Realizing Gaussian to flat-top beam shaping in traveling-wave amplification

Tianzhuo Zhao; Zhongwei Fan; Hong Xiao; Ke Huang; Zhenao Bai; Wenqi Ge; Hongbo Zhang

doi:10.1364/OE.25.033226

1. Introduction

Typically, laser beams are generated with a Gaussian intensity distribution, which is often undesirable for many practical applications. Flat-top laser beams with uniform beam intensity distribution and a sharp beam edge are more useful for laser welding, laser microfabrication, laser scanning, optical storage, optical image processing, laser radar, optical metrology, and nonlinear optics [1–3]. Consequently, various methods have been developed to convert the intensity profile of a Gaussian laser beam to a flat-top distribution. However, in most of these shaping methods, specific devices are required. For example, typical optical elements, such as a distributive afocal lens [4], a gradient-index lens [5], or a refractive optical system [6] are specifically designed to provide Gaussian to flat-top beam shaping. In addition, diffractive optical elements such as binary phase elements [7] and gratings [8], have been well used to perform shaping in laser systems. Beam modulation devices, such as liquid crystal spatial light modulators [9], liquid crystal light valves [10], and digital micromirror device [11] have also been employed for the benefits of real-time adjustable shaping and small-region compensable. Moreover, the uses of specific optical media, including fiber [12] and turbulent media [13] have also been broached to implement conversion.

In above mentioned flat-top beam applications, the required output laser power ranges up to thousand Watts, spurring considerable research in recent years. To meet these requirements, optical design of the laser system is a good method. Shaping output beam intensity by pump beam intensity modification has been reported by Igor [14], which proves an Ho:YLF end-pumped laser resonant cavity can realize output beam reshape by modulating pump beam shape of the Tm:YLF pump source. In this paper, we describe the theory of the directly performing Gaussian to flat-top shaping, and reports on the first experimental implementation in a master oscillator power amplifier (MOPA) system. Compared to existing beam shaping techniques, the use of a MOPA system to shape an input Gaussian seed laser beam into an amplified flat-top output laser beam via traveling-wave amplification has many advantages. First, the approach does not suffer from the problems of a laser damage threshold, any optical ghost points, or scattered laser light, which are especially prevalent in high-energy or power laser systems. Secondly, this shaping scheme does not decrease the power of the shaped beam. Using an afocal or gradient-index lens can result in several percent of insertion loss even with coatings, while the using a diffraction or liquid crystal modulation results in 10–50% of power loss. Compared with these traditional methods, direct shaping via amplification is highly efficient, and is also simple and of low cost.

Generally, in side-pumped amplifiers using small-diameter, rod-shaped working material and a reflection cell around the rod for multiple reflection, the gain distribution has a uniform cross section because of the rod’s short absorption length and the multiple absorptions of the reflected residual pump beam. However, for a larger rod working material, e.g., with diameter > 10 mm, a strong gradient in the cross-sectional gain distribution would induce non-uniform amplification of the input seed laser. Alternatively, by optimizing the gain distribution in the rod working material cross section, one can implement the effective beam shaping of an input seed laser.

In previous work, we have introduced how to design pumping energy distribution of a side-pumped amplifier and proved by experiments of fluorescence distribution measurement [15]. Here, the theory of direct Gaussian to flat-top conversion of a beam during amplification is discussed by laser gain theory, along with important optimization parameters including the absorption coefficient, pump radius, and working current. Finally, the theoretical approach is experimentally demonstrated using a 15-mm-diameter side-pumped amplifier.

2. Gain distribution analysis of the amplifier

A typical Gaussian seed laser signal can be written as

E_{i n} (r) = e^{- 2 (\frac{r^{2}}{R^{2}})} E_{i n} (0)

Here, $R$ is the radius of the rod working material in the side pumped amplifier, and when the seed laser beam has a diameter equal to the diameter of the rod, $E_{i n} (R) = 0.135 E_{i n} (0)$ . Within the range $0 < r < R$ , the normalized seed laser energy density distribution $E_{i n} (r)$ requires a different signal gain at different radial positions, and needs a specific absorbed pump energy distribution. If the gain at the center of the rod is $G (0)$ and at position $r$ is $G (r)$ , then

G (r) = \frac{E_{o u t} (r)}{E_{i n} (r)}

where

E_{i n} (r)

is the pulse energy density of the injected seed laser, and

E_{o u t} (r)

is the pulse energy density of the amplified output laser. Gaussian to flat-top beam shaping requires that

G (r) = \frac{E_{i n} (0)}{E_{i n} (r)} G (0)

Generally,

G (r)

can be written as (formula 4.11 of reference [16])

G (r) = \frac{E_{S}}{E_{i n} (r)} \ln {1 + [\exp (\frac{E_{i n} (r)}{E_{S}}) - 1] e^{g_{0} (r) L}}

where

g_{0} (r)

is the small-signal gain coefficient, and the saturation energy density

E_{S}

can be written as

E_{S} = \frac{h ν}{γ σ} = \frac{E_{S T} (r)}{γ g_{0} (r)}

The photon energy

h ν

for a laser photon with a wavelength of 1064 nm is 1.86 × 10⁻¹⁹J,

γ

is 1 for a four-level laser system, and the stimulated emission cross section

σ

is 2.8 × 10⁻¹⁹cm² for the Nd:YAG used as the rod working material. Therefore, the saturation energy density

E_{S}

is 0.664J/cm². Combining (1), (3), and (4),

\frac{G (r)}{G (0)} = \frac{E_{i n} (0) \ln {1 + [\exp (\frac{E_{i n} (r)}{E_{S}}) - 1] e^{g_{0} (r) L}}}{E_{i n} (r) \ln {1 + [\exp (\frac{E_{i n} (0)}{E_{S}}) - 1] e^{g_{0} (0) L}}} = e^{2 (\frac{r^{2}}{R^{2}})}

Gaussian to flat-top beam shaping requires that

[\exp (\frac{E_{i n} (r)}{E_{S}}) - 1] e^{g_{0} (r) L} = [\exp (\frac{E_{i n} (0)}{E_{S}}) - 1] e^{g_{0} (0) L}

and

g_{0} (r) = \frac{1}{L} \ln [\frac{\exp (\frac{E_{i n} (0)}{E_{S}}) - 1}{\exp (\frac{E_{i n} (r)}{E_{S}}) - 1}] + g_{0} (0)

Typically, the small-signal gain approximation is satisfied, meaning that

\frac{E_{i n} (r)}{E_{S}} < < 1

Then (formula 4.12 of reference [16])

G (r) ≅ e^{g_{0} (r) L}

Using (3) and (1),

\frac{G (r)}{G (0)} = e^{[g_{0} (r) - g_{0} (0)] L} = e^{2 (\frac{r^{2}}{R^{2}})}

Therefore, gain at the edge of the rod becomes

g_{0} (r) = \frac{2}{L} (\frac{r^{2}}{R^{2}}) + g_{0} (0)

Further, in a pulse amplification system, the storage energy density

E_{S T}

can be expressed as (formula 4.39a of reference [16])

E_{S T} (r) = η_{T} η_{A} η_{S} η_{Q} η_{B} η_{S T} η_{A S E} τ_{f} ρ_{P} (r)

In (13),

η_{T}

is the transmission efficiency,

η_{A}

is the material absorption efficiency,

η_{S}

is the Stokes efficiency,

η_{Q}

is the quantum efficiency,

η_{B}

is the mode matching efficiency,

η_{S T}

is the storage efficiency,

η_{A S E}

is the Amplified Spontaneous Emission (ASE) efficiency,

τ_{f}

is the fluorescence life time of the laser material, and where

ρ_{P} (r)

is the pump power density at

r

, defined as

ρ_{P} (r) = \frac{d P}{d V}

Where

P

is the pump power, and

V

is the volume of the working material. From (5) and (13),

g_{0} (r)

can then be written as

g_{0} (r) = \frac{η_{T} η_{A} η_{S} η_{Q} η_{B} η_{S T} η_{A S E} τ_{f}}{E_{S} γ} ρ_{P} (r)

For a typical side-pumped Nd:YAG amplifier,

η_{T} η_{A} η_{S} η_{Q} η_{B} η_{S T} η_{A S E} = 0.388

, when

η_{T} = 0.88

,

η_{A} = 0.85

,

η_{S} = 0.76

,

η_{Q} = 0.95

,

η_{B} = 1

,

η_{S T} = 0.80

,

η_{A S E} = 0.90

, and

τ_{f}

is 230 μs. Therefore,

g_{0} (r) =0 .134 ρ_{P} (r)

. From (8) and (12),

ρ_{P} (r) = \frac{E_{S} γ}{η_{T} η_{A} η_{S} η_{Q} η_{B} η_{S T} η_{A S E} τ_{f}} \frac{1}{L} \ln [\frac{\exp (\frac{E_{i n} (0)}{E_{S}}) - 1}{\exp (\frac{E_{i n} (r)}{E_{S}}) - 1}] + ρ_{P} (0)

Here,

ρ_{P} (0)

is pump-power density at the center of the rod. Again, using the small-signal gain approximation,

ρ_{P} (r) = \frac{E_{S} γ}{η_{T} η_{A} η_{S} η_{Q} η_{B} η_{S T} η_{A S E} τ_{f}} \frac{2}{L} (\frac{r^{2}}{R^{2}}) + ρ_{P} (0)

If the diameter of the rod is 15 mm, and the effective pumping length

L

is 100 mm,

g_{0} (R)

is expected to be

0.2 + g_{0} (0)

. For Gaussian to flat-top beam shaping with a single-pass amplification and no gain at the center of the rod, i.e.,

ρ_{P} (0) = 0

,

ρ_{P} (R)

should be 1.49kW/cm³. If

g_{0} (0)

is 0.135 cm⁻¹,

g_{0} (R)

will be 0.335 cm⁻¹, and the average is 0.273 cm⁻¹. In the amplifier used in the experiment, 15 bars were uniformly arranged around the rod in each of 8 rings placed along the length of the rod, with 120 bars in total. The slow axis of the bars paralleled the axis of the rod. From (9), the pump power density at the center of the rod was ~1.0 kW/cm³, and at the edge of the rod was ~2.49 kW/cm³. The average pump power density of the rod was ~2.04 kW/cm³, and with each bar at 300 W, the total power was 36.0 kW. If a dual-pass scheme were to be used for the same amplifier mentioned above,

g_{0} (R)

would be 0.235 cm⁻¹, which is only a factor of 1.75 larger than the central gain, because dual-pass amplification requires a significantly lower difference in gain density. Above mentioned equations base on rate equations, though there has other method to derive the formula [17, 18], the result is similar.

3. Parameters influencing pump-energy distribution

The software package ZEMAX was used to simulate the absorbed pump energy and gain distributions in the cross section of the rod working material of the side pump amplifier. Ten million rays were traced to improve the accuracy of the simulation. The parameters and configuration of the Nd:YAG rod, laser diode arrays, and side pump amplifier used in the simulation described in this section are the same as that mentioned above. The side pump amplifier configuration is shown in Fig. 1. During ray tracing, the laser diode module was assumed to have a Gaussian beam distribution profile, with full-width-at-half-maximum beam divergence on the fast axis of 40°, and on the slow axis of 10°. The emitter size was 1 μm × 150 μm, and there were 19 emitters on each bar. The module of laser diode arrays had 15 bars uniformly arranged in a ring around the rod, with 8 rings placed along the length of the rod to form the side pump amplifier. The amplifier was cooled by water circulating between the rod’s outer surface and the inner wall of the flow tube, which was made of fused silica. The thickness of the water layer around the rod was 1.5 mm, and the flow tube wall thickness was 2 mm. The diameter of the rod was 15 mm, and the effective pumping length $L$ , that is, the length of the rod shined by the pump light, was ~100 mm.

Fig. 1 Experimental side-pump amplifier structure and beam shaping scheme. (a) Structure of 15 LD bars uniformly arranged side-pump amplifier; (b) Directly shaping a Gaussian seed laser beam into a flat-top output laser beam with the Nd:YAG amplifier.

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In a side-pumped amplifier, the parameters of the pump structure and working material can affect the gain distribution in the cross section of the rod, and correspondingly affect shaping effect. Among these, the most important factor is the absorption coefficient, which is determined by the concentration of the Nd³⁺ dopant in the Nd:YAG rod. The absorption coefficient of the rod used was 0.230 mm⁻¹ when the Nd³⁺ adoption concentrations were 0.4 at%, 0.287 mm⁻¹ at 0.6 at%, 0.361 mm⁻¹ 0.8 at%, and 0.453 mm⁻¹ at 1.0 at%. These data were measured in the experiments. The relation between adoption concentration and absorbed pump energy distribution is plotted in Fig. 2. The cross-sectional absorbed pump energy distribution of the rod, located at the center of the rod along the rod length, is shown in the false color diagrams, while the horizontal distribution of the absorbed energy density though the center of the rod is plotted in the graph under each corresponding diagram. In these diagrams, the highest energy density region is colored red, and the lowest energy density region, which is zero, is colored blue. From Fig. 2, the absorption coefficient has a clear influence on the absorbed pump energy distribution. To precisely control the absorbed pump energy distribution, the adoption concentration of the working material should be chosen first. Typically, if a uniform gain is expected, 0.5 at% working materials should be used. However, if an amplifier is designed for Gaussian to flat-top beam shaping, 1.0 at% or higher Nd:YAG rod should be used. Further, the working temperature of laser diode arrays is another important factor which can affect the absorption coefficient by changing the spectrum of the laser diodes. Here, the pump spectrum of laser diode arrays was held at the absorption point of 808.6 nm when using the Nd:YAG rod.

Fig. 2 Influence of absorption coefficient on absorbed pump energy distribution. Top row: cross-sectional absorbed pump energy distribution of the rod. Bottom row: absorbed pump energy density distribution through the center of the rod with the cross section located at the center of the rod along its length. Absorption coefficient and adoption concentration: (a) 0.230 mm⁻¹, 0.4 at%; (b) 0.287 mm⁻¹, 0.6 at%; (c) 0.361 mm⁻¹, 0.8 at%; (d) 0.453 mm⁻¹, 1.0 at%.

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Another important factor affecting the absorbed pump energy distribution is the pump radius. The absorbed pump energy distributions at pump radii of 15 mm, 19 mm, 23 mm, and 27 mm were calculated and are summarized in Fig. 3. These diagrams show that the pump radius can affect the pump energy distribution, because, when the pump radius is small, the energy density is higher in the center. From (a) to (d), the normalized absorbed energy density at the edge of the rod is, respectively, 2.05, 2.77, 3.06, and 3.46 times that at the center. When the pump radius was 19 mm, as shown in Fig. 3(b), the absorbed pump-energy density meets the requirements mentioned above for implementing flat-top beam shaping. The side pump amplifier used in the experiment was designed with a pump radius of 19 mm. If the pump radius was larger, the difference of the absorbed pump energy density at the edge and center of the rod would also be larger. However, if the pump radius was larger than 40 mm, the difference would not change, revealing a limitation restricting possible implementation of Gaussian to flat-top beam shaping with side pump amplifier and small rod working material. The absorbed pump-energy density and pump radius, are related linearly, primarily because a smaller pump radius imposes a higher energy density on a smaller area in the center of cross-section of the rod.

Fig. 3 Simulation results of the absorbed pump energy distribution in the cross-section of a 15-mm-diameter, 1.0 at% Nd:YAG rod in a side pumped amplifier. Pump radii are: (a) 15 mm; (b) 19 mm; (c) 23 mm; (d) 27 mm.

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4. Experimental results and discussion

The amplifier used in the experiment reflects the optical ray-tracing model mentioned in the last section: the diameter of the rod was 15 mm; the effective pumping length was ~100 mm, with total length ~150 mm; 15 bars were uniformly arranged in a ring around the rod, with 8 rings along the rod for pumping. The water thickness around the rod was 1.5 mm, and the flow tube wall thickness was 2 mm. The pumping scheme of the side-pumped amplifier is shown in Fig. 1.

The experimental setup used for the fluorescence distribution measurement included an amplifier, focusing lens, and capture CCD. The focusing lens size was 25.4 mm, and focal length was 150 mm. The capture CCD was a silicon CCD camera (SP620U, Ophir Co. Ltd.), placed at the focus of the lens. The distance from the camera to the end surface of the rod was 935 mm. The wavelength of the pumping laser was monitored by a spectrometer (AVaSpec-3648, Avantes Co. Ltd.). As shown in Fig. 4, when the central pump wavelength was 804 nm, 805 nm, 806 nm, 807 nm, and 808 nm, the absorbed pump energy intensity in the center of the cross section of the rod was 58.6%, 53.7%, 47.6%, 35.8%, and 31.8%, respectively, of the edge intensity. In these diagrams, the highest energy density regions are colored by white, and the zero energy density regions are colored by black. Scale of the relative intensity bar is determined by contrast resolution, in the range from 0 to 65535. The solid line at the bottom shows relative intensity distribution though the center of the spot. Choosing an optimized cooling water temperature (25 degrees Celsius in our experiments) allowed control of the central pump wavelength, and as a result an expected relative intensity ratio was maintained. In addition, it is worth noting that nonlinear inhomogeneities in amplifier materials, thermal effects, coolant temperature and pressure, working current, and the consistency of laser diode bars are important factors that can affect the gain distribution.

Fig. 4 Fluorescence distribution at different pump central wavelength. (a) 804 nm; (b) 805 nm; (c) 806 nm; (d) 807 nm; (e) 808 nm.

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The experimental setup for the gain distribution measurement is shown in Fig. 5, and included a seed laser, beam expander, amplifier, and CCD camera (SP620U, Ophir Co. Ltd.). A polarizer, ½-wave plate, and reflection mirror (denoted M₃) constituted a dual-pass amplification structure. The seed laser output was at 1064 nm, with pulse width (full width at half maximum; FWHM) of 32 ns, frequency of 10 Hz, and a beam distribution similar to a Gaussian distribution. Figure 6 shows the measured amplified beam profile, demonstrating that Gaussian to flat-top beam shaping can be achieved in a range of ~50 A centered on 150 A. This is mainly because the working current determines thermal power on the laser diode bars, affects the pump wavelength, and further changes the absorption coefficient.

Fig. 5 Experimental scheme of Gaussian to flat-top beam shaping with dual-pass amplification.

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Fig. 6 Experimental results of the near-field distribution for Gaussian to flat-top beam shaping with simultaneous amplification. Working current: (a) 0 A; (b) 125 A; (c) 150 A; (d) 175 A; (e) 200 A.

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In order to verify behavior of the generated beam, we investigated its features in both the near and the far field. Observed ring ripples on the near-field spot are mainly caused by diffraction of the rod-edge aperture, as the seed laser beam has a full diameter similar to the diameter of the rod. However, in practical applications, the inclusion of a subsequent high-power amplifier with a flat-top distribution would be capable of diminishing these ripples.

Because these spots are not ideal round, different outline shape of spot will bring about different surface fitting error. Curve fitting on cross section of the spot is a good method to avoid this kind of error. Figure 7 shows a different view on the experimental data of Fig. 6(a) and (c), along with fitted curves. Figure 7(a) shows the parallel cross-sectional distribution for the seed laser beam intensity, and Fig. 7(b) shows it for the output beam intensity when the working current was 150 A. The curve fitting formula used was

Fig. 7 Fits of the cross-sectional distribution of (a) seed laser beam; (b) amplified beam with working current of 150 A.

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y = a e^{- {(\frac{x - b}{c})}^{n}}

For the signal laser distribution in Fig. 7(a), the super-Gaussian coefficient n = 2 denotes a typical Gaussian distribution. The other fitting parameters are a = 159.2, b = −1.782, c = 4.373, giving a goodness-of-fit R-squared of 0.8768 with root mean square error (RMSE) of 17.61. These fits show that the seed laser beam had a near Gaussian shape. Super-Gaussian fitting with n>2 is implemented in Fig. 7(b), and only super-Gaussian fitting with the best R-squared value and RMSE are shown here. With n = 4, 6, 8, 10, 12, fitting R-square are 0.8135, 0.8477, 0.8481, 0.8352, and 0.8164, respectively, and the RMSE is 14.77, 13.35, 13.33, 13.88, and 14.62, respectively. The best parameters are n = 8, a = 133, b = −1.49, c = −5.497. The curve fitting shows that Gaussian to flat-top beam shaping is appropriately realized.

To gather far-field spot characters, a focal lens is placed between mirror M₄ and CCD camera. Distance from the amplifier to the lens is 750mm, focal length of the lens is 250mm, and the camera is placed at the focusing point of the beam to gather Fig. 8. Figure 8 (a) shows the signal laser spot without amplified, and Fig. 8(b) shows the spot after dual-pass amplified. The changes of far-field spot to Airy-spot distribution further prove that the amplified beam has a flattop distribution.

Fig. 8 Experimental results of the far-field spot for Gaussian to flat-top beam shaping with simultaneous amplification. (a) seed without amplification; (b) amplified beam with working current of 150 A.

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The pulse energy of the signal laser was 10.7 mJ at the position of CCD camera when the amplifier was turned off, i.e., the working current was zero. At the working current of 125 A, 150 A, 175 A, and 200 A, the pulse energy was measured to be 56.5 mJ, 72.3 mJ, 90.1 mJ, and 107.2 mJ, respectively, with average gain of 5.28, 6.75, 8.42 and 10.02, respectively. The experimental results are lower than the theoretical predictions, which is due to two primary reasons. First, the pump spectrum width (FWHM) of the overall laser diode arrays was 6.3 nm, which is much wider than absorption peak width of the working material. Second, a 3° wedge angle was used between the axis of the rod and each end of the surface, which is a common design to avoid self-excited oscillations. The wedge angle prevents the full diameter beam from a complete transmission through the rod, which decreases the utilization ratio of the pump energy. Thermal effect also influences gain and shaping ability. The experimental results demonstrate that the working current determines the average thermal power generated in the working materials, and Gaussian to flat-top beam shaping can only be achieved in a certain range of working current. Nevertheless, if the amplifier can be sufficiently cooled, the range of working currents used could be wider.

5. Conclusions

We presented an approach of direct Gaussian to flat-top beam shaping using a side-pumped amplifier with a 15-mm-diameter Nd:YAG rod as the working material. By optimizing the absorbed pump energy cross-sectional distribution of the working material, a specific gain distribution through the rod can implement Gaussian to top-flat beam shaping of the transferred laser beam. Experimental demonstration showed that a full diameter Gaussian beam can be shaped to a flat-top distribution during amplification in a certain range of working currents.

Funding

National Key Scientific Instrument and Equipment Development Projects (2014YQ120351); National Natural Science Foundation of China (NSFC) (61675210); National Key Scientific and Research Equipment Development Project of China (ZDYZ2013-2); Youth Innovation Promotion Association of CAS (2014136); China Innovative Talent Promotion Plans for Innovation Team in Priority Fields (2014RA4051).

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Realizing Gaussian to flat-top beam shaping in traveling-wave amplification

Abstract

1. Introduction

2. Gain distribution analysis of the amplifier

3. Parameters influencing pump-energy distribution

4. Experimental results and discussion

5. Conclusions

Funding

References and links

Cited By

Figures (8)

Equations (18)

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