parameters as shown in the inset and the caption of Fig. 4, the views of the 3D intensity distribution of this general type beam are plotted along the propagation axis. Figs. 5–8 are plotted for k = 1.55 lm. For the discrete beams, in order to accurately observe the diffraction in the outer regions of the receiver planes, the number of pixels at the receiver planes are taken accordingly. Thus, the number of pixels taken for Fig. 5a–d, Fig. 6a and b, Fig. 7a and b, Fig. 8a and b are 71  71, 71  71, 141  141, 355  355, 71  71, 355  355, 71  71, 141  141, 71  71, 141  141 pixels, respectively. Taking the number of pixels high will naturally increase the accuracy of the intensity plots. For the plots reflecting the receiver plane intensities, our criterion for choosing the number of pixels is to observe the normalized intensity profile to a reasonable extent, under the appropriately chosen receiver plane size and within reasonable computer run time. In Fig. 5b and c however, we have intentionally chosen varying number of pixels so that we could see the interim

3221

details of the same picture with different sizes until we reach the full picture in Fig. 5d. For Fig. 5, we generate a chess board shaped beam at the source plane. Amplitude and phase information for each source pixel area are set at the source plane to generate the chess shaped beam. We note that at the source plane shown in Fig. 5a, which is divided into 71  71 pixels, each white and black square in the chess board screen is formed by 3  3 pixels. In Fig. 5, propagation of chess shaped beam in free space is examined through the intensity plot at the receiver planes with (b) 71  71 pixels, (c) 141  141 pixels and (d) 355  355 pixels for L = 1.5 km. For better evaluation, intensity plot of the beam is displayed perpendicular to the propagation axis, more brightness means higher intensity values. Fig. 5b–d shows the diffraction of the chess board shaped beam at different pixels and with increasing receiving plane areas. By the help of these plots, at a fixed resolution (number of pixels divided by the receiver plane area), the diffraction pattern at the receiver plane can be comprehensively investigated at the outer regions. From Fig. 5, it is seen that, the intensity level of this beam will be dense on the center of the receiver plane along the propagation axis. As a check case, in Fig. 6, diffraction pattern of the single white square located at the center of the chess board shaped beam is obtained by using 3  3 = 9 pixels to represent the central single white square. The diffraction pattern is obtained by using 355  355 pixels at the receiver plane. The length of each side of this square at the source plane, as seen in Fig. 6a, is around 0.845 cm and the diffraction pattern of this central white square in Fraunhofer region at the propagation distance of 1.5 km has a central lobe of size about 55 cm which can be seen in Fig. 6b. This diffraction size is in good agreement with the theory. Applying the relevant amplitudes and phases to Eqs. (1) and (2) we obtained an arbitrary source beam consisting of letters Ç.Ü. (the initials of our Çankaya University) and the receiver plane intensity of such a beam is represented in Fig. 7, along with the associated source plane intensity. As noted earlier, the receiver planes given in Figs. 7b and 8b are constructed by 141  141 pixels. According to Fig. 7, source beam field is diffracted by the propagation channel and at higher propagation lengths, spreading will increase. Finally, in Fig. 8, arbitrary beam is investigated where the source plane, as shown in Fig. 8a is composed 71  71 pixels out of which 41  41 pixel amplitude values are assigned at random using a normally distributed random variable with mean 0 and variance 1. The remaining pixel amplitudes in Fig. 8a are taken to be zero. The phase values of this source field are taken to be zero at all the pixels. Fig. 8 exhibits the intensity of such a beam at the source plane and at the receiver plane of L = 1.0 km. As seen in this figure, propagation in free space results in the redistribution of the intensity spots of the source beam. We note that to obtain the receiver intensity diagrams in a reasonable accuracy, source pixel sizes must approach zero and very large number of pixels must be taken into account, which in turn requires excessive computation time.

4. Conclusion

Fig. 8. Intensity distribution of an arbitrary beam with random source field amplitude at (a) the source plane with 71  71 pixels and (b) the receiver plane with 141  141 pixels for L = 1.0 km.

In this article, we introduce the source plane excitation of arbitrary beam fields and using such excitations and Huygens Fresnel diffraction integral, we formulate the intensity at the receiver plane located transversely on the propagation axis. Arbitrary source field profile is produced by decomposing the source into pixels. The received field after propagation through free space is obtained by superposing the contributions of all the fields originating from each source pixel. Utilizing our formulation, we evaluate the received intensity originating from any type of source field excitation and correctly checked with the already existing results in literature of cos-Gaussian, cosh-Gaussian and general type beams. Then, the received intensity distributions are obtained for

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some original arbitrary source field profiles. Our results can be applied in free space optics communication links, in problems of reflection from rough surfaces, medical surgical operations, optical cryptography and optical imaging systems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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