We solve the pulse propagation Eq. (1.122) in a reference frame moving with the velocity of light in vacuum c. We transform to new variables
τ =t−z
c, ξ =z. (1.123)
so that
∂
∂z + 1 c
∂
∂t = ∂
∂ξ, ∂
∂t = ∂
∂τ. (1.124)
In new frame, Eq. (1.122) simplifies into the following form
∂E
∂ξ = 2πikP. (1.125)
Eq. (1.125) governs the propagation of light pulse E(ξ, τ)through the atomic medium [54]. The polarization P(ξ, τ)in Eq. (1.125) is the source term which determines how the light pulse propagates through the material medium.
1.4.2 Propagation equation for light beam
We now study the propagation equation of a light beam. In case of a continuous wave quasi-monochromatic field, the amplitude E(x, y, z) does not vary with time i.e.
∂E/∂t= 0. Therefore, the propagation equation for a light beam is given by 1
2ik∇2⊥E +∂E
∂z = 2πikP. (1.126)
The above form of wave equation is known as paraxial wave equation. We can further expressed Eq. (1.126) in terms of Rabi frequency Ω as shown below
∂Ω
∂z = i
2k∇2⊥Ω + 2πikχΩ, (1.127) where χ is the susceptibility of the medium. The first term on right hand side is a second order partial derivative in the xy plane i.e. ∇2⊥ = (∂2/∂x2+∂2/∂y2) which describes inherent optical diffraction of the light beam. The second term on right hand side incorporates the dispersion and absorption profile of the medium.
Fig. 1.8 (a) Intensity profile of a Gaussian beam in transverse plane. (b) The variation of Gaussian beam width w(z) along the propagation axis. The minimum beam waist isw0. The Rayleigh lengthzR of the Gaussian beam is the distance from the minimum beam waist where the beam widthw(z)is increased by a factor of the square root of 2 i.e. w(zR) = √
2w0. The divergence angle of the beam is given by Θ.
electric field. Thus the paraxial wave equation in free-space is given by
∂2E
∂x2 + ∂2E
∂y2 + 2ik∂E
∂z = 0. (1.128)
In Eq. (1.128), we consider that the variation of electric fieldE(x, y, z)along the z-axis are very small as compared to transverse directions (x, y) within a distance of the order of a wavelength. This consideration can be demonstrated mathematically in the following form
∂2E
∂z2
≪
∂2E
∂x2 ,
∂2E
∂y2 ,
k∂E
∂z
(1.129) Eq. (1.129) is known as paraxial wave approximation. Eq. (1.128) offers an infinite set of functions as a solution such as Gaussian, Laguerre-Gaussian and Hermite-Gaussian modes. Next, we study some of these solutions of the paraxial wave equation in great detail.
Gaussian mode
The Gaussian mode is a basic solution of the paraxial wave equation. In Fig. 1.8(a), we show the intensity profile of a Gaussian mode in transverse plane. The complex
electric field amplitude of Gaussian mode can be expressed as E(x, y, z) =E0 w0
w(z)exp
−(x2+y2)
w2(z) +ik(x2+y2)
2R(z) −ikz−itan−1 z
zR
, w(z) =w0
s 1 +
z zR
2
, R(z) =z+ zR2
z , zR =πw20/λ. (1.130) In Eq. (1.130), w(z)represents the spot size of the beam. The minimum value of spot size is w0 which occurs at z = 0. The quantityR(z) is known as radius of curvature of the beam’s wavefront. The parameter zR is called Rayleigh length. Its value implies the distance on z-axis from minimum beam waist to the point at which the beam width w(z)becomes √
2w0. Rayleigh length depends on the minimum beam waist w0 and wavelength of light beam λ. In Fig. 1.8(b), we portray the variation of Gaussian beam width w(z)along the propagation axis. Fig. 1.8(b) clearly indicates that size of the beam w(z)increases with distance as the Gaussian mode propagates along z-axis.
The divergence angle of a Gaussian mode for a distance z ≫zR is given by Θ = 2θ = 2w(z)
z ,
≈2w0 zR
= 2λ πw0
. (1.131)
Eq. (1.131) clearly shows that divergence angle Θ is inversely proportional to the minimum beam waist w0. Therefore, a Gaussian mode with a narrow spot size spreads rapidly as it moves away from the beam waist w0.
Hermite-Gaussian mode
The paraxial wave equation in Cartesian coordinate offers a family of solutions known as Hermite-Gaussian modes. The transverse electric field distribution of Hermite-Gaussian mode is given by the product of a Gaussian function and a Hermite polynomial along with a phase term
E(x, y, z) = E0 w0 w(z)Hn
√2x w(z)
! Hm
√2y w(z)
! exp
−(x2+y2) w2(z)
×exp
−i(n+m+ 1)tan−1 z
zR
, (1.132)
where Hn and Hm are the nth and mth order Hermite polynomials. The indices m and n are non-negative integers. The first few Hermite polynomials are shown as a
Fig. 1.9 Intensity profiles of various Hermite-Gaussian modes HGmn are plotted at z = 0. The mode numbers n and m determine the shape of a specific profile in x and y plane respectively.
function of ζ =√
2x/w(z)
H0(ζ) = 1, H1(ζ) = 2ζ, H2(ζ) = 4ζ2−2,
H3(ζ) = 8ζ3−12ζ. (1.133) The polynomials shown in Eq. (1.133) determine the shape of the beam profile in the x and y direction. Intensity profiles of various Hermite-Gaussian modes HGmn are shown in Fig. 1.9. The intensity distribution clearly indicates that each mode has n nodes along the vertical direction and m nodes along the horizontal direction. For n = 0 andm = 0, we obtain the intensity distribution of a Gaussian mode. This HG00 mode is also called the fundamental mode.
Fig. 1.10 Intensity profiles of various Laguerre-Gaussian modes LGlm are plotted at z = 0. The mode numbersm and l determine the shape of a specific intensity profile in x and y plane respectively. LG00 mode represents intensity distribution of the fundamental mode. The other LGlm modes display concentric ring-shaped intensity distribution. A particular LGlm mode includes (m+ 1) rings except LG00 mode.
Laguerre-Gaussian mode
Laguerre-Gaussian modes are the general solution of paraxial wave equation in cylin- drical coordinate. The paraxial wave equation in cylindrical coordinate (r, ϕ, z)can be written as
1 r
∂
∂r
r∂E
∂r
+ 1 r2
∂2E
∂ϕ2 + 2ik∂E
∂z = 0, (1.134)
where we have considered the following relation
∂2
∂x2 + ∂2
∂y2 ≡ 1 r
∂
∂r
r ∂
∂r
+ 1 r2
∂2
∂ϕ2. (1.135)
Now, the solution of Eq. (1.134) is expressed as
Fig. 1.11 Phase profiles of various Laguerre-Gaussian modes LGlm are plotted atz = 0.
The mode numbers m and l determine the shape of a specific phase profile inx andy plane respectively. The total phase change in the transverse plane is given by 2πl.
E(r, ϕ, z) = E0
w0 w(z)
r√ 2 w(z)
!|l|
exp
− r2 w2(z)
Llm
2r2 w2(z)
eilϕ
×exp
ikr2 2R(z)
exp
−i(2m+|l|+ 1) tan−1 z
z0
; r =p
x2+y2, ϕ= tan−1y x
, (1.136)
where Llm are the generalized Laguerre polynomials with l is an integer and m is a non-negative integer. A few Laguerre polynomials are shown below
Ll0(ζ) = 1,
Ll1(ζ) =−ζ+l+ 1, Ll2(ζ) = 1
2[ζ2−2(l+ 2)ζ+ (l+ 1)(l+ 2)]. (1.137) In Fig. 1.10, we show the intensity profiles of various Laguerre-Gaussian modes LGlm at z = 0. The phase profiles of the corresponding Laguerre-Gaussian modes LGlm can be found in Fig. 1.11. The Laguerre-Gaussian modes display concentric ring-shaped
intensity distribution. The number of ring is determined by the mode index m. Any particular LGlm mode in Fig. 1.10 includes (m+ 1) rings except LG00 mode which represents intensity distribution of the fundamental mode like HG00 mode. The other mode index l decides the structure of azimuthal phase termeilϕ as shown in Fig. 1.11.
The total phase change in the transverse plane is given by2πl.