by Simin Feng, Herbert G. Winful Opt. Lett. 26, 485-487 (2001) http://smos.sogang.ac.r April 18, 2014
Introduction What is the Gouy phase shift? For Gaussian beam or TEM 00 mode, ( w 0 r 2 E(r, z) = E 0 w(z) exp w(z) 2 iz i r 2 ) 2R(z) + iφ G(z) (1) Along its propagation direction, a Gaussian beam acquires a phase shift which differs from that for a plane wave with the same optical frequency. This difference is called the Gouy phase shift, named after L. G. Gouy who first discovered this phenomenon. φ G (z) = arctan(z/z R ) (2) where z R = πw0 2 /λ is the Rayleigh length and z = 0 corresponds to the position of the beam waist.
Discovered more than a century ago - but where is it from? L. G. Gouy Although Gouy made his discovery more than 100 years ago, efforts are still being made to provide a simple and satisfying physical interpretation of this phase anomaly. In this letter, the authors provide a simple intuitive explanation of the physical origin of the Gouy phase shift.
Derivation - for Gaussian beam Consider a monochromatic wave of frequency ω and wavenumber = ω/c along z direction. If it is an infinite plane wave, the momentum has no transverse components. A finite beam, however, consists transverse components of momentum. The wavenumber can be written as: 2 = 2 x + 2 y + 2 z (3) Since the wave-vector components are finitely spread, it is appropriate to deal with expectation values (or simply averages) defined by: ξ = ξ f (ξ) 2 dξ f (ξ) 2 dξ (4) where f (ξ) is the distribution of the variable ξ
Derivation - for Gaussian beam The effective axial propagation constant can be defined as z = 2 z = 2 x 2 y (5) which is associated with the overall propagation phase φ(z) through z = φ(z)/ z. By integrating with respect to z we get φ(z) = z 1 z { x 2 + y 2 }dz (6) Here the first term stands for phase evolution of infinite plane wave, and the second gives rise to the Gouy phase shift φ G = 1 z { x 2 + y 2 }dz (7)
Derivation - for Gaussian beam φ G = 1 z { 2 x + 2 y }dz (8) Hence, the Gouy shift is due to the transverse momentum spread. It can be well formulated since we now transverse distribution exactly. For Gaussian beam, transverse distribution is given by: 2 1 f (x, y) = [ π w(z) exp x 2 + y 2 ] w 2 (z) [ ( ) ] 2 Here w 2 (z) = w0 2 1 + z z R is beam radius and w 0 is the minimum spot size at z = 0. The Rayleigh length z R is defined by z R = πw 2 0 /λ (9)
Derivation - for Gaussian beam The distribution of transverse wavevector components is given by the Fourier transform. F ( x, y ) = 1 2π 1 1 = 2π 3 w(z) = w(z) [ exp 2π f (x, y)e ix x e iy y dxdy (10) w 2 (z) (x 2 + y 2 ) 4 exp [ x 2 + y 2 w 2 (z) ] ] e ix x e iy y dxdy We can see this is also Gaussian and centered about x = y = 0. And both of the distribution are normalized such that f (x, y)dxdy = F ( x, y )d x d y = 1
Derivation - for Gaussian beam Now, we have 2 x = = x 2 = w 2 (z) 2π x 2 F ( x, y ) 2 d x d y w 2 [ (z) 2π exp w 2 (z) 2 [ x 2 exp w 2 ] (z) x 2 d x 2 = 1 w 2 (z) = 2 y ] (x 2 + y 2 ) d x d y [ exp w 2 (z) y 2 2 ] d y The Gouy phase shift for the Gaussian beam is then: φ G = 1 z { x 2 + y 2 }dz = 2 z 1 w 2 (z) dz = arctan(z/z R) (11)
Generalization - for higher-order transverse modes We now show that our equation predicts the phase anomaly not only for fundamental Gaussian beams but also for higher-order transverse modes and hence is valid for arbitrary field distributions. One complete set of transverse modes is described by Hermite-Gaussian beams. [ ] [ ] 2 2x 2y f mn (x, y) = C mn w(z) Θ m Θ n w(z) w(z) (12) where Θ( m (ξ) = H m (ξ) ) exp( ξ 2 /2) is the Hermite-Gaussian of mth order and 1/2 C mn = is normalization coefficient. 1 π2 m+n m!n!
Generalization - for higher-order transverse modes The Fourier transform of Hermite-Gaussian distribution is [ ] [ ] F mn ( x, y ) = ( i) m+n w(z) w(z)x w(z)y C mn Θ m Θ n 2 2 2 (13) Utilizing the recursion relation H n+1 2ξH n + 2nH n 1 = 0 for the Hermite polynomials, one can derive the expectation values. x 2 mn = 2 ( w 2 m + 1 ) (z) 2 y 2 mn = 2 ( w 2 n + 1 ) (z) 2 Hence, Gouy shift for TEM mn modes are formulated by: φ G,mn (z) = (m + n + 1) arctan(z/z R ) (14)
Conclusion A general expression and physical explanation of the Gouy phase shift is discussed, by showing that the Gouy phase can be derived from the transverse momenta. Gouy shift of finite beams stems from transverse spatial confinement, which consequently causes transverse momentum component. This approach is valid for higher transverse modes, as well as for fundamental transverse (Gaussian) mode.