Self-focusing effect of annular beams propagating in the atmosphere

Yuqiu Zhang; Xiaoling Ji; Xiaoqing Li; Qiang Li; Hong Yu

doi:10.1364/OE.25.021329

1. Introduction

When a powerful laser beam propagates through the atmosphere, the beam filamentation and collapse result in breaking of the beam if the beam power is over a threshold critical power [1,2]. However, it was demonstrated that the beam filamentation can be suppressed and the powerful laser beam can be compressed as a whole due to the self-focusing effect when it propagates from orbit through Earth’s inhomogeneous atmosphere to the ground [3]. Recently, we studied the effect of spherical aberration and the effect of spatial coherence on the laser beam self-focusing in the atmosphere [4,5]. It was shown that the negative spherical aberration may be beneficial to compress the laser beam due to the self-focusing effect [4], and a partially coherent beam is less sensitive to the self-focusing effect than a fully one [5].

The telescopes that create an annular beam are used as the transmitter in high-power laser systems. Therefore, it is very important to study the self-focusing effect of annular beams propagating in the atmosphere to assist delivering powerful laser beams from orbit to the ground. In recent years, the influence of the atmospheric turbulence on the beam quality of annular beams has attracted many attentions [6–10], but the self-focusing effect on the beam quality of annular beams hasn’t been investigated. On the other hand, it is known that the self-focusing is related to the beam profile. Thus, an interesting question arises: What are the differences of self-focusing effect between annular, flat-topped and Gaussian beams? Furthermore, the B integral is a very useful parameter to characterize quantitatively the beam modulation caused by the self-focusing effect. Until now, the B integral has been studied only for Gaussian beams [11]. In this paper, we investigate the B integral of annular beams and the rule related to the B integral of annular beams for maximal compression as a whole from orbit to the ground. Furthermore, the reliability of this rule is also examined. The results obtained in this paper are very useful for laser power transportation from orbit to the ground.

2. Theoretical model

Consider the laser beam propagation along the vertical direction z from the orbit to the ground, and z = 0 is the sea level. The nonlinear Schrödinger equation is usually used to describe the features of diffraction and Kerr nonlinearity of laser beams propagating in a nonlinear medium. Under the standard paraxial approximation for the envelope of the electric field A, the nonlinear Schrödinger equation reads as [3]:

2 i k \frac{\partial A}{\partial z} + \nabla_{⊥}^{2} A + 2 k^{2} \frac{n_{2}}{n_{0}} {| A |}^{2} A = 0,

where

k = 2 π / λ

is the wave-number related to the wave length

λ

,

\nabla_{⊥}^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

is the two-dimensional transverse Laplace operator to describe transverse diffraction of the electric field, n₀ is the linear refractive index, and n₂ is the nonlinear refractive index. In the atmosphere, n₂ is related to altitude and is expressed as [3]

n_{2} (z) = n_{2} (0) e x p [- (z / h)],

where

n_{2} (0) = 5.6 \times 10^{- 19} {cm}^{2} / W

is the refractive index on the ground, and h = 6 km is adopted by interpolating method as exponential [3]. When the altitude from the ground is high enough, the value of n₂ is so small that the nonlinear term in Eq. (1) can be negligible. Usually, self-focusing effect in the atmosphere is considered just below approximately 20 km. In a homogeneous medium, the self-focusing starts when the beam power exceeds the critical value

P_{cr} = 0.93 λ^{2} / (2 π n_{0} n_{2})

. For the

λ = 0.8 μm

light adopted in this paper, we have

P_{cr} = 4.6 GW

when

z = h = 6 km

.

An analytical solution of Eq. (1) is very difficult to obtain but numerical results can be generated. Equation (1) can be rewritten as [12]

\frac{\partial A}{\partial z} = (\hat{D} + \hat{N}) A,

where

\hat{D}

represents the differential operator that accounts for the transverse diffraction in a linear medium, and

\hat{N}

is the nonlinear operator that governs the nonlinear self-focusing effect. Equation (3) indicates that the diffraction effect and the nonlinear self-focusing effect can be dealt with by using the split-step Fourier method [12, 13].

The beam profile and phase waveform of annular beams at the height of the orbit (i.e., $z = F$ ) is expressed as [7]

A (z = F, r) = A_{1} \exp [- \frac{i C_{0}}{w_{0}^{2}} r^{2}] \sum_{u = 1}^{N} \frac{{(- 1)}^{u - 1}}{N} (\begin{matrix} N \\ u \end{matrix}) [\exp (- \frac{u r^{2}}{w_{0}^{2}}) - \exp (- \frac{u r^{2}}{ε w_{0}^{2}})],

where

(\begin{matrix} N \\ u \end{matrix})

is the binomial coefficient, w₀ is the outer radius of the annular beam, N is the beam order, u is a positive integer. ε = S / S₀ represents the obscure ratio of an annular beam, where

S = π {(\sqrt{ε} w_{0})}^{2}

,

S_{0} = π w_{0}^{2}

and 0 < ε < 1.

C_{0} = k w_{0}^{2} / 2 F

, and F is the focal distance (i.e., the height of orbit in this paper). The well-known relationship between the beam power P and the electric field A can be expressed as [14]

P = \int_{0}^{2 π} d θ \int_{0}^{\infty} {| A |}^{2} r d r = const,

which is called the Talanov theorem [14]. Substituting Eq. (4) into Eq. (5), the expression of the coefficient A₁ can be obtained, i.e.,

A_{1} = \sqrt{\frac{P}{2 π w_{0}^{2} \sum_{u = 1}^{N} \sum_{v = 1}^{N} \frac{{(- 1)}^{u + v - 2}}{N^{2}} (\begin{matrix} N \\ u \end{matrix}) (\begin{matrix} N \\ v \end{matrix}) [\frac{1 + ε}{2 (u + v)} - \frac{ε}{2 (u + ε v)} - \frac{ε}{2 (ε u + v)}]}} .

where v is a positive integer.

Under the same values of the initial power P and the outer radius w₀ (e.g., P = 50 P_cr, w₀ = 0.3 m), at the initial plane z = F, the two-dimensional (2D) intensity distributions and the contour lines of 3D intensity of annular beams with different values of the beam obscure ratio ε are shown in Figs. 1 and 2 respectively. It can be seen that, as ε increases, the area of the dark region across annular beams increases (see Figs. 1 and 2), and the maximum intensity decreases (see Fig. 1).

Fig. 1 At the initial plane z = F, 2D initial intensity distributions I(x, y = 0, z = F) of annular beams with different values of the obscure ratio ε.

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Fig. 2 At the initial plane z = F, contour lines of 3D intensity of annular beams with different values of the obscure ratio ε.

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On the other hand, the beam profile and phase waveform of flat-topped beams at z = F can be expressed as

A (z = F, r) = A_{2} \exp [- \frac{(i C_{0})}{w_{0}^{2}} r^{2}] \sum_{u = 1}^{N} \frac{{(- 1)}^{u - 1}}{N} (\frac{N}{u}) \exp (\frac{u r^{2}}{w_{0}^{2}}) .

Substituting Eq. (7) into Eq. (5), the expression of the coefficient A₂ can be obtained, i.e.,

A_{2} = \sqrt{\frac{P}{2 π w_{0}^{2} \sum_{u = 1}^{N} \sum_{v = 1}^{N} \frac{{(- 1)}^{u + v - 2}}{N^{2}} (\begin{matrix} N \\ u \end{matrix}) (\begin{matrix} N \\ v \end{matrix}) [\frac{1}{2 (u + v)}]}} .

In this paper, N = 10 is adopted for annular and flat-topped beams. When N = 1, Eq. (7) reduces to the expression of Gaussian beams, i.e.,

A (z = F, r) = \sqrt{\frac{2 P}{π w_{0}^{2}}} \exp [- \frac{(1+i C_{0})}{w_{0}^{2}} r^{2}] .

Under the same values of the initial power P and the outer radius w₀ (e.g., P = 2 P_cr, w₀ = 0.8 m), at the initial plane z = F, the 2D intensity distributions and the contour lines of 3D intensity of annular (ε = 0.7), flat-topped and Gaussian beams are shown in Figs. 3 and 4 respectively.

Fig. 3 At the initial plane z = F, 2D initial intensity distributions I(x, y = 0, z = F) of annular, flat-topped and Gaussian beams.

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Fig. 4 At the initial plane z = F, contour lines of 3D intensity of annular, flat-topped and Gaussian beams.

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3. Self-focusing effect of annular beams

Assume that energy transport from a low earth orbit where an annular beam is focused by an orbital mirror with a focal distance F = 500 km and propagates from height of 500 km to the ground in all modeling of this paper. The influence of the self-focusing effect on the characteristics of annular beams is shown in Figs. 5-10, in which w₀ = 0.3 m is adopted.

Fig. 5 Intensity distributions I(x, y = 0, z) versus the propagation distance z, P = 50 P_cr, ε = 0.9.

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Fig. 6 2D intensity distributions I(x, y = 0, z) for different values of the obscure ratio ε on the ground, P = 50 P_cr.

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Fig. 7 3D intensity distributions for different values of the obscure ratio ε on the ground in the atmosphere, P = 50 P_cr.

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Fig. 8 For different values of initial power P, (a) maximum intensity I_max, and (b) S_R versus obscure ratio ε on the ground.

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Fig. 9 For different values of initial power P, (a) w, and (b) w / w_free versus obscure ratio ε on the ground.

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Fig. 10 For different values of initial power P, (a) w_86.5%, and (b) w_86.5%/w_86.5%free versus obscure ratio ε on the ground.

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The changes of the intensity distribution with the propagation distance z from a height of 20 km to the ground in free space and in the atmosphere are shown in Figs. 5(a) and 5(b), respectively. Figure 5 indicates that the dark hollow intensity distribution disappears when an annular beam enters the atmosphere at z = 20 km, and it becomes a Gaussian-like main lobe surrounded by a ring side lobe. Figure 5(a) shows that the intensity maximum decreases in free space as the propagation distance increases (i.e., as the value of z decreases). On comparing Fig. 5(b) with Fig. 5(a), one can see that in the atmosphere the laser beam is compressed as a whole without running into filamentation and nonlinear beam breakup even if the beam power P is much larger than the critical value P_cr (e.g., P = 50 P_cr), the intensity maximum increases due to self-focusing effect as z decreases, and on the ground the main lobe becomes the leptokurtic beam profile. However, the side lobe is nearly unchanged on propagation both in free space and in the atmosphere. It implies that the effect of self-focusing on the main lobe is much stronger than that on the side lobe.

The 2D and the 3D intensity distributions of annular beams with different values of the beam obscure ratio ε on the ground are shown in Figs. 6 and 7, respectively. It can be seen that the intensity maximum increase as ε increases in free space (see Fig. 6(a)). Figures 6(b) and 7 indicate that the main lobe is compressed more strongly due to the self-focusing effect as ε increases. Therefore, the annular beam with larger obscure ratio ε is more sensitive to the self-focusing effect. In addition, comparing Fig. 6(b) with Fig. 6(a), it can be seen that the side lobes in the atmosphere remain nearly unchanged as those in free space (see subfigures in Fig. 6(b) and Fig. 6(a)).

In this paper, the Strehl ratio S_R is applied to describe the influence of the self-focusing effect on the maximum intensity, which is defined as S_R = I_max / I_0max [15], where I_max and I_0max are the maximum intensity in the atmosphere and in free space respectively. The larger S_R is, the stronger the self-focusing effect is. For different values of initial power P, the maximum intensity I_max and I_0max, and Strehl ratio S_R on the ground versus the beam obscure ratio ε are shown in Figs. 8(a) and 8(b) respectively. Figure 8(a) shows that, in free space I_0max increases slightly as ε increases whatever the value of P is, but in the atmosphere I_max increases rapidly as ε increases when the value of P is large. Figure 8(b) indicates that S_R increases as ε and P increase. It means that the self-focusing effect on the maximum intensity becomes stronger as ε and P increase.

From Figs. 6-8 above we see that the self-focusing effect on the main lobe is much stronger than that on the side lobe. So, changes of the beam radius w at half maximum intensity of the main lobe are studied in this paper. For different values of initial power P, the beam radius w and relative beam radius w / w_free on the ground versus the obscure ratio ε are shown in Figs. 9(a) and 9(b) respectively, where w and w_free are the beam radius in the atmosphere and in free space respectively. The w is smaller than w_free (see Fig. 9(a)), and w and w / w_free decrease as ε and P increase (see Figs. 9(a) and 9(b)). It means that the main lobe is compressed more strongly due to self-focusing as ε and P increase.

On the other hand, the power in the bucket (PIB) is a measure of laser energy focusability in the far field, which is defined as _{$PIB = \int_{0}^{b} {| A (r, z) |}^{2} r d r / \int_{0}^{\infty} {| A (r, z) |}^{2} r d r$} [16], where the b is the bucket half-width chosen. Changes of the beam width w_86.5% and relative beam width w_86.5% / w_86.5%free are studied in this paper (see Fig. 10), where w_86.5% and w_86.5%free are the bucket half-width when PIB = 86.5% in the atmosphere and in free space respectively. On comparing Fig. 10 with Fig. 9, one can see that the law of w_86.5% versus ε and P is the same as that of w, but the difference of w_86.5% between different values of P is much smaller than that of w. It implies that the self-focusing effect on w_86.5% is weak. The physical reason is that the self-focusing effect on the main lobe is much stronger than that on the side lobe, and the beam width w_86.5% is large enough to include the whole main lobe for different values of P.

4. B integral of annular beams

The beam modulation caused by the self-focusing effect can be quantitatively described by the B integral [17–20]. The B integral represents the nonlinear phase shift acquired after propagation through nonlinear medium [20]. The nonlinear phase shift is related to the beam profile, i.e., the B integral of annular beams is different from that of Gaussian beams. The B integral is a very important characteristic parameter to describe the beam quality, e.g., the value of the B integral usually should not exceed several units in order to avoid filamentation [18]. The B integral of laser beams propagating through the atmosphere can be expressed as [19,20]

B = k \int_{0}^{z_{0}} I_{0} n_{2} (z)d z,

where z₀ is the thickness of atmosphere (z₀ = 20 km in this paper), and I₀ is the initial maximum intensity entering the atmosphere. The larger value of the B integral means the laser beam is more heavily modulated by the self-focusing effect.

Substituting Eq. (2) into Eq. (10), the B integral can be expressed as

B = k I_{0} n_{2} (0) h [1 - \exp (- \frac{z_{0}}{h})] .

In the linear theory, based on the extended Huygens–Fresnel principle, one can get the axial field of annular beams focused by an orbital mirror and propagating L distance in free space, i.e.,

\begin{array}{l} A (z = L, r = 0) = A_{1} \frac{i k}{2 L} \exp (- i k L) \sum_{u = 1}^{N} \frac{{(- 1)}^{u - 1}}{N} (\begin{matrix} N \\ u \end{matrix}) \\ \times {[\frac{u}{w_{0}^{2}} + \frac{i k {(1 - L / F)}^{- 1}}{2 L}] - [\frac{u}{ε w_{0}^{2}} + \frac{i k {(1 - L / F)}^{- 1}}{2 L}]} . \end{array}

For our case (i.e., L = 480 km, L ≈F), from Eq. (12) one can obtain the maximum intensity I₀ of annular beams entering atmosphere, i.e.,

I_{0} = A_{1}^{2} {(\frac{k}{2 L})}^{2} w_{0}^{4} \sum_{u = 1}^{N} \sum_{v = 1}^{N} \frac{{(- 1)}^{u + v - 2}}{N^{2}} (\begin{matrix} N \\ u \end{matrix}) (\begin{matrix} N \\ v \end{matrix}) [\frac{{(1 - ε)}^{2}}{u v}] .

Substituting Eq. (13) into Eq. (11), we can obtain the expression of the B integral of annular beams on the ground, i.e.,

B = k n_{2} (0) h [1 - \exp (- \frac{z_{0}}{h})] A_{1}^{2} {(\frac{k}{2 L})}^{2} w_{0}^{4} \sum_{u = 1}^{N} \sum_{v = 1}^{N} \frac{{(- 1)}^{u + v - 2}}{N^{2}} (\begin{matrix} N \\ u \end{matrix}) (\begin{matrix} N \\ v \end{matrix}) [\frac{{(1 - ε)}^{2}}{u v}] .

Equation (14) indicates that the B integral increases as P and w₀ increase, i.e., the self-focusing effect becomes stronger as P and w₀ increase. The change of the B integral versus the obscure ratio ε is shown in Fig. 11. It can be seen that the B integral increases as the beam obscure ratio ε increases, which is in agreement with the result obtained in section 3.

Fig. 11 B integral versus obscure ratio ε on the ground.

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In this paper, assume that the threshold critical power reaches when a laser beam can’t be compressed as a whole and starts the filamentation. Changes of the maximum B integral (B_max) of annular beams for maximal compression without filamentation on the ground versus the outer radius w₀ and the obscure ratio ε are shown in Fig. 12 (see the red dots in Fig. 12). It can be seen that the B_max decreases at first and then keeps almost unchanged as w₀ increases. However, the B_max varies slightly with ε for the same value of w₀. In particular, for w₀ = 0.7-1.3, we have B_max ≈2.1.

Fig. 12 Maximum B integral (B_max) versus the outer radius w₀ and the obscure ratio ε on the ground. (Red dots: numerical simulation results; Curve surface: fitting surface by using Eq. (15)).

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Based on the red dots in Fig. 12, the fitting equation of the B_max of annular beams for maximal compression without filamentation on the ground can be obtained, i.e.,

B_{\max} = \frac{{C+D}_{01} ε + F_{01} w_{0} + F_{02} w_{0}^{2} + G_{02} ε w_{0}}{{1+D}_{1} ε + F_{1} w_{0} + D_{2} ε^{2} + F_{2} w_{0}^{2} + G_{2} ε w_{0}},

where the coefficients are shown in Table 1. According to Eq. (15), the fitting surface is also given in Fig. 12. It can be seen that the fitting surface is well matched with the numerical simulation results (red dots). When B < B_max, annular beams can be compressed as a whole due to the self-focusing effect without filamentation.

Table 1. Values of the coefficients in Eq. (15).

View Table

It is known that the good quality of a fitting model can be tested with analysis of variance (ANOVA), which employs statistical coefficients to describe the model. The Adjusted R-Squared ( $R_{Adj}^{2}$ ) is a measure of the amount of variation around the mean explained by the model, which is adjusted for the number of terms in the model, and $R_{Adj}^{2} < 1$ [21]. The value of $R_{Adj}^{2}$ is closer to 1 means the fitting model is better [21]. It is shown_{$R_{Adj}^{2} = 0.999$}for our fitting Eq. (15). Thus, the fitting Eq. (15) is reliable, and the expression of Eq. (15) is simple enough. The fitting Eq. (15) presents an effective design rule for maximal compression without filamentation of transported annular beams.

5. Comparison of self-focusing effect between annular, flat-topped and Gaussian beams

In this section, we compare the self-focusing effect between annular, flat-topped and Gaussian beams under the same value of w₀ (e.g., w₀ = 0.8 m in Figs. 13-16). 3D intensity distributions for annular (ε = 0.7), flat-topped and Gaussian beams on the ground are shown in Figs. 13(a)-13(c) respectively. On comparing the atmosphere case with the free space case, one can see that the self-focusing effect on the annular beam is stronger than that on flat-topped and Gaussian beams. The effect of self-focusing on the Gaussian beam is the weakest.

Fig. 13 3D intensity distributions for annular, flat-topped and Gaussian beams on the ground, P = 2 P_cr.

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Fig. 14 Relative intensity distributions I(x, y = 0, z)/I_0max for annular, flat-topped and Gaussian beams on the ground, P = 2 P_cr.

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Fig. 15 Beam radius w versus the propagation distance z for annular, flat-topped and Gaussian beams, P = 2 P_cr.

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Fig. 16 Beam radius w on the ground versus P/P_cr for annular, flat-topped and Gaussian beams.

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The relative intensity I(x, y = 0, z) / I_0max distributions for annular (ε = 0.7), flat-topped and Gaussian beams on the ground are shown in Fig. 14, where I_0max is the corresponding maximum intensity for the free space case. From Fig. 14 it can be seen that the annular beam is more strongly compressed than flat-topped and Gaussian beams. The Gaussian beam is most weakly compressed. In addition, the annular beam has the largest I_max / I_0max, followed by the flat-topped beam. The Gaussian beam has the smallest I_max / I_0max.

Changes of beam radius w with the propagation distance z for annular (ε = 0.7), flat-topped and Gaussian beams both in free space and in the atmosphere are shown in Fig. 15. It is clear that the three type beams entering the atmosphere are all compressed due to self-focusing. From Fig. 15, we have w / w_free = 0.55, 0.65 and 0.89 on the ground for annular, flat-topped and Gaussian beams respectively. Namely, the self-focusing effect on annular, flat-topped and Gaussian beams should be in sequence from strong to weak. In addition, Fig. 15 indicates that the w of a flat-topped beam may be smaller than w_free of an annular beam due to self-focusing.

Changes of beam radius w on the ground with the relative initial power P / P_cr for annular (ε = 0.7), flat-topped and Gaussian beams are shown in Fig. 16. Figure 16 shows that the annular beam has the smallest w, followed by the flat-topped beam. The Gaussian beam has the largest w. However, the Gaussian beam has the largest threshold critical power value, followed by the flat-topped beam. The annular beam has the smallest threshold critical power value.

6. Conclusions

In this paper, we study the self-focusing effect of annular beams propagating in the atmosphere to assist delivering powerful laser beams from orbit to the ground. The dark hollow intensity distribution disappears when an annular beam enters the atmosphere, and it becomes a Gaussian-like main lobe surrounded by a ring side lobe. It is found the self-focusing effect on the main lobe is much stronger than that on the side lobe, and the annular beam is compressed more strongly due to the self-focusing as the beam obscure ratio increases. On the other hand, we compare the self-focusing effect between annular, flat-topped and Gaussian beams. It is shown that the self-focusing effect on the annular beam is stronger than that on flat-topped and Gaussian beams. The effect of self-focusing on the Gaussian beam is the weakest. However, the threshold critical power values of annular, flat-topped and Gaussian beams should be in sequence from small to large. Furthermore, we derive the expression of the B integral of annular beams propagating from orbit to the ground in the atmosphere, and present the fitting equation related to the B integral of annular beams for maximal compression without filamentation. The results obtained in this paper are useful for laser power transportation from orbit to the ground.

Funding

National Natural Science Foundation of China (NSFC) (61775152, 61475105, 61505130).

Acknowledgments

The authors are very thankful to the reviewers for their valuable comments.

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Coefficient	C	D₀₁	F₀₁	F₀₂	G₀₂
Value	−9.08282	3.21759	7.76183	−8.12504	−0.40872
Coefficient	D₁	F₁	D₂	F₂	G₂
Value	−1.13783	−5.93073	0.45978	0.38321	1.92003

Self-focusing effect of annular beams propagating in the atmosphere

Abstract

1. Introduction

2. Theoretical model

3. Self-focusing effect of annular beams

4. B integral of annular beams

5. Comparison of self-focusing effect between annular, flat-topped and Gaussian beams

6. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (16)

Tables (1)

Equations (15)

Optics Express