A laser beam quality definition based on induced temperature rise

Harold C. Miller

doi:10.1364/OE.20.028819

1. Introduction

High power laser beam characterization data often include a far-field power-in-the-bucket (PIB) curve [1–3]. The PIB data is compared with an ideal curve calculated assuming a perfect (i.e. uniform phase and intensity) near-field distribution with the same circumscribed profile and power as the beam being measured. A beam quality number is assigned by comparing the curves using some choice of an acceptable bucket width or useful power level. Much more common in the laser community is the use of camera, knife-edge, or other beam-size measurement approaches to acquire data that are fit to Gaussian-beam propagation formulas to arrive at an M² or related beam quality metric. The primary issue with the current situation is the lack of consistency in defining laser beam quality and the possibility that a reported beam quality number may not adequately describe how the power in a beam is distributed. For example, it is possible for a PIB beam quality number to disregard a large fraction of the power in a beam, or for a second-moment measurement to misrepresent the tangible size of an intensity distribution. Rather than using beam-size measurements or arbitrary comparisons of PIB curves, it might be useful to instead arrive at a beam quality number by looking at how a far-field beam would interact with a material. The following sections motivate an alternative beam quality definition based on a consideration of the maximum-possible laser-induced temperature rise in an ideal absorbing volume.

2. Power-in-the-bucket

The PIB method reduces a 2D far-field intensity distribution to a one-dimensional integrated-intensity curve, providing a simple way to visualize the distribution of power in a beam. Formally, the PIB is an area integral over a far-field intensity distribution behind a real or imagined “bucket” (aperture) of a given size. Restricting ourselves to circular buckets,

P I B (θ) = \int_{0}^{2 π} \int_{0}^{θ} I (θ', φ) θ' d θ' d φ = 2 π \int_{0}^{θ} I_{a v e} (θ') θ' d θ',

where I(θ ^× ,ϕ) is any far-field intensity distribution (rectangular, elliptical, etc.). ϕ is the tangential coordinate. In practice, a far-field intensity distribution might be measured using a CCD array positioned at the transform plane of lens. Alternatively, PIB data can be directly acquired by measuring the transmission through a series of circular apertures [3]. In order to keep the analysis independent of the physical size of the far-field distribution, the bucket radii are usually expressed in terms of an angle coordinate (θ), instead of the radial coordinate, r [cm]. θ ^× in Eq. (1) is the corresponding integration variable. The scaling of θ depends on the nominal diameter (D) of the near-field beam, the laser wavelength (λ), and the far-field distance (f). For example; an ideal Gaussian beam distribution can be expressed in terms of the angular coordinate using,

θ (r) = \frac{r}{f} = \frac{2 r λ}{π w_{0} D},

where w₀ is the ideal second-moment beam waist radius in the far-field. For an ideal top-hat beam emitted from a circular aperture,

θ (r) = \frac{1.22 r λ}{r_{0} D},

where r₀ is the first zero in the far-field Airy diffraction pattern. Usually, PIB curves are simply graphed as a function of θ in units of λ/D, with an understanding of how the calculated ideal far-field intensity distribution depends on the size and shape of the particular near-field profile [2]. For Gaussian-like beams, the size of the near-field can be assessed using a second-moment, knife-edge, or other standard measurement approach. However, applying beam-size metrics to arbitrary beam shapes (arrays, annular beams, etc.) can lead to an improper description of the near-field profile. In such cases, it is preferable that the near-field beam profile be defined by a physical aperture, whose size and shape can be optimized for best far-field performance [3]. It is important to note that the ideal far-field intensity distribution and a corresponding beam quality number will always depend on how the ideal near-field beam is defined. For example, a particular application might seek to compare a measured annular beam with a top-hat beam instead of an ideal annular beam.

I_ave in Eq. (1) is the azimuthally-averaged intensity at a particular angle,

I_{a v e} (θ) = \frac{1}{2 π} \int_{0}^{2 π} I (θ, φ) d φ .

I_ave is a derived axially-symmetric intensity distribution that is related to a PIB curve by,

I_{a v e} (θ) = \frac{1}{2 π θ} \frac{d P I B (θ)}{d θ} .

For an arbitrary beam, certain details about the shape of the intensity distribution may be lost in the PIB data reduction process. Still, both the PIB curve and I_ave remain representative of how the power is distributed radially across the beam.

3. Laser-induced temperature rise

In the pioneering work of Lax [4], the heat equation was used to calculate the steady-state temperature rise of a semi-infinite absorbing volume exposed to an axially-symmetric laser beam. At the time, the interest was in light-induced semiconductor material degradation. For the purposes of assessing beam quality, our imaginary absorber will be placed at a far-field plane of the laser beam. The geometry is defined such that a beam propagates in air along the z-axis from a near-field plane at z = f<0 before striking an infinite flat absorbing surface on the x-y plane at z = 0. The material occupies all points z≥0.

Lax found that the steady-state induced temperature-rise distribution in the material could be written,

Δ T (R, Z, W) = Δ T_{\max} N (R, Z, W),

where N(R,Z,W) is a smoothly-varying function of normalized radial (R), depth (Z) and material absorption coefficient (W) coordinates. The normalization factor is a nominal beam size parameter (w) with R = r/w, Z = z/w, and W = αw, where r and z are the radial and axial coordinates, respectively, and α is the absorption coefficient. The idealized boundary condition,

\frac{d Δ T (R, 0, W)}{d z} = 0,

provides that there is no heat flow from the flat input surface of the volume to the air. Because the absorber is infinite, the temperature rise goes to zero as R or Z approach infinity regardless of the power absorbed. In Lax’s analysis, the thermal conductivity, κ, of the absorber is temperature-independent. A later paper addresses the κ = κ(T) situation [5].

ΔT_max is the temperature rise at the origin in the extreme case when all of the power is absorbed by an infinitesimally thin surface layer, meaning that,

N (0, 0, \infty) = 1.

For finite beam penetration, the actual temperature rise is reduced by N(0,0,W) < 1. Lax also showed that ΔT_max only depends on the laser beam power, P, the thermal conductivity of the material, and the mean value of the intensity-weighted inverse distance from the center of the beam to each differential heat source element on the surface. Using Lax’s notation,

Δ T_{\max} = \frac{P}{2 π κ} 〈 \frac{1}{r} 〉,

where,

〈 \frac{1}{r} 〉 = \int_{0}^{\infty} (\frac{1}{r}) f (R) r d r {(\int_{0}^{\infty} f (R) r d r)}^{- 1} .

f (R) is a distribution function that describes the “shape” of the beam. If we assume that all of the power in the beam is absorbed by the surface (no reflection), one can show that,

Δ T_{\max} = \frac{1}{κ} \int_{0}^{\infty} I (r) d r,

where I(r) is the laser intensity distribution.

It turns out that the underlying formalism is applicable to all beam shapes and not just to those with axial symmetry. This is due to the fact that temperature rise, ΔT(x,y,z), at any point in the material point depends on the superposition of the contributions of each differential laser-induced heat source element, regardless of where it is located. The heat equation, which reduces to a Poisson equation in steady-state, can be written,

\nabla^{2} Δ T (x, y, z) = - \frac{S (x, y, z)}{κ},

where S(x,y,z) is the heat source in units of power per unit volume. Using the Green’s function method, one arrives at the familiar general solution to Eq. (12) for the semi-infinite geometry [6],

Δ T (x, y, z) = \frac{1}{2 π κ} \underset{V}{∭} \frac{S (x', y', z') d x' d y' d z'}{\sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}},

where the integral is over the volume, V, containing the heat source,

S (x', y', z') = α I (x', y', 0) \exp (- α z'),

where I(x ^× ,y ^× ,0) is the incident intensity distribution.

When all of the power is absorbed on the input surface, the heat source has the form,

S (x', y', z') = I (x', y', 0) δ (z') .

Inserting Eq. (15) into Eq. (13), one finds that the maximum attainable temperature rise is,

Δ T_{\max} = Δ T (0, 0, 0) = \frac{1}{2 π κ} \int_{0}^{\infty} \int_{0}^{\infty} \frac{I (x', y') d x' d y'}{\sqrt{x'^{2} + y'^{2}}} .

With complex far-field distributions, Eq. (13) and Eq. (15) can be used to locate the temperature-based center of the beam. In cylindrical coordinates, the maximum temperature is given by,

Δ T_{\max} = \frac{1}{2 π κ} \int_{0}^{\infty} \int_{0}^{2 π} \frac{I (r, φ) r d r d φ}{r} = \frac{1}{κ} \int_{0}^{\infty} I_{a v e} (r) d r .

One can normalize an intensity distribution to extract the laser power dependence. Therefore, the maximum attainable surface temperature can always be expressed using Eq. (9), scaling as the intensity-weighted average inverse distance from the center of the beam to each differential heat source element on the surface. It is not necessary for the purposes of this paper to solve Eq. (13) for the temperature profile throughout the entire absorbing volume. Lax and others have performed the calculation for a variety of configurations associated with materials processing [4–8].

4. Laser beam quality definition

Laser beam interactions with materials can be complex, involving details like reflectance, temperature dependencies of the material parameters, melting, cooling, exposure times, etc. However, the intrinsic ability of a far-field distribution to induce the maximum-possible centerline temperature-rise in an ideal absorber is a property that is related by Eq. (9) to an effective spot-size of the beam. Our semi-infinite absorber is “ideal” in the sense that 1) its material properties do not change with temperature, 2) all of the power in the beam is absorbed, 3) there is no loss of heat across any boundary, and 4) the rapid flow of heat produces a steady-state temperature profile, regardless of the energy absorbed. In such an absorber, the scale factor ΔT_max depends explicitly on the shape of the intensity distribution and scales linearly with the laser power. Also, the dependence of ΔT_max on the beam-shape remains the same regardless of the material properties, κ and W. For these reasons, one might consider the alternative beam quality definition,

B Q = \frac{Δ T_{\max - ideal}}{Δ T_{\max - m e a s u r e d}} .

The beam quality (BQ) definition in Eq. (18) depends only on the measured and ideal beam-shape functions and not on the beam power or any property of the absorbing material. The definition can be used with far-field beams of any shape. For unclipped circular Gaussian beams, it is easy to show using Eq. (17) and Eq. (18) that BQ = M², consistent with the standard second-moment-based definition. For general beam shapes, BQ≠M², so a more general kind of beam analysis is required. From the induced temperature rise perspective, the effective far-field performance of any beam scales as BQ⁻¹. The definition captures the contributions due to all of the power (heat sources) in the beam. Therefore, a beam quality number determined using Eq. (18) only depends on how the ideal near-field distribution is defined.

5. Interpretation of power-in-the-bucket data

Since BQ = M² for axially-symmetric Gaussian beams, standard measurement methodologies and instrumentation are entirely consistent with the induced-temperature-rise beam quality definition. PIB analysis using circular buckets is compatible with all beams since the method samples the far-field distribution in a radial fashion, consistent with the definition of ΔT_max. Using Eq. (5), Eq. (17) and Eq. (18), the beam quality can be expressed,

B Q = \frac{\int_{0}^{\infty} \frac{d P I B {(θ)}_{i d e a l}}{θ}}{\int_{0}^{\infty} \frac{d P I B {(θ)}_{m e a s u r e d}}{θ}} .

If appropriate, the BQ can be calculated directly from the measured and ideal far-field intensity distributions using Eq. (16) or Eq. (17). When analyzing a measured curve using Eq. (19), attention must be paid to ensure that scatter in the data does not upset the natural convergence of the integrands to a finite number as θ approaches zero. A simple curve fit using, for example, a Gaussian-based PIB function could be used in this region. The θ ⁻¹ dependence in Eq. (19) reduces the contribution due to power at large radial distances from the center of the beam. Therefore, in cases where the entire beam profile is not captured by a PIB measurement [2], a reasonable extrapolation (i.e. linear, Gaussian-based PIB) of the PIB curves will usually introduce a small and predictable level of uncertainty to a BQ determination.

6. Comparison with conventional beam quality definitions

In this section, a simulated “measured” beam is used to review how conventional PIB analysis is used to arrive at beam quality numbers. We then demonstrate how the various metrics behave as the shape of a far-field beam is changed. The ideal PIB curve in Fig. 1 was generated by assuming that the near-field is a circular top-hat beam of diameter, D, producing an Airy diffraction pattern in the far-field. The measured PIB curve is derived from a “core-pedestal” far-field distribution. The core is approximated by a narrow Gaussian beam with width θ_c = 0.85 λ/D, which is close to the width of the central lobe of the Airy pattern, while the pedestal is a much broader Gaussian (θ_p = 12.2 λ/D). The measured PIB curve in Fig. 1 assumes that 25% of the total power is in the core beam and 75% of the power is in the pedestal.

Fig. 1 Calculated PIB curves. The upper curve is for an ideal Airy pattern with its first zero at 1.22λ/D. The lower curve is the PIB of a core-pedestal beam approximated using two Gaussian beams with widths of 0.85 λ/D and 12.2 λ/D. In this example, 25% of the power is in the narrower core beam.

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The conventional vertical beam quality (VBQ) metric [3] defined in Eq. (20) below is often used to compare the amount of power contained within in the first zero (θ₀ = 1.22 λ/D) of the ideal-beam Airy pattern. For the curves shown in Fig. 1, the VBQ number would be,

V B Q = \sqrt{\frac{P I B {(θ_{0})}_{i d e a l}}{P I B {(θ_{0})}_{m e a s}}} \approx \sqrt{\frac{0.84}{0.26}} \approx 1.8.

The far-field performance of a beam is generally thought to scale like the ratio of the average intensity in the bucket or VBQ²≈3.24. The horizontal beam quality (HBQ) [3] metric specifies an arbitrary threshold power level (PIB_th), then compares the bucket radii (θ_meas(PIB_th) and θ_ideal(PIB_th)) required to capture the specified power. With PIB_th set at 0.5, the HBQ of our simulated measured beam is, by inspection of the curves in Fig. 1,

H B Q = \frac{θ_{m e a s} (P I B_{t h})}{θ_{i d e a l} (P I B_{t h})} \approx \frac{5.5}{0.5} \approx 11.

Applying Eq. (19) to the same curves gives BQ≈3. The M² of the beam cannot be calculated since the second-moment width of the Airy distribution is infinite. From this simple example, it is clear how a choice of metric, bucket size or threshold power level can dramatically change the interpretation of a beam quality number.

To further clarify the behavior of the various definitions, calculations were done on an even simpler case where the ideal beam is a Gaussian distribution with width, w₀ = 1.0 (arbitrary units) and the measured beam is again fabricated using the sum of a broad pedestal (w_p = 10.0) and a narrow core (w_c = 1.0) Gaussian beam. The near-field is assumed to be a circular Gaussian sized such that w₀ = 1.0. By varying the fractional power in the core beam, one can scan the shape (and beam quality) of the simulated measured far-field intensity distribution. Curves showing the trends in the various beam quality definitions as the fractional power in the core beam are shown in Fig. 2 . PIB_th used in the HBQ calculation was again set at 50% of the total power. The bucket radius used for the VBQ determination was arbitrarily set at w = 1.22.

Fig. 2 Comparison of beam quality definitions versus far-field beam profile. The ideal beam is a Gaussian with a width of 1.0 (in arbitrary units), while the test beam is a core-pedestal beam formed using the sum of two Gaussians with widths of 1.0 (core) and 10.0 (pedestal). The horizontal axis is the fractional power in the core lobe of the test beam. The HBQ threshold power level is 50% and the VBQ bucket radius is arbitrarily set at 1.22. The BQ curve is calculated using Eq. (19).

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Note the convergence in Fig. 2 of the VBQ², HBQ and BQ curves above about 60% of the power in the core beam. The VBQ metric clearly over-predicts beam performance from both the average intensity (VBQ²) and temperature-rise (BQ) perspectives. The VBQ² curve diverges quite dramatically from the BQ curve as the fractional power in the core is reduced. Although not shown Fig. 2, the VBQ² converges to a limiting value of ~30 as core-beam power goes to zero. This value is consistent with the peak intensity level of the pedestal Gaussian averaged over the small bucket area. At this point, less that 1.5% of the total power in the beam is considered by the metric.

The HBQ curve in Fig. 2 displays a dramatic inflection as the total power in the core region falls below PIB_th. Since the beam intensity distribution is changing smoothly, this behavior demonstrates how inconsistencies in an HBQ measurement can arise if PIB_th is not well-matched to the shape of the beam being measured. As the core power approaches zero, the HBQ converges toward a more sensible value determined only by the width of the Gaussian pedestal beam.

The M² curve is generated by using a second-moment calculation on the core-pedestal beam to determine an effective far-field beam waist (w_meas), which is compared with the ideal Gaussian beam waist using M² = w_meas/w₀. The curve agrees with expected beam quality at the endpoints where the far-field beam is a single Gaussian. However, the rest of the curve clearly demonstrates how second-moment-based beam size determinations on unusually-shaped distributions can over-predict beam sizes and lead to incorrect assessments of beam quality.

The BQ curve calculated using Eq. (19) is the only metric that appears to remain consistent over the entire range of beam shapes. On both endpoints, the metric agrees with the M² of the respective Gaussians, while in between there are no inflections or divergent behavior as the shape of the far-field is scanned. The BQ also displays good agreement with the conventional HBQ and VBQ² metrics in the core-dominated region where we would expect them to have the most validity.

7. Elliptical Gaussian beam example

This section will show how the temperature-based beam quality is connected with the invariant beam propagation ratio in Gaussian beams with non-circular far-field profiles. The treatment of general astigmatic beams and testing methods is outlined in an International Standards Organization (ISO) document [9]. The approach supports the determination the ten second-order moments to arrive at an effective beam propagation ratio, the intrinsic astigmatism, and the twist parameter. For simple astigmatic beams, the invariant effective beam propagation ratio (usually considered the beam quality) is given by,

M_{e f f}^{2} = \sqrt{M_{x}^{2} M_{y}^{2}},

where

M_{x}^{2}

and

M_{y}^{2}

are the beam propagation ratios along the two principal axes of the beam.

We first consider the case of a Gaussian beam propagated from a circular near-field but having an elliptical far-field intensity distribution due to a difference in M² along the two principal axes of the beam. Any elliptical Gaussian intensity distribution can be written,

I (x, y) = \frac{2 P}{π x_{0} y_{0}} \exp (\frac{- 2 x^{2}}{x_{0}^{2}}) \exp (\frac{- 2 y^{2}}{y_{0}^{2}}),

where x₀ and y₀ are the second-moment beam sizes along the two principal axes. Inserting Eq. (23) into Eq. (17), one arrives at the expression [7,8],

Δ T_{\max} = \frac{\sqrt{2} P}{π^{\frac{3}{2}} κ y_{0}} K (\sqrt{1 - β^{2}}),

where β = x₀/y₀ describes the beam asymmetry and K is the complete elliptic integral of the first kind. The major axis (y₀ in this case) is always chosen such that β≤1. For a circular near-field distribution, the waist sizes in the far-field are proportional to the beam propagation ratios, so that β =

M_{x}^{2}

/

M_{y}^{2}

(for

M_{x}^{2}

<

M_{y}^{2}

). The beam quality as defined in Eq. (19) for a circular near-field beam using Eq. (24) is given by,

B Q = \frac{\frac{π}{2}}{\frac{1}{M_{y}^{2}} K (\sqrt{1 - {(\frac{M_{x}^{2}}{M_{y}^{2}})}^{2}})},

where K(0) = π/2. The fact that BQ in Eq. (25) does not agree with the prediction of Eq. (22) is expected, since the ability of a beam to raise the temperature of a material is not, by definition, a beam invariant. Intuitively, one would expect the centerline heating to be reduced by beam asymmetry. Therefore, it is instructive to consider the ratio,

η = \frac{Δ T_{\max - c i r c u l a r}}{Δ T_{\max - e l l i p t i c a l}} = \frac{(\frac{π}{2})}{\sqrt{β} K (\sqrt{1 - β^{2}})},

describing how the induced heating by any Gaussian beam is reduced solely due to far-field asymmetry. One way to produce such asymmetry would be to change the near-field beam aspect ratio while keeping the effective beam area constant. The area constraint keeps the average intensity in the far-field constant. Using Eq. (25) and Eq. (26), the beam quality can now be written,

B Q = η \sqrt{M_{x}^{2} M_{y}^{2}} .

Equation (27) provides a connection between the BQ the beam invariant for a circular near-field.

In the more general case, where both the measured and the ideal beams are elliptical in the far-field plane, Eq. (24) can again be used to arrive at a formula for the beam quality. Using similar arguments as above, one can also show that impact on heating due to beam asymmetry is now, more generally,

η = \frac{\sqrt{β_{i d e a l}} K (\sqrt{1 - β_{i d e a l}^{2}})}{\sqrt{β_{m e a s}} K (\sqrt{1 - β_{m e a s}^{2}})},

where β_ideal and β_meas are the ideal and measured beam asymmetries in the far-field, respectively. Note that η does not depend on the beam sizes or orientation, but only on the measured and ideal asymmetries. The determination of η simply uses a consideration of how the heating by any circular beam is reduced by making it elliptical at constant intensity. Equation (27) provides that if

M_{x}^{2}

and

M_{y}^{2}

are measured using standard techniques, the BQ can be calculated using Eq. (28).

8. Conclusions

An alternative laser beam quality definition (BQ) has been proposed that offers an unambiguous way to compare a measured far-field beam with an ideal beam. The definition is based on the intrinsic ability of a laser beam to induce the maximum-possible steady-state temperature rise in an ideal semi-infinite absorbing volume. The definition weights the contributions to the temperature rise over the entire far-field beam profile, which can be of any shape. The BQ of an axially-symmetric Gaussian beam under the new definition is equal to the invariant beam propagation ratio (M²). For elliptical Gaussian beams, the beam invariant is related to the BQ through the far-field beam asymmetries. For general beam shapes, the definition supports the use of circular far-field power-in-the-bucket analysis to determine laser beam quality. The concepts also provide a straightforward way to extract a BQ number from power-in-the-bucket data.

Acknowledgments

The author would like to thank Jack Slater and Alan Paxton for their useful comments.

References and links

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A laser beam quality definition based on induced temperature rise

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