Defect edge steepness dependence of multiple nonlinear hot-image formation from a single phase defect

Hongchang Wang; Hongchang Wang; Zhaoyang Jiao; Yanli Zhang; Mingying Sun; Jianqiang Zhu

doi:10.1364/OE.399044

1. Introduction

With the increase in operational laser fluences, the laser-induced damage [1–5] of optical components is one of the bottlenecks of high-power laser systems, particularly in the final optics assembly [6]. This systematic damage problem is complicated, and it is related to the beam quality, component quality, and working environment [6,7]. The defect-induced downstream light intensification is primarily responsible for the optical damage in high-power laser systems [8–13]. Among light intensification mechanisms, the formation of nonlinear hot image due to Kerr nonlinearity [14–17] is one of the key processes limiting the output performance of high-power laser systems. The hot image located downstream is mainly formed by the nonlinear self-focusing effect [18,19]. The laser beam is first modulated by an upstream defect. After propagating through a distance in free space, the beam goes through a third-order nonlinear medium; thereafter, an intensified holographic image of the defect is produced at the conjugate plane of the upstream defect [18,20–23]. The peak intensity of the hot image may be several times higher than that of the initial background beam [24]; therefore, it can easily damage the optical components. Some valuable studies [20,25–27] have revealed the basic mechanism and characteristics of nonlinear hot-image formation. The properties of a hot image are closely related to the characteristics of the corresponding defect and the B integral [14]. Moreover, the hot image formed from a phase defect can be more intense than that formed from an amplitude one. Thus, the phase defect is mainly considered in this study. For the design of high-power laser systems, it is important to identify the downstream location and intensity of the peak light intensification caused by upstream defects [28]. Theoretically, we easily attempt to place the optical component away from the hot-image plane to avoid the risk of damage if the strong light modulation only exists on the conjugate plane.

Unfortunately, multiple hot image peaks at different axial positions occur during the hot-image formation under certain conditions, which makes the laser system design against strong light intensification more complicated and difficult. A second-order hot image can be formed from a phase defect, and its distance would be half that of the hot image [27]. Therefore, optical components tend to be placed behind the conjugate plane to avoid the effects of the high-order hot image. For multiple defects, besides the formation of the original hot image, more phenomena occur during the beam propagation due to the complicated interplay between the scatter waves from the defects. For example, when the beam is modulated by two or three parallel defects, a middle-line second-order hot image [29], an interference hot image [30], a pseudo-second-order hot image [31], an intense double image [32], a middle-line hot image [33], and a strong fringe at the downstream position of the hot image [34] can be formed under different conditions. For the cascaded nonlinear medium, some new phenomena also occur. When the laser beam is modulated by single or double defects and subsequently transmitted through the cascaded Kerr medium, multiple hot images are formed [35]. The intensity of the new hot image peak can be higher than that of the hot image. It is suggested that hot images generated via cascaded Kerr medium disks may be suppressed by widening the space between two neighboring disks [36].

In general, thus far, the multiple hot image peaks reported in previous works are formed under the condition of multiple defects or cascaded Kerr medium disks. However, few studies have been conducted on multiple hot image peaks, generated when the laser is modulated by a single defect and subsequently passes through a single Kerr medium. Meanwhile, the characteristics of the hot image are deeply related to the nonlinear medium parameters and the defect property. We found that the thickness of the nonlinear medium influences the structure evolution of the axial intensity distribution for a single defect [37]. The influence of the defect size, shape, and modulation depth on the hot image has already been investigated [28]. Another important property of a defect pertains to its fine structure, such as the edge steepness. However, to the best of our knowledge, few studies have examined the influence of the defect edge steepness on the nonlinear hot-image formation. When considering different edge steepnesses of a single defect, will the hot image appear at the expected position? Will new strong hot image peaks, besides those on the hot-image plane, be observed? Evidently, it is worthwhile to examine the above questions because some new findings may come to light. Therefore, we could enhance our understanding of the nonlinear hot-image effect and exploit it to improve the performance of high-power laser systems.

This study is devoted to investigating the influence of the edge steepness on the hot-image formation by the super-Gaussian defect model. We demonstrate that when the defect size and super-Gaussian order are larger than a certain value, a single hot image peak gradually evolves into two peaks. One moves forward and the other moves backward. This is a phenomenon different from previous traditional models. This paper is structured as follows: in Section 2, the optical path model is described; in Section 3, the formation of the double peak is demonstrated, and the propagation process for the formation is analyzed; in Section 4, the factors that influence the hot image peak are discussed; in Section 5, the results are discussed, and Section 6 presents the conclusion.

2. Optical path model

The optical path model and the beam propagation process discussed in this paper are shown in Fig. 1. The incident super-Gaussian beam propagates along the z-axis. The beam center is at the following point: x = 0, y = 0. The defect plane contains one defect, which is located at the center of the beam. The z coordinate of the defect plane is at 0.

Fig. 1. The optical path model. P₁ represents the plane of first hot image peak. P₂ represents the conjugated plane. P₃ represents the plane of second hot image peak.

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The distance from the defect plane to the incident surface of the Kerr medium slab is defined as the object distance (d₀). The thickness of the Kerr medium slab is expressed as L. The distance from the plane of interest to the exit plane of the Kerr medium is marked as d. P₂, at d = d₂, represents the conjugated plane. According to the traditional hot-image theory, d₂= d₀ when the Kerr medium is thin. P₁, at d = d₁, represents the plane of first hot image peak. P₃, at d = d₃, represents the plane of second hot image peak investigated in this study.

The incident beam is modulated by the defect; thereafter, it penetrates the Kerr medium, after transmitting through the free space of distance d₀. Upon exiting the Kerr medium, the beam passes through the free space behind the Kerr medium. The complex amplitude envelope of the electric field of the incident beam at d = 0 can be described by the following equation:

(1)$${\boldsymbol E}({{\boldsymbol x},{\boldsymbol y}} )= {{\textbf E}_{\textbf 0}}{\textbf {exp}}\left[ { - {{\left( {\frac{{{{\boldsymbol x}^{\textbf 2}} + {{\boldsymbol y}^{\textbf 2}}}}{{{\boldsymbol w}_{\textbf 0}^{\textbf 2}}}} \right)}^{{{\boldsymbol m}_{\textbf 1}}}}} \right],$$

where ${E_0}$ represents the peak amplitude; ${m_1}$, the super-Gaussian order; and w₀, the waist radius. The phase defects discussed in this paper are local defects that are caused by the local bump on the surface, the inclusion, or the coating flaw of an optical element. The super-Gaussian model with different orders is used to model different edge steepnesses of the defects. The transmitted function T (x, y) of a phase defect can be described using the following equation:

(2)$${\boldsymbol T}({{\boldsymbol x},{\boldsymbol y}} )= {\textbf{exp}}\left\{ {{\textbf i}{\boldsymbol \varphi }{\textbf{exp}}\left[ { - {{\left( {\frac{{{{\boldsymbol x}^{\textbf 2}} + {{\boldsymbol y}^{\textbf 2}}}}{{{{\boldsymbol a}^{\textbf 2}}}}} \right)}^{{{\boldsymbol m}_{\textbf 2}}}}} \right]} \right\},$$

where a represents the size of the defect; φ, the defect modulation depth; and m₂, the super-Gaussian order. The defects in different super-Gaussian orders are shown in Fig. 2. When m₂ = 1, the defect is a Gaussian defect. We can observe that when m₂ is large, the defect has a relatively steep edge. The width of the curves that exceeds 99% of the peak value is defined as the inner diameter, denoted as r₀, while the width of the curves that exceeds 1% of the peak value is defined as the outer diameter, denoted as r₁, as shown in Fig. 3(a). The degree of hard edges is described by the hardening factor, H:

(3)$${\boldsymbol H} = \frac{{{{\boldsymbol r}_{\textbf 1}}}}{{{{\boldsymbol r}_{\textbf 1}} - {{\boldsymbol r}_{\textbf 0}}}}.$$

The higher the hardening factor, the more extreme the edge steepness. Different hardening factors in the super-Gaussian orders are shown in Fig. 3(b). The hardening factor increases as the super-Gaussian order increases. The beam modulated by the defect can be described using the following equation:

(4)$${\boldsymbol E^{\prime}}({\boldsymbol r} )= {\boldsymbol E}({\boldsymbol r} )\; \times \; {\boldsymbol T}({\boldsymbol r} ).$$

If A = E/E₀, then the free space beam propagation can be described by the partial differential equation:

(5)$$\frac{{\mathbf{\partial}} {\boldsymbol A}}{{\mathbf{\partial}} {\boldsymbol z}} = {\boldsymbol i}\frac{\textbf 1}{{{\textbf 2}{{\boldsymbol k}_1}}}\; \mathbf{\nabla} _ \bot ^{\textbf 2}\; {\boldsymbol A}.$$

The beam propagation in the nonlinear medium can be described by the nonlinear Schrödinger (NLS) equation:

(6)$$\frac{{\mathbf{\partial} {\boldsymbol A}}}{{\mathbf{\partial} {\boldsymbol z}}} = {\boldsymbol i}\frac{\textbf 1}{{{\textbf 2}{{\boldsymbol k}_{\textbf 2}}}}\; {\mathbf \nabla} _ \bot ^{\textbf 2}\; {\boldsymbol A} + {\boldsymbol i}{{\boldsymbol B}_0}{|{\boldsymbol A} |^{\textbf 2}}{\boldsymbol A},$$

where $\nabla _ \bot ^2 = \left( {\frac{{{\mathbf{\partial}^2}}}{{\mathbf{\partial} {x^2}}}} \right) + \left( {\frac{{{\mathbf{\partial}^2}}}{{\mathbf{\partial} {y^2}}}} \right)$; k₁ and k₂ are the wavenumbers in the free space and the Kerr medium, respectively; B₀ = k₂n₂|E₀|²/n₀, where n₀ and n₂ are the linear refractive index and the nonlinear refractive index coefficients, respectively. Therefore, the B-integral value is given as B = B₀L. We calculate the propagation with a computer simulation using the standard split-step Fourier method. For the simulation, the parameters are mainly acquired from the high-power laser facility in our lab. Unless otherwise noted, the default parameters are as follows: for the incident beam, λ₀ = 351 nm, m₁ = 6, w₀ = 0.3 cm; for the defects, a = 200 μm, m₂ = 3, φ = 0.1π; for the Kerr medium, n₀ = 1.48, n₂= 4.2 × 10⁻²⁰, L = 40 mm; d₀ = 0.8 m. The average intensity of the incident beam is 3 GW/cm², which is the design point of 9 J/cm² @ 3ns. The transverse sampling points are 1024 × 1024. The size of the calculation window is 12mm×12mm.

3. Formation of the multiple nonlinear hot images

3.1 Downstream double hot image peaks

According to the traditional hot-image theory, the hot-image plane should be on the conjugate plane (located 0.8 m downstream of the nonlinear medium). To investigate the influence of the defect edge steepness on the hot-image formation, we first analyze the downstream modulation evolution. We define the maximum intensity in the beam profile as I_max. The evolution of I_max starting from the rear surface of the Kerr medium for the super-Gaussian defect of different orders is shown in Fig. 4. Evidently, there are double hot image peaks under certain conditions, which implies the formation of multiple hot images from a single defect.

Fig. 2. Defects in different super-Gaussian orders.

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Fig. 3. (a) Inner diameter and outer diameter in the super-Gaussian defects. (b) The hardening factor versus the super-Gaussian order.

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Fig. 4. The evolution of I_max from the rear surface of the nonlinear medium for defects of different super-Gaussian orders. The defect size is 200 μm, and the modulation depth is 0.1π.

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It can be observed in Fig. 4 that the downstream two-peak intensity is several times higher than the average incident beam intensity. In addition, a notable transition from a single hot image peak to double hot image peaks with the increase in the super-Gaussian order can be observed. For the sub-Gaussian cases (m₂ ≤ 1), a single hot image formed from the defect appears near the conjugate plane, according to the traditional hot-image theory. The single peak evolves into two peaks when the super-Gaussian order increases continually (m₂> 1). The first peak moves toward the Kerr medium, while the second peak moves away from the Kerr medium. When m₂= 3, the intensity of the first peak reaches 7.2 GW/cm², which appears at ∼0.68 m. The intensity of the second peak reaches 6.5 GW/cm², which appears at ∼0.91 m. This implies that there is one more dangerous location with strong peak light intensity, which was thought to be safe, and the associated risk has not been recognized in traditional models.

The beam intensity profiles of the two hot image peaks are shown in Fig. 5. As can be observed, there is an intense spot at the center of the field. The hot images at d = 0.68 m and d = 0.91 m have similar characteristics. This kind of intense spot is not same as the traditional hot-image formation mechanism but via a new propagation phenomenon, and it is defined as “double hot image peaks” in the following section.

Fig. 5. Beam intensity profile of the third order super-Gaussian defect at (a) d = 0.68 m and (b) d = 0.91 m; (c) the corresponding variation of the intensities with x at y = 0.

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3.2. Analysis of the propagation process for double hot image peaks

In this section, we analyze the propagation process to elucidate the formation mechanism of the double hot image peaks. The top view of the evolution of the transversal beam intensity distribution with the propagation distance is shown in Fig. 6. In these top-view transmission images, the common logarithm of the intensity is determined to magnify the rippled structure.

Fig. 6. The top-view images of the evolution of the transversal beam intensity distribution with the propagation distance for the 200 μm defect and 0.1π modulation depth. The common logarithm of the intensity is taken. (a) The third order super-Gaussian defect. (b) Gaussian defect

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Fig. 7. (a) Beam intensity profile on the exit surface of the Kerr medium for the super-Gaussian and the Gaussian cases. The common logarithm of the intensity is taken. (b) The evolution of I_max from the rear surface of the nonlinear medium for the defect of the super-Gaussian case, Gaussian case, high pass of the super-Gaussian case, low pass of the super-Gaussian case, high pass of the Gaussian case, and low pass of the Gaussian case. To observe the corresponding position of the peak, the data is normalized. The top view of the variation of the transversal beam intensity distribution with the propagation distance for (c) the high pass of the super-Gaussian case, (d) the low pass of the super-Gaussian case, (e) the high pass of the Gaussian case, and (f) the low pass of the super-Gaussian case. The common logarithm of the intensity is taken in (c–f).

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Figure 6(a) shows the case when the super-Gaussian order is three. The two left solid red lines denote the front and rear surfaces of the Kerr medium. The two right red dotted lines represent the locations of the first and the second peaks. We notice that there are some light tracks with high intensities on the exit surface of the defect plane. We can observe that the two peaks are both on the path of the bright light bands. As a comparison, the evolution of the beam intensity distribution for the Gaussian defect is shown in Fig. 6(b). In the case of Gaussian defect, H is smaller than the super-Gaussian case. There are no light bands in the exit surface of the Kerr medium slab in Fig. 6(b), compared with that in Fig. 6(a). The beam intensity profiles on the exit surface of the Kerr medium for both the third-order super-Gaussian and the Gaussian defect cases are shown in Fig. 7(a). We notice that there are some irregular ripples outside the two vertical red dotted lines in Fig. 7(a). We try to use the boundary of the main lobe to determine the high pass and low pass cases. The optical distribution evolution of the blocking of different areas in the super-Gaussian and Gaussian cases are shown in Figs. 7(c)–(f). The two solid red lines in Figs. 7(c)–(f) are the front and rear surfaces of the Kerr medium. In the high pass case (see Fig. 7(a)), the inside area of the two vertical red dotted lines in Fig. 7(a) is blocked, while in the low pass case, the outside area of the two vertical red dotted lines in Fig. 7(a) is blocked. It can be observed that in the low pass cases, the evolution of I_max is similar to those in the Gaussian cases. In the low pass cases of Gaussian case and super-Gaussian case in Figs. 7(d) and (f), the top view of the variation of the transversal beam intensity distribution with the propagation distance is similar. That’s because the light field in the low pass area for the two cases is almost the same as showed in Fig. 7(a). Without the irregular ripples outside the two vertical red dotted lines, there is only one peak left even for the super-Gaussian defect. Thus, the cause of multiple hot image peaks for the super-Gaussian defect is believed to be the appearance of multiple light bands during transmission. Therefore, the hard-edge diffraction is primarily responsible for the double hot image peaks for a single defect. Interestingly, in the high pass of the super-Gaussian cases, there is a two-valley-like structure at the position corresponding to the double peak in the super-Gaussian case, which may be due to the loss of most of the coherent superposition enhancing energy from the central area of the beam.

4. Further analysis of the double hot image peaks characteristics

The phenomenon of the double hot image peaks has been demonstrated, and the mechanism behind it has been revealed above. In this section, we discuss this phenomenon in detail. The influence of the super-Gaussian order, defect size, modulation depth, and Kerr medium thickness on the location and intensity of the double hot image peaks are investigated.

4.1 Influence of the super-Gaussian order

Since the formation of the two peaks is mainly caused by the hard-edge diffraction, this phenomenon should be related to the super-Gaussian order, i.e., the edge steepness. The distance of the maximum intensity from the rear surface of the nonlinear medium and the double peak intensity versus the super-Gaussian order are shown in Fig. 8. With the increase in the super-Gaussian order, the position of the two peaks from the rear surface of the nonlinear medium is roughly stable: the first peak, 0.68 m; the second peak, 0.91 m. The intensity of the hot image peaks increases as the super-Gaussian order increases. The intensity of the first peak increases from 6.5 GW/cm² to 8.2 GW/cm². The intensity of the second peak increases from 5.7 GW/cm² to 7.6 GW/cm². The growth trends of the two peaks are similar. They increase rapidly in the low super-Gaussian order and stabilize in the high super-Gaussian order.

Fig. 8. Distance of the maximum intensity and intensity versus the Super-Gaussian order, when the defect size is 200 μm, modulation depth is 0.1π, thickness of the medium (L) is 40 mm, refractive index (n₀) is 1.48, nonlinear index coefficient (γ) is 3.1 × 10⁻⁷ cm²/GW, and the input intensity is 3 GW/cm².

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The double hot image peaks are mainly attributed to the hard-edge diffraction. With the increase in the super-Gaussian order, the hardening factor H increases, so is the edge steepens and the strength of the hard-edge diffraction. The modulation of the light field on the exit surface of the Kerr medium is mainly caused by the interference of scattered waves of the defect and the background field. From the perspective of the angular spectrum, as m₂ increases, the high-frequency components in the light field gradually increase and become stronger. According to the B-T theory, the nonlinear growth is mainly the enhancement of the high frequencies. Thus, the intensity of the double peaks increases. When the super-Gaussian order is high, the structure of the hard edge changes slightly; thus, the intensity of the double hot image peaks changes and gradually stabilizes as the super-Gaussian order increases.

4.2 Influence of the defect size

There are different sizes of defects on the optical components. It is necessary to discuss the influence of the defect size on the two peaks. The defect radius range of 50–500 μm is adopted, which is the range utilized in the high power laser system. The distance of the double hot image peaks away from the Kerr medium and the intensity of the double hot image peaks for different defect radius are shown in Fig. 9. We can observe that with the increase in the defect radius, the distance of the first peak decreases from 0.8m to 0.2 m. The intensity of the first peak increases from 5.1 GW/cm² to 8.3 GW/cm² and decreases to 7.5 GW/cm² when the defect radius is 450 μm. The distance of the second peak increases from 0.8m to 1.3 m. The intensity of the first peak increases from 5.1 GW/cm² to 7.5 GW/cm².

Fig. 9. Distance of the maximum intensity and intensity of the two peaks versus the defect radius when the modulation depth is 0.1π, super-Gaussian order is three, thickness of the medium (L) is 40 mm, refractive index (n₀) is 1.48, nonlinear index coefficient (γ) is 3.1 × 10⁻⁷ cm²/GW, and input intensity is 3 GW/cm².

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We notice that the location of the second peak is 0.8 m when the defect radius is 50 μm. This is the predicted location of the hot image in traditional models. Actually, there is only one hot-image peak for the 50 μm defect here. Interestingly, it is found that for the super-Gaussian defects, double hot image peaks would not always be generated. When the defect is small enough for a fixed super-Gaussian order, only one peak is obtained in the downstream propagation. The evolution of I_max for different defect sizes and different super-Gaussian orders is shown in Fig. 10. When the defect radius increases to 100 μm, the single peak is divided into two peaks, as shown in Fig. 10. The defect radius is 90 μm when the two peaks are about to separate.

Fig. 10. I_max versus the propagating distance away from the nonlinear medium for different defect sizes when the super-Gaussian order is three.

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Once there are two hot image peaks, the first peak moves toward the Kerr medium, while the second peak moves away from the Kerr medium. The relative displacements from the conjugate plane for the first peak and the second peak at different defect sizes are shown in Fig. 11(a). As the size increases, the two peaks shift relative to the hot-image plane. The first peak and the second peak are roughly symmetrical to the traditional hot-image plane when the defect is small. The first peak shifts more than the second peak when the defect is large. The displacement of the second peak is denoted as L. L versus 0.9 × a²/λ is shown in Fig. 11(b) for two different wavelengths, where a is the defect radius. We find that the fitting (L = 0.9 × a²/λ) matches the simulation results quite well. Here, we observe that the displacement of the second peak is very close to the Rayleigh distance for a spot with diameter a [38]. In a sense, this further implies that the double peaks are the result of hard-edge diffraction.

Fig. 11. (a) The relative displacement from the conjugate plane for the first peak and the second peak at different defect radius. (b) The displacement of the second peak for different wavelengths.

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4.3 Influence of the modulation depth

In the above description, it is assumed that the modulation depth is 0.1π. In this section, we discuss the influence of the modulation depth on the double hot image peaks. We change the modulation depth from 0.2π to π at intervals of 0.1π. The distance of the double hot image peaks from the Kerr medium and the intensity for different modulation depths are shown in Fig. 12.

Fig. 12. Distance of the maximum intensity and intensity versus the modulation depth when the defect size is 200 μm, super-Gaussian order is three, thickness of the medium (L) is 40 mm, refractive index (n₀) is 1.48, nonlinear index coefficient (γ) is 3.1 × 10⁻⁷ cm²/GW, and input intensity is 3 GW/cm².

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The distance of the first peak is about 0.67 m with different modulation depths. The distance of the second peak is about 0.9 m with different modulation depths. The modulation depth slightly affects the positions of the two peaks. The intensity of the first peak initially increases and subsequently stabilizes, which is different from the traditional hot-image property. As the modulation depth increases, the light intensity of the second peak initially increases and subsequently decreases.

4.4 Influence of the Kerr medium thickness

The hot image is formed by the nonlinear Kerr effect. Thus, the thickness of the Kerr medium is another important factor. In the above description, it is assumed that the Kerr medium thickness is 40 mm according to the optical component parameter in our facility. In this section, we discuss the influence of the Kerr medium thickness on the double hot image peaks. We change the Kerr medium thickness from 20 mm to 100 mm at intervals of 10 mm. The distance of the two peaks away from the Kerr medium and the intensity versus the Kerr medium thickness are shown in Fig. 13. We can observe that the distance of the two peaks varies slightly for different medium thicknesses. The intensities of the two peaks increase with the Kerr medium thickness. The increase of the Kerr medium thickness means the increase of B-integral. In traditional B-integral theory, the increase of B-integral will enhance the nonlinear process, thus bringing a much stronger hot image. So in the double hot image peaks, it is consistent with the B-integral theory [14].

Fig. 13. Distance of the maximum intensity and intensity versus the Kerr medium thickness when the defect size is 200 μm, super-Gaussian order is three, modulation depth is 0.1π, refractive index (n₀) is 1.48, nonlinear index coefficient (γ) is 3.1 × 10⁻⁷ cm²/GW, and input intensity is 3 GW/cm².

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5. Discussion

In this paper, we describe the hot-image effect with more than one hot image peak, formed from a single defect with a single nonlinear medium. When the defect is a super-Gaussian defect, double hot image peaks are observed in the downstream propagation. According to the simulation results, the hard-edge diffraction is primarily responsible for the double hot image peaks. To investigate the factors influencing the two peaks, different super-Gaussian orders, modulation depths, defect sizes, and Kerr medium thicknesses are simulated. The simulation results revealed that the defect size has a significant effect on the position of the second peak. The other factors have little effect on the position of the two peaks; however, have a great effect on the intensity of the two peaks.

One super-Gaussian defect can form two hot image peaks. However, this occurs only under certain conditions. When the defect size is adequately small or the super-Gaussian order is adequately low, a second peak is not generated. The transition condition between the single peak and double peaks for different defect sizes and super-Gaussian orders is shown in Fig. 14. The dots represent the super-Gaussian order when two peaks occur at different sizes. The curve is fitted to the dots, which is the dividing line between the single hot image peak and the double hot image peaks. When the size and the super-Gaussian order are in the solid area, as shown in Fig. 14, two peaks are observed downstream; when the size and the super-Gaussian order are chosen in the grid area in Fig. 14, one peak is observed downstream. The fitting function for the curve is as follows:

(7)$${{\boldsymbol m}_\textbf{2}} = \frac{\textbf{19.25}}{{{\boldsymbol a} - \textbf{82.64}}} \textbf{+ 1.0256},$$

where m₂ represents the super-Gaussian order, and a represents the defect size. As shown in Fig. 14, for a certain super-Gaussian order, when the defect size decreases, a single peak appears gradually. This is because when the defect is overly small, it is equivalent to a point source for the nonlinear medium in the diffraction process; consequently, it loses its fine structure. Therefore, as the defect size decreases, it becomes increasingly difficult to observe the double hot image peaks. For a certain small defect, as the super-Gaussian order increases, a double-peak would gradually appear. This is because when the super-Gaussian order increases, the hardening factor H increases, as shown in Fig. 3(b). Consequently, the hard-edge diffraction effect increases. Therefore, it is relatively easy to observe the double hot image peaks for large super-Gaussian orders.

Fig. 14. Transition condition between the single peak and double peaks for different defect sizes and super-Gaussian orders.

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According to Eq. (7), the critical size is 82.64μm, and the critical super-Gaussian order is 1.0256. However, the current wavelength depends on the high power laser in our lab. Moreover, characteristics of hot image depend on the light field fluctuation distribution at the Kerr medium. This fluctuation distribution largely depends on the linear diffraction process from the defect to the Kerr medium. The light field distribution is determined by the Fresnel number. Therefore, the critical size a, wavelength λ and object distance d that causes double hot image peaks should meet a particular Fresnel number ($\frac{{{a^2}}}{{\lambda d}}$). According to equivalent Fresnel number, the critical size will increase as the wavelength increases. It will also increase as the object distance increases.

For the formation of multiple hot image peaks, previous researches have focused on multiple defects or cascaded mediums. Our research paper presents a new finding, that a single defect can form double hot image peaks under certain conditions when considering the defect edge steepness. In addition, the hot image peaks behind the conjugate plane are totally unexpected for a single defect. The analysis results revealed that the double hot image peaks are formed by the hard-edge diffraction effect. K. You [37] found that the Kerr medium thickness can influence the structure of the axial hot image peak; multiple peaks were observed for a thin Kerr medium, while a single peak was observed for a thick Kerr medium. In our study, we found that the double hot image peaks are formed mainly via the hard-edge diffraction. When the defect is a Gaussian defect, only a single peak will appear even under the same thin Kerr medium condition in Ref. [37]. Therefore, the defect edge steepness-induced hard-edge diffraction is a considerably essential process for the formation of multiple hot images from a single-phase defect. Actually, this paper provides a new dimension for studying the properties of hot image in high-power laser systems, which enriches the basic physical images of hot-image formation laws.

Moreover, the new rules presented in this paper are important for optical arrangement in system design to avoid high modulation-induced damage. For the actual optical systems, the influence of soft edge defect is relatively easy to control, even though it may produce a strong hot image. We can avoid placing optical components near the hot-image to reduce its influence because the hot image appears in a single fixed axial location that can be precisely predicted regardless of its size. However, for a hard-edge defect, there are multiple hot image peaks, which are not located at the fixed position of traditional hot image. It is even more complicated that the location of the two peaks varies with the defect size, which makes the hot image more complicated and unpredictable. The axial area affected by the hot image is greatly expanded for the hard-edge defect. Therefore, it is possible to place optical components at these locations without realizing the potential danger. This explains many of the expected filamentation damage sustained by the final optical components of high-power laser facilities, such as the National Ignition Facility, the Laser Mega Joule, and the SG laser facility [6]. With the rules revealed in this paper, we can better handle the influence of the nonlinear intensification caused by various defects of different edge steepnesses in the real world. The research on the double hot image peaks formation is of great significance to the arrangement of optical components and for avoiding hot-image-induced optical damages. Incidentally, the nonlinear manipulation of the light field can be achieved by controlling the defect edge property according to this study. Further, this approach can potentially be utilized for the design of bifocal diffraction lenses and the generation of multi-focus arrays in the future.

6. Conclusion

Here, the effect of the defect edge steepness on the formation of hot image was studied. We discovered a new hot-image phenomenon: when the defect is a super-Gaussian defect, double hot image peaks are formed in the downstream propagation under certain conditions. The intensity of the two peaks can be several times higher than the average intensity. The analysis results indicate that the hard-edge diffraction is mainly responsible for the double hot image peaks formation. The influence of different factors, including the super-Gaussian order, defect size, modulation depth, and Kerr medium thickness, on the double hot image peaks intensity and location were systematically investigated. The results show that with the increase in the super-Gaussian order, the intensity of the second-peak hot image increases gradually. The defect size exerts a great influence on the position of the two peaks. The modulation depth and thickness of the Kerr medium influence the intensity of the two peaks; however, they have less impact on the peak location. Moreover, the defect edge steepness and size dependences of the formation of multiple nonlinear hot images from a single-phase defect are further revealed in the discussion. The critical conditions for the formation of the double hot image peaks are also given.

The double hot image peaks are fatal to the optical components in high-power laser systems; in particular, the hot image peak behind the conjugate position is totally unexpected for a single defect. This work provides a new dimension for studying the properties of hot image in high-power laser systems. It also facilitates further understanding of the basic physical picture and laws of hot-image formation. The results can provide important guidance for optical defect specification evaluation and optical component layout design, as well as for beam quality control, in high-power laser systems.

Funding

Natural Science Foundation of Shanghai (19ZR1464400); National Natural Science Foundation of China (61975218); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2018282); The Strategic Priority Research Program of Chinese Academy of Sciences (XDA25020202).

Disclosures

The authors declare no conflicts of interest.

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Defect edge steepness dependence of multiple nonlinear hot-image formation from a single phase defect

Abstract

1. Introduction

2. Optical path model

3. Formation of the multiple nonlinear hot images

3.1 Downstream double hot image peaks

3.2. Analysis of the propagation process for double hot image peaks

4. Further analysis of the double hot image peaks characteristics

4.1 Influence of the super-Gaussian order

4.2 Influence of the defect size

4.3 Influence of the modulation depth

4.4 Influence of the Kerr medium thickness

5. Discussion

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (14)

Equations (7)

Optics Express