An initial experiment was performed with the beam expander arm blocked. This meant that the expanded HG mode was not observed and the intensity structure of the LG modes themselves could be observed. After selecting a high order HG mode from the laser, the intensity distribution of the beam after it was transmitted by the mode converter was examined. These intensity distributions were observed simply by placing a screen in the beam path. They were also recorded by using a CCD (charge-coupled-device) array interfaced to a framegrabbing card running on a computer so that they could be recorded.
HG mode
Mode Transformation using Cylindrical Lens Mode ConverterCorresponding
^
LG mode
HG 5,0
LG 5
HG 3,1
LG 2
Figure IV.c Photographs of input HG modes and the output LG modes after mode transformation by the cylindrical lens mode converter.
Basically, these images confirmed the results presented in the work by Beijersbergen et al [3]. At this stage, it was a case of checking that the cylindrical lens mode converter was transforming the HG modes into LG modes correctly. Figure IV.c shows images taken of two input HG modes and the resulting output LG modes from the mode converter.
Once the various intensity patterns of the input HG and output LG modes had been observed and confirmed, the beam telescope arm of the Mach- Zehnder interferometer was unblocked. This allowed the LG mode to be superimposed onto the expanded HG mode, aligned such that the LG mode fell within one of the greatly expanded HG modal lobes. The resulting interference pattern could be observed on the screen. The expanded HG mode acted as a plane-wave reference and the observed interference fringes revealed the relative phase variations of the LG mode being studied.
Figure IV.d shows images of the intensity and interference patterns obtained for several higher order HG modes and their corresponding LG modes after mode conversion. In all cases the number of radial fringes was equal to I
and the number of radial nodes was equal to p+1. For the LG modes with multiple ring amplitudes (e.g. the LG^i, which is obtained from the HG^ ^), the azim uthal position of the fringe maxima and minima is reversed between successive rings, corresponding to the change of phase between successive rings of the Laguerre-Gaussian distribution. Note also that in the special case of the m = n HG mode, there is an on-axis intensity for the corresponding LG m ode, and no azim uthal phase dependence. Consequently, these fringes are circular and not spiral demonstrating a constant azimuthal phase. The radii of the fringes can be calculated from the wavefront curvature of the beam and are analogous to Newton's rings.
HG mode ---^ LG mode
Interferogram between
Mode
LG mode and plane wave
Conversion
HG 1,0
LG 1
H G 5 , 0LG5o
H G 3 , 1LG 2
# # # #HG 3,3
LGO3Figure IV.d Photographs of HG modes, the corresponding LG modes and the interferograms between LG modes and a plane-wave reference.
The propagation of the LG modes was also investigated. When the CCD array was moved along the propagating beam, the interference patterns were observed at various positions about the beam waist (figure IV.e). At the beam waist itself, the 6n azimuthal phase change around the LG^^ mode gave rise to three dark radial fringes. No spiral fringes were observed as there was no radial phase dependence. Away from the beam waist, spiral fringes were observed.
These arise from the combination of the radial phase variation due to w avefront curvature, and the tangential phase variation due to the azimuthal phase dependence. The sense of the spiral is reversed either side of the waist in accordance with the curvature of the wavefronts, thus changing the radial phase dependence.
The sense of the spirals could also be reversed by changing the sense of the azimuthal phase variation. This is most readily achieved by rotating the mode converter about the optical axis by 90° (i.e. from +45° to -45° with respect to the input mode). Alternatively, the input HG mode could be rotated by 90°, changing its orientation with respect to the cylindrical lens mode converter. Either of these procedures changed the sense of the spirals.
HG 3 ,0 mode Mode Conversion LG 3 0 mode Interferogram of LG 3 0 mode and
plane wave reference before beam waist
Interferogram of LG 3 0 mode and
plane wave reference at the beam waist
Direction of
propagation LG 3 Interferogram of
0 mode and
plane wave reference after beam waist
Figure IV.e Photographs of input HG^ ^ mode and corresponding LG^^ mode and the interferograms as it propagates through the beam waist.
IV. 3
Second harmonic generation using LG
laser modes
This section will describe an experiment that we performed to investigate the second harmonic generation using Laguerre-Gaussian laser modes. I performed the initial proof-of-principle experiment, while the bulk of the subsequent results and analysis were undertaken by Kishan Dholakia. The experimental details and the results are presented here for completeness and were published in the Physical Review A in November 1996 as a rapid communication [4]. Our work was also presented as a Postdeadline Paper at
QELS '96 in Anaheim, USA [5] and also at the lEEE/LEOS Meeting of the Scottish Chapter at Heriot-Watt University [6]. The experiment sought to
observe the second harmonic generation (SHG) of non-zero order LG light beams and show that the modes become converted to a higher order. This is consistent with the fact that the modes possess orbital angular momentum which is conserved in the nonlinear process.
This type of experiment was initially carried out in 1993 by Basistiy et al [7], who frequency doubled a beam that contained a phase dislocation along its axis to a beam containing two phase dislocations. At the time they failed to understand the mechanism of this process and the work we have carried out investigated more fully this process with respect to the orbital angular momentum of light.
IV. 3.1
SHG and orbital angular momentum of light
LG modes have been shown theoretically to possess a well-defined orbital angular momentum oi ifi per photon [8] where i is the azimuthal phase
index of the mode. This is in addition to any spin angular momentum due to their state of polarisation. Upon frequency doubling, the resulting higher mode order can be explained by considering the orbital angular momentum content of the input and output beams.
If the input beam, of frequency co, has an amplitude E®, then the second harmonic of frequency 2o), will have an amplitude proportional to the
^2co
V i (4.2)
which for a fundamental Gaussian mode means that the spot size is reduced by a factor of V2 compared to the input fundamental beam's spot size.
CO" yj = 4 Gxp
-2 r (4.3)
If this same approach is considered for a Laguerre-Gaussian laser beam, whose amplitude is given by Equation 2.3, then for a simplified case at the beam waist for a p=0 mode, the amplitude is
A^exp —r
CO" ex "V
2
(4.4) Upon squaring for the second harmonic generation process, the harmonic is a p=0 LG mode with the £ index of the azimuthal phase term becoming 2£ as follows.
= £[&xp —^ exp(-/70)
y c o j = expy2œ^ jexp(-/2/0 ) (4.5)
Not only has the frequency doubled but so has the angular momentum per photon {£% » 2£ht). This can be thought of as two photons combining their energies of fico to form a single photon of energy 2fico. At the same time, the two units oi £fi orbital angular momentum have combined to form a photon with orbital angular momentum of 2£H. This is not possible for circularly polarised light as a second harmonic can't have spin angular momentum of
2h.