Medical, Pharma, Engineering, Science, Technology and Business

^{1}Council for Scientific and Industrial Research, PO Box 395, Pretoria, 0001, South Africa

^{2}University of KwaZulu-Natal, Private Bag X 54001, Durban 4000, South Africa

- *Corresponding Author:
- Bell T

Council for Scientific and Industrial Research

Council for Scientific and Industrial Research

Pretoria, GP, South Africa

**Tel:**+27 31 260 8596

**E-mail:**[email protected]

**Received Date**: November 28, 2016; **Accepted Date:** January 02, 2017; **Published Date**: January 15, 2017

**Citation: **Bell T, Ngcobo S (2017) Selective Excitation of Higher-radial-order Laguerre-Gaussian Beams Using a Solid-state Digital Laser. J Laser Opt Photonics 3: 144. doi: 10.4172/2469-410X.1000144

**Copyright:** © 2017 Bell T, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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In this paper, we use a digital laser to generate high-radial-order Laguerre-Gaussian, LGp,0 modes by loading digital holograms on a phase-only spatial light modulator that act as an end mirror of a diode-end-pumped laser resonator. The digital holograms were encoded with an amplitude ring mask, which contained absorption rings that match the p-zeros of the Laguerre polynomial. We demonstrate the generation of high-quality LGp,0 modes with a mode volume that is directly proportional to the mode order, p. This work demonstrates the possible of using the digital laser as a tool for simulating optical elements that will be used in pursuing high brightness lasers.

Lasers; Laser resonators; Spatial light modulators; Holographic optical elements; Laser beam shaping; Laser beam characterization

Since high power solid-state lasers are the most developing branches of laser science and they have become an increasingly important tool for modern technology inside the end pumped solid-state laser resonator [1]. A new design of these solid-state lasers in a form of a digital laser was recently discovered by South African researchers [2]. This technology may lead into a new era in laser manufacturing and their applications in the industry where this digital laser will be used as simulation [3] and testing tool [4,5] in many future laser manufacturing processes. The solid-state digital laser is pumped using a diode laser, due to better frequency stability, higher efficiency, higher brightness and long operational lifetime that is provided by diode lasers [6]. The digital laser comprises of an intra-cavity phase-only spatial light modulator (SLM) that acts as an end mirror of the laser resonator cavity. The SLM is used to introduce a phase mask and/or amplitude mask inside the laser resonator by digitally loading a grey-scale hologram image that represents a mask of interest. The digital laser is limited in the output power it can produce, when compared to standard diode-end-pumped solid-state lasers [7-9] due to low damage threshold of the SLM. Nevertheless, the digital laser is found to be a stable intra-cavity beam shaping tool, which can be used to simulate any desired intra-cavity optical element prior to it being manufactured. In this paper we show how to generate high-radial-order Laguerre-Gaussian (LGp,0) modes from zero up a radial order, p=3, by selectively encoding and loading amplitude digital holograms mask on a phase-only SLM that contain absorbing rings that match the p-zeros of the Laguerre polynomial. We show that the generated LGp,0 modes results are very consistent with similar previous [10] experimental results there were obtained using a physical diffractive optical element that was lithographically engraved with aluminium absorbing rings and inserted inside the resonator cavity. We show that the results obtained in this paper are far more superior because of the better design of the amplitude mask that allows for better mode matching of the laser the pump and the absorbing rings.

The electric field of Radial Laguerre-Gaussian modes, LGp0, where p is the radial order geometries can be mathematically represented by the following equation:

(1)

Where r is the radial coordinates, and L_{0} is the Laguerre polynomial.
All other parameters have their usual meaning [9,11]. LGp,0 consist
of central peak that surrounded by p concentric rings. The Laguerre
polynomial (Lp(X), letting X=2r^{2}/w^{2}) for the rings is define in **Table 1**.

p | Lp(X) |

0 | 1 |

1 | 1-X |

2 | X^{2}/2−2X+1 |

3 | −X^{3}/6+3X2/2−3X+1 |

**Table 1:** Laguerre Polynomials.

The spot size w is to only define the fundamental Gaussian beam, LG0,0 where the radial order, p, equals to zero. For higher-radial-order Laguerre-Gaussian, LGp,0, modes their intensity profiles consist of a central peak that is surrounded by p concentric rings of light and dark rings. The intensity profile of the dark concentric rings of the (LGp,0) modes matches the Laguerre polynomial p-zeros. The intensity profile of the LGp,0 modes is defined by the absolute square of Eq. 1 and the LGp,0 beam

Width, wp, which is based on a second moment radius is given as:

(2)

and the propagation factor, M2, of such modes is given as:

(3)

To force the laser to generate LGp,0 modes, a mask containing p
absorbing rings that have a geometry which closely follows the location of the Laguerre Polynomial p-zeros as illustrated in **Table 2** [12], were
digitally encoded to the SLM of the digital laser [2], which will then
allow for the generations of such LGp,0 modes.

p | r_{i}/w |
|||

0 | 0.707106 | |||

1 | 0.541195 | 1.306562 | ||

2 | 0.455946 | 1.071046 | 1.773407 | |

3 | 0.401589 | 0.934280 | 1.506090 | 2.167379 |

**Table 2:** Roots of Laguerre polynomials.

To generate radial-order Laguerre-Gaussian, LGp0, modes we
consider a planoconcave solid-state digital laser resonator that is
end-pumped with a multi-mode fibre coupled diode laser, where
the Hamamatsu spatial light modulator (SLM 1) is encoded with an
amplitude mask that will function as an end mirror of the resonator.
The amplitude mask is encoded to have p absorbing rings of varying
width thickness that will 80% match each null of the LGp,0 mode, for
radial-order p=1-3. A schematic of the experimental setup is presented
in **Figure 1**.

The gain medium Nd:YAG rod crystal (4 mm × 25 mm) had a 1.1%
neodymium concentration and was anti-reflection coated for 808 nm
to minimise pump reflections. The laser crystal was mounted inside
a 21°C water-cooled copper block. The pump diode laser (Jenoptic,
JOLD-75-CPXF-2P) had a maximum output power of 75 Watts at an
emission wavelength of 808 nm (at an operating temperature of 25°C).
The pump diode laser output was coupled into a fibre with a core
diameter of 400 μm and was fast axis collimated and a lens coupled
to end-pump the Nd:YAG crystal by using a 25.4 mm (L1) and 150
mm (L2) focal length spherical lenses, respectively. A gain area with a
diameter of 2 mm was then excited within the centre of the Nd:YAG
rod crystal. The planoconcave cavity comprised of a phase-only spatial
light modulator (SLM 1) (Hamamatsu) which acted as an end mirror
of the cavity with a reflectivity of 95% and a curved output coupler
mirror with a radius of curvature of 400 mm and a reflectivity of 90%.
The resonator was designed to form a Z-shape (in order to avoid
illuminating the SLM with the residual pump light) by including a 45° mirrors (M2) within the cavity that were highly reflective for 1064
nm and highly transmissive for 808 nm. The resonator length was
chosen to be 173 mm and the Nd:YAG crystal centre was positioned
120 mm from the SLM 1 [2]. The Line Filter (LF) was introduced to
only transmit 1064 nm and block the 808 nm pump. The laser beam
was transmitted out of the cavity through an output coupler mirror
(M3 on **Figure 1**) and was 1:1 relay imaged using two 125 mm lenses
(L3 and L4) to a Photon ModeScan Meter for measuring the beam
quality factor M2. The emitted beam was also 1:1 relay imaged using
two 125 mm lenses (L3 and L5) to the Meadowlark optics spatial light
modulator (SLM 2) to perform modal decomposition [13,14] into the
Laguerre-Gaussian basis and the measurement of signal at the origin of
the Fourier plane using a 125 mm lens (L6) was captured using a CCD
camera (Spiricon, LBA-USB).

We perform a numerical calculation of the fundamental mode of
the resonator that will contain an intra-cavity amplitude mask. The
simulation is based on the expansion of the resonant field on the basis
of the eigenmodes of the bare cavity (without any diffracting object).
This method will not be given here since it has been already described
elsewhere for the case of a Plano-concave cavity including an absorbing
ring on the plane mirror [12]. The modelling gave in ref. [12] can be
easily adapted to the case of an amplitude mask made up of concentric
absorbing rings just by evaluating the overlapping integral (Eq. A10 of
Ref. [12]) upon all the regions of transparency of the mask. The width
of the absorbing rings is simulated to be a 98% match of the LGp,0
mode p-zero nulls that will be oscillating inside the resonator. This
allows for better mode selection since each absorbing ring width will
be tailor designed to match the oscillating LGp,0 mode p-zero nulls
inside the resonator as shown in **Figure 2**. The simulated LGp,0 modes
of p=1 − 3 cross-section profiles are shown in **Figure 2**. The thickness
of the absorbing rings for the p=3 mode increases from the inner ring
to the outer ring. The minimum thickness of the ring is 20 μm which
corresponds to the pixel pitch of the SLM 1.

**Figure 2:** Intensity cross section of numerically simulated lowest-loss
eigenmodes for p=1 to 3 when a mask containing an absorbing p rings is
inserted inside the resonator. The mask is inserted such that the high-loss
absorbing rings coincide with the intensity p-zero nulls for an example for p=3
Laguerre-Gaussian mode.

The results of selectively generating LGp0 modes with an appropriate grey scale masks that contain absorbing rings are shown in **Figure 3**. The grey scale images in **Figure 3a-3d** which are also termed
the digital holograms that contains absorbing rings and aperture, were
encoded on the SLM using complex amplitude modulation [15] The
laser beam profiles of the generated LGp,0 modes in **Figure 3**.

3 (e-h) clearly match the corresponding grey scale digital holograms.
This is what we were expecting for the generated LG single modes of
orders p=0 − 3. It is also clear that in **Figure 3e-3h** the width of the p-zero
intensity nulls on the generated LGp,0 modes increases as the order of
the mode increases, which allows for better mode matching with the
encoded digital hologram absorbing rings. Since the generated highorder
LGp,0 modes come from a stable resonator cavity, their beam
radius, w, and beam propagation factor, M2, of the modes are known
analytically, they can be compared to experimental results as there
are summarised in **Figure 4a**. It is evident that the cavity is selecting
desired modes with appropriate beam sizes and beam propagation
factors which are all in good agreement with the theory. It is evident
that the cavity is selecting desired modes with appropriate beam sizes
and beam propagation factors which are all in good agreement with the
theory. Furthermore, the modal decomposition [13] was performed to
determine the mode purity of the generated high-order LGp,0 modes.
In **Figure 4b** the mode purity is greater than 90% for modes p=0 and
p=2, and then dropping to just below 95% to 90% for p=1 and p=3 respectively. The output power from a laser is defined to be linearly
proportional to the mode volume, Vp, where the volume of the pth
radial mode is found from:

(4)

Where l0 is the length of the gain medium and V0 is the mode
volume of the p=0 (Gaussian) mode **(Figure 4)**.

**Figure 4:** (a) Beam width at the output coupler as a function of mode order, with theoretical (Eq.2) results represented by a back solid line and beam Propagation factor (M2) as a function of mode order, with theoretical (Eq. 3) results represented by a red solid line. (b) Contour plot of modal decomposition results with intensity normalized to 1, for p=0 − 3.

In the limit that the length of the crystal is much smaller than the
Rayleigh range of the beam, then Eq. 4 simplifies to Vp=M 2V0. The
results of the output power versus input power for the LG0,0 mode
and LG3,0 mode are shown in Figure 5a. The results for slope efficiency
and pump power threshold **(Figure 5)** of the generated LGp,0 modes
of p=0 − 3 are shown in **Figure 5b**. The digital aperture in **Figure 3a-3d** was encoded because we wanted to minimize the losses, an increased
number of transverse modes will go below a certain loss threshold
required to maintain a steady-state oscillation, and selectively excite
required LGp 0. Since the beam radius is proportional to Eq. 2, it is a
reasonable assumption that the diffraction losses of the LGp,0 mode
will exhibit the highest slopes if the aperture radius is varied around
a value of about ×4 the Gaussian beam radius. The threshold pump
power show in **Figure 5b**, indicated by a blue solid line is required to
reach a gain factor G of more than 1.0, for modes to be excited. The
detailed knowledge of the diffraction losses of the transverse modes is
crucial for the optimized design of a laser resonator, especially for lasers
with low gain media. The pump power required to reach threshold is
determined by the losses the radiation experiences in a round trip.
The gain factor per transit G0 has to compensate the losses generated
by output coupling and diffraction. The threshold condition reads as
follow:

(5)

Where R_{1} and R_{2} are the reflectance of SLM 1 and mirror M3, and
V is loss factor per round trip. In addition, the efficiency of a laser is
heavily affected by internal losses in the laser resonator. The output
power is also inversely proportional to the round-trip losses. This
suggests that the higher-order radial modes have an output power that
may be expressed by Eq. 6, this is evident from **Figure 5b**:

(6)

where the subscripts p and 0 refer to the radial mode orders and the
round trip losses are denoted by σ. It is clear that higher order modes
extract more power compared to mode order of p=0, this is due to an
increase in mode volume, as shown by Eq. 4. This will be possible if the
round trip losses can be kept to a minimum compared to the extracted
gain, as shown by Eq. 6. This is evident in **Figure 5a** that shows a graph
of the output power versus input power of the LG0,0 and LG3,0 modes.
At the input power (Pin) of above 18.2 W, the power extracted from
LGp,0 mode of order p=3 exceeds that of the fundamental Gaussian
mode, p=0, by a factor of 1.2, despite the fact that the LG3,0 mode have
a higher loss. This is because the extra gain compensates for the extra
losses when the laser resonator generates the LG30 mode. These results
show that it is possible to select a higher-order LGp,0 mode of the very
high beam quality factor and mode purity, with an aim of extracting more power within a laser resonator that contains an intra-cavity SLM
encoded with amplitude absorbing rings musk that operates as an end
mirror of the resonator.

We have shown that it is possible to selectively excite high radial order LG modes of p=0−3, which are in very good agreement with the theoretically expected results for the beam profiles, sizes and quality factor. Our results show that one can digitally excite a single high order LGp,0 mode, by loading a grey-scale digital hologram that represents the mode of interest with an aim of extracting more power from the laser gain medium. We have shown that generating an LG3,0 mode, we can extract more power from the digital laser gain medium than generating the lowest order, LG0,0, mode when using an amplitude mask that is introduced on intra-cavity SLM that acts an end mirror of the resonator. Our results suggest that a digital laser can be used as a simulation test tool for manufacturing amplitude masks for the development of the high-brightness laser that will operate by selective excitation of a single higher-order LGp,0 laser modes above the critical input power.

This work is based on the research supported by the National Research Foundation (NRF), Republic of South Africa (RSA).

**Funding:** National Research Foundation (NRF) (PDP2014).

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