Theoretical Study of Electromagnetic Wave Propagation: Gaussian Bean Method

Applied Mathematics, 2013, 4, 1466-1470 http://dx.doi.org/10.4236/am.2013.410198 Published Online October 2013 (http://www.scirp.org/journal/am)

E. I. Ugwu, J. E. Ekpe, E. Nnaji, E. H. Uguru Department of Industrial Physics, Ebonyi State University, Abakaliki, Nigeria Email: [email protected]

Received March 21, 2013; revised April 21, 2013; accepted April 29, 2013

Copyright © 2013 E. I. Ugwu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT In this work, we present the study of electromagnetic wave propagation through a medium with a variable dielectric 2 E function using the concept of Gaussian Beam. First of all, we start with wave equation 22E   0 with r t 2 which we obtain the solution in terms of the electric field and intensity distributions approximate to Gaussian Function, 2r2  2    Ixy,  e . With this, we analyze the dependency of r on Gaussian beam distribution spread, the distant from the 1 axis at which the intensity of the beam distribution begins to fall at a given estimate of its peak value. The influ- e2 ence of the optimum beam waist wo and the beam spread on the intensity distribution will also be analyzed.

Keywords: Electromagnetic Wave; Wave Equation; Dielectric Medium; Distribution; Gaussian Function; Intensity; Electric Field; Beam Waist; Wave Propagation

1. Introduction angular deflection on reflection of a Gaussian beam at a dielectric interface with two or more layer [7,8]. Along with the rapid evolution of fiber optics, integrated However in this paper, we intend to study electromag- optics and application of laser in both medicine and netic wave propagation using the concept of Gaussian technology, there has been a growing interest in the study beam starting from general wave equation with which of Gaussian beam propagation. This is based on the fact obtain the Gaussian function in which waves operating that Gaussian beam has to do with focusing and modifi- on the fundamental transverse mode is approximated to cation of shape of propagating electromagnetic wave or Gaussian profile. The distribution profile is analyzed with laser beam. From the earlier finding on the research on intent to observe the influence of the beam waist on the laser beam, it has been found that laser beam propagation profile. can be approximated by an ideal Gaussian beam intensity profile [1,2]. Understanding of the basic properties of 2. Theoretical Framework Gaussian beam has been specifically found to be very vital. I select the best optics for practical application [3]. In this case we start with wave equation in terms of elec- Sequel to this, lots of scientists had worked on Gaussian tric field given as beam applications in electromagnetic wave propagation 2 22 E and in optics. For instance, a work has been carried out  E r 0  0 (1) t 2 on nonspecular phenomena for beam reflection at mono- layer and multilayered dielectric interface respectively where  r is the dielectric constant as a function wave from where it has revealed that under various conditions solution of the form nonspecular beam phenomena is more realizable [4-6]. EExyz,,,exp  itkr r  (2) Tamir on his own presented a unified and simplified analysis of the lateral and longitudinal displacement with Which if we consider medium with non-uniform re-

  fractive index, we have where pz and qz are the complex conjugate of 22 pz and qz respectively hence E r 0  0 (3) ikr 2 11 where the propagation constant in the medium is given  (12) EEexp ipzz p  exp   by z zqq z z

k   r 0 (4) Considering two real beam parameters that Rz and With this, Equation (2) becomes  relating to qz by depend on z 22 11i EkrE (5)  , (13) qRw2 In general, if the medium is absorbing or exhibit gain, z the its dielectric constant  r is considered to have real pz is that describes the axial distance from the Gaus- and imaginary part, but in a situation where the medium sian beam waist. conducts with conductivity  . then the complex propa- We can express Equation (11) as gation vector is introduce which obeys. 2  ikr i i  EE exp 22 (14) 22 2 ww kir 1 [9]. (6) r 0 2r 2  I  exp (15) However, a simple solution of this type is inadequate xy1 2  w to describe the field distributions of transverse mode. As a result, we seek solution to the plane wave equation of the Signifying the dependency of Gaussian beam intensity form. to r while w signifies the distance from the axis at z 1 EExyz 0 ,, exp ikz (7) where the intensity of the beam falls to of it its peak e2 Propagating in the z-direction and localized the z-axis. value on the axis pz total power of the beam with Thus the idea is to obtain a solution of the wave equation these parameters Equation (10) is now written as that gives phase front that can be approximated over a kr2 1 i  narrow region. Thus substituting Equation (7) into Equa- Eikzpexp   (16) z 2  tion (4) we obtain.  2 R  2 22  E0 EE  E pz is evaluate to give 2222ik 0 (8) xyz z qq 0  z (17)

Since we are looking for a paraxial beam-like solution, where q0 is a constant of integration the expresses the then E0 varies slowly with z and Equation (8) becomes value of the beam parameter at the plane z  0 for 22 dpi EE E0 which when we consider  , we obtain 22zik 0 (9) xy z dzqqzo 

With a solution given as pizqz In 0  iqIn 0 (18) k where   iqIn defines a constant of integration sub- Exyz,, exp ipz  r2 (10) 0 0  stituting Equation (18) into (16) we have wz  2 where rxy222 is the square distant of the points x,  zkri1   Eikziexp  In 1  2  y from the axis of propagation where p and q qzR z z 0  are beam parameters Equation (10) gives the fundamen- (19) tal Gaussian beam of time-independent wave equation. The factor exp pi is a constant phase factor. Thus

3. Intensity of Gaussian Beam   zkri2 1   Eikziexp  In 1  2  (20) The intensity of I of Gaussian beam is given by qzR x, y   0   IEExy,  4. Diffraction Effect  (11) kk2 2 exp ip z rexp   ip r The limitations of Gaussian beam based on the fact that 2qz z 2q  z even if the wave fronts were made flat at some plane, it

Copyright © 2013 SciRes. AM 1468 E. I. UGWU ET AL. quickly acquires curvature and begin to spread in accor- 0.8 dance with 2 w2 Rz1 0  (21) 0.6 z  z  and 1 0.4 2 2  z ww1  (22) z 0 w2 0 0.2 where z is the distance propagated from the plane where the wave-front is flat   wavelength of light, w is the radius of the 1 irradiance contour at the e2 0 10 5 0 5 10 plane where wave-front is flat, wz is the radius of the 1 contour after the wave has propagated a distance Figure 1. Gaussian distribution profile when y = 2, w = 3. e2 z . 0.8 Considering the optimum starting beam radius for a 12  z distance z , we have w0  , Equation (22) can 0.6  be reduced to w 3 w (23) z 2 0 0.4 With the irradiance distance distribution of the Gaus- sian beam described as 2r222 p 2r 0.2 IIr exp exp  (24) 0  22w  2 where ww and p is the total power in the beam z 0 which is the same at all cross sections of the beams we 10 5 0 5 10 also obtain that Figure 2. Gaussian distribution profile when y = 2, w = 4. I 2 r 2r 2 30 In2 2r (25) 1 I0 w 2

Ir In 3 x22 y  (26) I 0  0 at optimism starting beam radius for a given distance, gx() 0 z . 2 p I  w (27) 0w2 0

5. Discussion 1 From the displayed results in Figure 1, the distribution 10 5 0 5 10 profile when y = 2, x = 2 and w = 2, the distribution beam x width ranged within −5 to 5, In Figure 2, for y = 2 and x Figure 3. Gaussian distribution profile when y = 2, w = 5. = 3 when w = 2, the width of the beam ranged from −6.5 to 6.5. In Figures 3 and 4 the distribution profiles dis- y = 2 when w = 10, the departure became more pro- played is found to be departed from normal distribution nounced pattern as shown in Figure 5. However, the shape as in the other figures when w is increased to 5 for distribution profile of percentage irradiance as function

2 wz  is over a distant z and a particular wavelength. z Figure 8 1 6. Conclusions fx() Gaussian intensity profiles as observed from this study seem to manifest one feature that is based on the fact that 0 beam waist, w plays a role in determining the distribution profile. This explains the fact that the far-field divergence

1 10 5 0 5 10 x Figure 4. Gaussian distribution profile when y = 2, w = 5.

fx() 0

Figure 6. Percentage irradiance.

1 10 5 0 5 10 x Figure 5. Gaussian distribution profile when λ = 0.50 m, y = 2, w = 10. contour radius as displayed in Figure 6 indicate the fact that as the contour decreases, the beam distribution tap- pers showing that there is a point known as spot size 1 where the intensity of the distribution is fallen to . e This indicates that Gaussian distribution function depends on r as shown in Equation (16) that depicts char- Figure 7. Gaussian distribution intensity as function of pro- acteristic of normal distribution function. file width, w. However, the shape of the distribution depends gener- 1 ally on diameter at which the intensity has fallen of e as in Figure 7 when x = y coupled with the optimum 12 z beam waist that is given as wo optimum  ,  axial distance and Raleigh range which is given in Equa-

wo tion (22) with zR  [9] These values provide the  best combination for minimum starting beam diameter Figure 8. Gaussian beam width w(z) as a function of the for good minimum spread according to [10] whose ratio axial distance z.

Copyright © 2013 SciRes. AM 1470 E. I. UGWU ET AL. must be measured at a distance greater than the half width ences Books, 1986. of intensity distribution while the near or mid-field di- [2] S. A. Self, “Focusing of Spherical Gaussian Beam,” Ap- vergence values are obtained by measuring a distance less plied Optics, Vol. 22, No. 5, 1983, pp. 658-661. than the half width of the intensity distribution in order to [3] H. Sun, “Thin Lens Equation for a Real Laser Beam with obtain a good normal distribution profile. Therefore in Weak Lens Aperture Truncation,” Optical Engineering, order to achieve optimum beam spread, and collimation Vol. 37, No. 11, 1998, pp. 2903-2913. over a distance, the optimum starting beam radium/waist http://dx.doi.org/10.1117/1.601877 must be determined. Though one should note that this [4] J. W. Ra, H. L. Bertoni and L. B. Felsen, “Reflection and depends strictly on the type of beam that is whether the Transmission of Beam at Dielectric Interface,” SIAM Journal on Applied Mathematics, Vol. 24, No. 3, 1973, beam is coherent or not, the general expression for opti- pp. 396-413. mum starting beam radius for any given distance, z is 12 [5] M. McGuirik and C. K. Carniglia, “An Angular Spectrum  z Representation Approach to the Goos-Hänchen Shift,” w0 optimum  .  Journal of the Optical Society of America, Vol. 67, No. 1, This study also reveals that the concept of Gaussian 1977, pp. 103-107. http://dx.doi.org/10.1364/JOSA.67.000103 beam is realizable when it is focused on a small spot as it spreads out rapidly as it propagate away from the spot [6] R. P. Riesz and R. Simon, “Reflection of a Gaussian Beam from a Dielectric Slab,” Journal of the Optical So- From Figures 3 to 6, it is clearly shown that at certain ciety of America A, Vol. 2, No. 11, 1985, pp. 558-565. value of the beam waist, the distribution profile changes. [7] T. Tamir, “Nonspecular Phenomena in Beam Field Re- Therefore for a beam to be well collimated, it must have flected by Multilayered Media,” Journal of the Optical a large diameter. This relation is as a result of diffraction. Society of America A, Vol. 3, No. 4, 1986, pp. 586-594. Thus in order to achieve normal Gaussian beam distribu- [8] S. Z. Zhang and C. C. Fan, “Nonspecilar Phenomena on tion, the product of the width and divergence of the beam Gaussian Beam Reflection at Dielectric Interface,” Jour- profile must be as small as possible to be achieved when nal of the Optical Society of America A, Vol. 5, No. 9, paraxial approximation is considered. Thus this implies 1988, pp. 1407-1411. that the idea of Gaussian beam model is realizable and http://dx.doi.org/10.1364/JOSAA.5.001407 valid only for beam whose width is larger than that of [9] F. Pampaloni and J. Enderlein, “Gaussian, Hermite- equation. Gaussian, and Laguerre Gaussian Beam: A Primer,” 2004, 29p. [10] J. Ralton, “Gaussian Beams and the Propagation of Sin- REFERENCES gularity, Studies in Partial Differential Equations,” MAA [1] A. Siegman, “Lasers Sensitization,” C.A University Sci- Studies in Mathematics, Vol. 23, 1982, pp. 206-248.