1.4 Maxwell’s wave equation in a dielectric medium
1.4.6 Laguerre-Gaussian beam
If the laser’s optical resonator is cylindrically symmetric, the natural solutions of the paraxial wave equation result in Laguerre-Gaussian modes [140]. Thus, the solution of the paraxial Helmholtz equation in cylindrical coordinates (ρ, φ,z) leads to the LaguerreGaussian modes.
The complex amplitude of the Laguerre-Gaussian beam is expressed as
Epl(ρ, φ,z)=E0 w0 w(z)
Ãρ√ 2 w(z)
!|l|
L|pl| Ã 2ρ2
w2(z)
!|l|
exp
"
ikρ2
2R(z) − ρ2 w2(z)
#
×exp
"
−i(2p+l+1) tan−1 Ã z
zR
! +ilφ
#
, (1.172)
where L|pl| is the generalized Laguerre polynomial function, p is the radial index and l is the azimuthal index. The lowest-order Laguerre-Gaussian beam l = p = 0 coincides with the Gaussian beam. The other beam parameters are same as a Gaussian beam case. The Laguerre- Gaussian beam can be produced using a computer generated holograms [141, 142], or all- optical spatial light modulator in coherent media [143] and digital micro-mirror device [144].
Di ff ractionless optical cloning via single dark states
In this chapter, we theoretically explore the possibility of cloning of an arbitrary image car- ried on control field to probe field with high resolution. For this purpose, we utilize the EIT and CPT resonances in presence of spatially dependent coherent fields to obtain higher spa- tial resolution. A rigorous literature survey suggested the possibility of an alternative way of imaging through various theoretical and experimental approaches based on EIT and CPT schemes. Mitsunaga et al. have suggested and experimentally demonstrated an EIT based absorption imaging technique in cold sodium atoms [145]. They found that a bright spot of signal beam is transmitted through an opaque atomic cloud at the points where coupling beam was present. In their experimental set up, they achieved signal beam transmission upto 200%
which is due to cross-focusing effect induced by spatial dependent coupling beam. Based on CPT mechanism, Agarwal et al. have predicted that the atom can be localized at sub- wavelength scale [146]. They have used standing-wave and tightly focused beams to spatially localize the atomic population in subwavelength domain. In an extension of this CPT based scheme, Yavuz and colleagues have discussed and reported the atom localization by spatial dependent dark state [147]. They predict that a fluorescence shadow image of a nanometer sized object implanted into an atomic medium can be obtained by scanning the focusing lens around the object. In an early work, Kapale and Agarwal [148] have found a new optical microscopy technique by using CPT mechanism. In this scheme, three-level atom is driven by an amplitude modulated probe field and a spatially dependent coupling field to localize the
population. Additionally, they have predicted that localization scheme can be used to get a shadow image of nanometer sized object embedded into the atomic medium.
More recently, Li et al. [153] experimentally demonstrated that the spatial shape of a control beam can be cast onto a weak probe beam via CPT in a three-level lambda atomic system. In their experiment, the transmitted intensity of the probe beam had a similar spatial profile as that of the control beam, no matter what the input probe is. Moreover, the size of the transmitted probe beam was half of that of the diffraction-limited input probe.
These studies motivate us to explore other possibility of mapping the spatial shape of control beam onto probe beam. The spatial profile of the strong control beam makes the probe’s susceptibility inhomogeneous along the transverse direction. This inhomogeneity in susceptibility leads to spatial variations in both absorption and refractive index of probe beam. Key idea behind cloning mechanism is that the absorption and refractive index profiles are dependent on shape and intensity of control beam. Thus, the absorption and refractive index profiles can be flexibly engineered along the transverse direction with desired shape of control beam.
In our study, both control and probe fields are coupled to a three-level atomic lambda system to form a CPT configuration. It is important to note that the probe field is typically treated as a weak and control field as a strong, such that perturbation theory can be employed to derive the linear effect of the atomic medium on the probe field propagation. There have been theoretical and experimental studies where the probe is not necessarily weak [149–151].
In this situation, the effect of the atomic coherence on the propagation dynamics of the control field need to be taken into account [152]. We assume the two laser fields to be of comparable strength, such that perturbation theory for the probe field is not valid any more to describe the effect of the atomic medium on the two fields. We start by calculating the susceptibilities including linear and nonlinear effects for both fields by solving the related density matrix equations. As expected, we find that a spatially-dependent refractive index for the probe field is generated, structured by the spatial intensity profile of the control beam. In particular, the generated structures enable one to transfer the transverse distribution of the control field onto the transmission profile of the probe field. In order to study the full propagation dynamics, we then numerically solve the paraxial propagation equations for both fields by using a higher order split operator method. We begin our analysis with a Gaussian control and a super- Gaussian probe field and observe the gradual mapping of the control field onto the probe field
throughout the propagation. We find in particular that in the case of a strong probe field, the transmitted probe beam is focused more tightly by a factor of two compared to the weak probe field case. Next, we consider a control field with a spatial two-peaked Hermite-Gaussian profile, and demonstrated cloning of the profile onto the probe beam with feature size reduced by a factor of about 2.5. In order to verify that our method can serve as an universal tool for cloning of arbitrary image, we finally simulate the three-dimensional light propagations for both fields, in which the spatial profile of the control field carries the three letters “CPT”.
We show that also this structure can be cloned onto the probe beam which initially has a simple plane-wave profile, even though the control field is severely distorted throughout the propagation due to diffraction. Again, we observe a reduction of the feature size by a factor of about 2 in the probe field.