Design and analysis of unequal aperture off-axis optical integrator for solar simulators

Siwen Chen; Gaofei Sun; Gaofei Sun; Gaofei Sun; Shi Liu; Shi Liu; Shi Liu; Guoyu Zhang; Guoyu Zhang; Guoyu Zhang; Jierui Zhang; Jiali Chen; Ran Zhang

doi:10.1364/OE.503844

1. Introduction

The sun provides the most common source of radiation for remote sensing applications. However, due to the difficulty of repeated observation of solar irradiation in nature due to the influence of test time and season, the difficulty and cost of related experiments have risen sharply [1,2]. Irradiation uniformity is often one of the important indicators for reliability evaluation of solar simulators [3,4], but limited by the parameters of the optical system itself [5], light source, light concentrating system and light homogenizing system will all have an impact on irradiation uniformity, and there are many difficulties in the study of solar simulators with high irradiation uniformity.

V. Esen [6] and M. Tawfik [7] studied the light source, discussed the advantages and disadvantages of various solar simulator light sources, and determined that xenon lamp has a dominant position in the design of collimating solar simulator. For the uniform light system, researchers have tried to apply a variety of uniform light devices [8–11] in solar simulators, among which the optical integrator based on Kohler lighting principle has been chosen by more researchers for its high light energy utilization and large area uniform illumination, and its related research has been more extensive [12–23].The main factors affecting the uniform effect of optical integrator are its optical parameters and geometric shape. To this end, Katsuki [12] and Udekok [13] optimized the design of the exit pupil optical integrator and the compact optical integrator by increasing the focal length and increasing the aperture respectively. However, the increase of focal length leads to the decrease of edge irradiance, and the increase of aperture makes the relative aperture difficult to match. T. Lv [14] and S. Liu [15] have studied the geometry of the optical integrator, and the results show that the regular hexagonal sub-eye lens has a high energy utilization rate, and the edge compensation lens can improve the edge irradiance of the irradiation surface. H. Jiang [16] applied the free-form surface with finer aberration correction to the optical integrator, but it was difficult to design and process heat-resistant materials [17]. To this end,H. Peng [18] adopted variable curvature design to optimize the optical integrator, which reduced the difficulty of optimization, but still faced the problem of aspherical surface manufacturing.

To sum up, an unequal aperture off-axis optical integrator design method is proposed in this paper. An unequal aperture off-axis optical integrator uniform light model is established to compensate for the decrease of uniformity ability caused by aberration, uneven incident light distribution and incident light angle. The unequal aperture off-axis optical integrator is designed to break through the limitations of aberrations and incident light spots on the optical integrator's uniformity ability and solve the problem that it is difficult to improve the optical integrator's uniformity ability. To achieve the goal of improving the optical integrator's evenness without greatly affecting the irradiance or introducing the aspherical design.

2. Analysis of factors affecting the uniformity ability of optical integrator

In the traditional optical integrator design, the lens group is mostly regarded as a thin lens (there is no thickness of the lens). In this case, the Gaussian image plane and the ideal image plane are the same plane. When the near parallel light incident on the optical integrator, the radiation of all the sub-eye lens can be fully superimposed on the ideal image plane. The principle of uniform light is shown in the red line in Fig. 1 [19].

Fig. 1. The principle of irradiation uniformity by optical integrator

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However, in the actual optical system, aberrations occur due to the thickness of the lens. The Gaussian image plane is curved and does not completely coincide with the ideal image plane, resulting in sidelobe effect, that is, the projection of sub-eye lens in different ring bands are not fully superimposed. At the same time, due to the existence of incidence angle α and the Gauss distribution of irradiance before optical integrator [20], the image plane still presents a distribution state with high center and low edge after irradiation superposition. This distribution state of high center and low edge is further expanded under the action of aberration, thus reducing the improvement of irradiation uniformity. At this time, the principle of smoothing light is shown in the blue line in Fig. 1.

3. Composition and the working principle of unequal aperture off-axis optical integrator

Aiming at improving the optical integral smoothing ability, an unequal aperture off-axis optical integrator composed of front and rear fly-eye lens array and focusing lens was designed, as shown in Fig. 2. Among them, the front and rear fly-eye lenses are mirror symmetric, and are composed of a glass backplane and a multi-ring band sub-eye lens arranged in a ring band on it. The yellow part of the picture is the gap among the lenses, which is blackened to prevent light leakage, and the blue part is sub-eye lens. All the optical axes of sub-eye lens are located on the line between their geometric center and the main optical axis of the integrator but not overlaps with the geometric center, and the focal length is the same. The caliber of sub-eye lens is the same in the same belt, and the caliber of sub-eye lens is different in different belts.

Fig. 2. The principle of irradiation uniformity by unequal aperture off-axis optical integrator

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The light with a certain incidence angle entering the unequal aperture off-axis optical integrator is divided by the front fly-eye lens, then superimposed by the rear fly-eye lens, and finally imitated to the image plane through the focusing lens. Because the sub-eye lens aperture D₁ and D₂ are different, but their focal lengths are the same, the projected area of the edge sub-eye lens is larger, thus improving the irradiation uniformity of its projection in the effective irradiation surface S_I. Since there is a certain off-axis value Δhi (i = 1,2) in the ring band fly-eye lens, the aberration caused by the lens thickness is compensated by the off-axis value, so that the outgoing rays A'A, A"A, B'B, B"B near the main optical axis of the ring band sub-eye lens coincide with the edge of the effective irradiation surface SI. Since the irradiation distribution of the incident light is Gauss like, this shift can improve the effective irradiation surface SI edge irradiation. The unequal aperture off-axis optical integrator compensates for the difficult increase in irradiation uniformity caused by the design of the concentrator system, while providing more uniform irradiation for the subsequent collimation system, thus solving the difficult improve in irradiation uniformity of the solar simulator.

4. Unequal aperture off-axis optical integrator design

By analyzing the uniformity of unequal aperture off-axis optical integrator, the factors that affect irradiation uniformity in the unequal aperture off-axis optical integrator are determined. A design method for unequal aperture off-axis optical integrator is proposed, and specific design parameters for the unequal aperture off-axis optical integrator are determined.

4.1 Principle of irradiation uniformity by unequal aperture off-axis optical integrator

Based on scalar diffraction theory, the uniform light principle of unequal aperture off-axis optical integrator is analyzed. By analyzing the intensity distribution function of each lens surface when light propagates in unequal aperture off-axis optical integrator, the corresponding relation between optical intensity distribution on the image surface after optical integrator is convergent through the condenser lens and optical field distribution at the front light source of optical integrator is established.

A key component of scalar diffraction theory is the Huygens-Fresnel principle, which states that every surface source on a wave front can be viewed as a secondary perturbation center that produces spherical waves, and that the position of the wavefront at the next moment is the envelope of all these wavelet fronts. Therefore, for a light field with wavelength λ emitted by a point (x_0, y₀, z) in space, the Fresnel diffraction function at a point (x, y, z) in the wave front at a later time can be expressed as

(1)$$U({x,y} )= \frac{{{e^{({jkz} )}}}}{{j\lambda kz}}\int {\int\limits_{ - \infty }^\infty {U({{x_0},{y_0}} ){e^{\left\{ {j\frac{k}{{2z}}[{{{({x - {x_0}} )}^2} + {{({y - {y_0}} )}^2}} ]} \right\}}}d{x_0}d{y_0}} } $$

where, j is the imaginary part; λ is the incident wavelength; k is the wave number.

Since the irradiation distribution after the fly-eye lens of unequal aperture off-axis optical integrator is the fresnel function of the incident light field relative to the fly-eye lens emerging light angle spectrum, in order to realize the matching between spatial propagation function and the transmittance function of unequal aperture off-axis optical integrator, Eq. (2) is obtained by fresnel transform

(2)$$U\left( {x,y} \right) = \frac{{{e^{\left( {jkz} \right)}}}}{{j\lambda kz}}{e^{\left\{ {j\frac{k}{{2z}}\left[ {{{\left( {x - {x_0}} \right)}^2} + {{\left( {y - {y_0}} \right)}^2}} \right]} \right\}}}F\left[ {U\left( {{x_0},{y_0}} \right){\textrm{e}^{j\frac{k}{{2z}}\left( {x_\textrm{0}^\textrm{2} + \textrm{y}_\textrm{0}^\textrm{2}} \right)}}} \right]$$

where, the F (U, j, k, z, x₀, y₀) function represents the fresnel transform.

Since the arrangement of sub-eye lens in unequal aperture off-axis optical integrator is centrosymmetric, the sub-eye lens arrangement can be analyzed in one dimension and then generalized to a two-dimensional state by it. Since the fly-eye lens array is densely composed of sub-eye lens, the transmittance function of the fly-eye lens array can be regarded as the convolution of the transmittance function of sub-eye lens and the comb function of the lens array, and the transmittance function ${\tau _\textrm{m}}(y )$ of the fly-eye lens array in the one-dimensional direction is

(3)$${\tau _\textrm{m}}(y )= \sum\limits_{n ={-} N}^N {\delta ({y - np} )\otimes \left[ {rect\left( {\frac{y}{p}} \right){e^{ - \lambda k\frac{{{y^2}}}{{2f}}}}} \right]} $$

where, N is the number of axial lenses of the integrator; p is sub-eye lens aperture; f is the focal length of sub-eye lens.

The one-dimensional form of Eqs. (1), (2), that is, the Huygens-Finnier diffraction function $U(y )$ at any subsequent point (y₀, z) for a light field of wavelength λ emitted by a point (y, 0) can be expressed as

(4)$$U(y )= \frac{{{e^{({jkz} )}}}}{{j\lambda kz}}\int\limits_{ - \infty }^\infty {U({{y_0}} ){e^{\left\{ {j\frac{k}{{2z}}{{({y - {y_0}} )}^2}} \right\}}}d{y_0}} $$

(5)$$U(y )= \frac{{{e^{({jkz} )}}}}{{j\lambda kz}}{e^{\left\{ {j\frac{k}{{2z}}{{({y - {y_0}} )}^2}} \right\}}}F\left[ {U({{y_0}} ){\textrm{e}^{j\frac{k}{{2z}}\textrm{y}_\textrm{0}^\textrm{2}}}} \right]$$

By using Eqs. (3) and (5), the optical field distribution function(${U_1}({{y_1}} )$) of the surface light field after the front fly-eye lens in the optical integrator irradiation uniformity process, the optical field distribution function(${U_{12}}({{\textrm{y}_\textrm{2}}} )$) of rear fly-eye lens front surface light field, the optical field distribution function(${U_\textrm{2}}({{y_\textrm{2}}} )$) of rear fly-eye lens rear surface light field, the optical field distribution function(${U_{\textrm{2}F}}({{\textrm{y}_\textrm{F}}} )$) of focusing lens front surface light field, the optical field distribution function(${U_\textrm{F}}({{\textrm{y}_\textrm{F}}} )$) of focusing lens rear surface light field, the optical field distribution function(${U_\textrm{s}}({{\textrm{y}_\textrm{s}}} )$) of focusing lens mirror surface can be inferred as Eqs. (6)–(11)

(6)$${U_1}({{y_1}} )= {U_0}({{y_1}} ){\tau _{\textrm{m1}}} = {U_0}({{y_1}} )\sum\limits_{n ={-} N}^N {\delta ({{y_1} - np} )\otimes \left[ {rect\left( {\frac{{{y_1}}}{p}} \right){e^{ - jk\frac{{{y_1}^2}}{{2f}}}}} \right]} $$

(7)$${U_{12}}({{\textrm{y}_\textrm{2}}} )= \frac{{{e^{jk{d_1}}}}}{{j\lambda k{d_1}}}\int\limits_{ - \infty }^\infty {{U_1}({{y_1}} ){e^{j\frac{k}{{2{d_1}}}{{({{y_2} - {y_1}} )}^2}}}d{y_1}} $$

(8)$${U_\textrm{2}}({{y_\textrm{2}}} )= {U_{\textrm{12}}}({{y_\textrm{2}}} ){\tau _{\textrm{m2}}} = {U_{\textrm{12}}}({{y_\textrm{2}}} )\sum\limits_{n ={-} N}^N {\delta ({{y_\textrm{2}} - np} )\otimes \left[ {rect\left( {\frac{{{y_\textrm{2}}}}{p}} \right){e^{ - ik\frac{{{y_\textrm{2}}^2}}{{2f}}}}} \right]} $$

(9)$${U_{\textrm{2}F}}({{\textrm{y}_\textrm{F}}} )= \frac{{{e^{jk{d_L}}}}}{{j\lambda k{d_L}}}\int\limits_{ - \infty }^\infty {{U_2}({{y_2}} ){e^{j\frac{k}{{2{d_L}}}{{({{y_F} - {y_2}} )}^2}}}d{y_2}} $$

(10)$${U_\textrm{F}}({{\textrm{y}_\textrm{F}}} )= \frac{{{e^{jkz}}}}{{j\lambda kz}}\int\limits_\infty ^{ - \infty } {{U_{\textrm{2}F}}({{y_F}} ){e^{\left[ {j\frac{k}{{2z}}{{({{y_s} - {y_F}} )}^2}} \right]}}} d{y_F}$$

(11)$${U_\textrm{s}}({{\textrm{y}_\textrm{s}}} )= \frac{{{e^{jk{f_2}}}{e^{jk\frac{{{y_F}^2}}{{{f_2}}}}}}}{{j\lambda k{f_2}}}F\{ {U_F}({{y_F}} )\} $$

where, ${U_0}({{y_1}} )$ is the light field distribution function at the light source; j is the imaginary part; λ is the incident wavelength; k is the wave number; d₁ is the interval between the front and rear fly-eye lens groups, d₁= f₁; z is the distance between the irradiation surface and the focusing mirror, namely the focal length of the focusing mirror, z = f₂.

The relationship between the optical field distribution function (${U_\textrm{s}}({{\textrm{y}_\textrm{s}}} )$) at the focusing mirror surface and the optical field distribution function (${U_0}({{y_1}} )$) at the light source can be established by substituting equation (3)–(10) into Eq. (11) in turn. Since the optical intensity distribution function at the focusing mirror surface is the product of the optical field distribution function(${U_\textrm{s}}({{\textrm{y}_\textrm{s}}} )$) at the focusing mirror surface and its conjugate functions(${U_\textrm{s}}^\mathrm{\ast }({{\textrm{y}_\textrm{s}}} )$), the phase factor of the constant term can be abandoned by this product to define the optical intensity distribution function I at the irradiation surface

(12)$$I \propto {\left|{\sum\limits_{m ={-} \infty }^\infty {\delta \left( {{y_s} - m\frac{{\lambda {f_2}}}{p}} \right)\left[ {\sin c\left( {\frac{{p{y_s}}}{{\lambda {f_2}}}} \right) \otimes rect\left( {\frac{{{y_s}}}{{{{{F_p}} / {{f_s}}}}}} \right)} \right]} } \right|^2}$$

We can see from Eq. (12)

1) The intensity distribution function of the irradiance surface is actually the product of the convolution of the two fly-eye lens correlation functions and the sub-eye lens array arrangement function, so the intensity distribution of the irradiance surface can be changed by changing the sub-eye lens arrangement form;
2) The convolution term has the same propagation effect as sub-eye lens single channel, so the irradiation uniformity of the entire irradiation surface can be improved by changing the irradiation uniformity of any channel;
3) The sub-eye lens arrangement function is affected by the difference between the aperture of sub-eye lens and the wavelength function, and the aperture of sub-eye lens in the solar simulator is much larger than the wavelength, that is, diffraction phenomena will only occur at the edge of sub-eye lens, and the diffraction will only adjust the edge intensity of the image plane. Therefore, the edge irradiation of the image plane can be improved by changing the edge irradiation of sub-eye lens.

Therefore, the unequal aperture off-axis optical integrator was designed. The arrangement of sub-eye lens was changed by unequal aperture design and the edge irradiation of sub-eye lens was changed by off-axis design. The highest point of the projection energy of the sub-eye lens at the ring band is made close to the main optical axis, so as to coincide with the edge of the effective irradiation surface. The uniformity ability of sub-eye lens is improved by the combination of the increase of irradiation uniformity in sub-eye lens and the shift of the highest energy point of sub-eye lens.

4.2 Sub-eye lens aperture value and arrangement method

The aperture and arrangement of sub-eye lens have a great impact on the uniformity ability of unequal aperture off-axis optical integrator and the energy utilization rate of the system. Therefore, in order to determine the aperture value and the arrangement mode of sub-eye lens, the energy efficiency of the system and the light uniformity ability of the unequal aperture off-axis optical integrator should be combined to analyze.

Under the premise of the same aperture, regular hexagonal sub-eye lenses have better uniformity ability [14], so in this paper, regular hexagonal sub-eye lenses are chosen. Based on the principle of unequal aperture off-axis optical integrator, it can be seen that sub-eye lenses are symmetrically distributed in circular ring band on the glass back plate. Therefore, the layout problem of sub-eye lenses can refer to the optimization problem of geometric layout in circular space [20], to simplify the problem, in the unequal aperture off-axis optical integrator, the fly-eye lens only has two aperture values (when there are three or more sub-eye lens aperture values, it can be considered as multiple iterations of the sub-eye lens aperture value determination problem under two aperture values). At this time, the sub-eye lens layout problem can be seen as placing several large regular hexagons and several small regular hexagons into a large circle, and the regular hexagons do not overlap with each other, to define the sub-eye lens distribution constraint, the minimum distance between any regular hexagon and the boundary should satisfy

(13)$$\sum\limits_{i \ne j} {\max \left[ \begin{array}{l} {\left( {0,{{({\raise0.7ex\hbox{${\sqrt {3} }$} \!\mathord{\left/ {\vphantom {{\sqrt {3} } {4}}}\right.}\!\lower0.7ex\hbox{${4}$}}D_B^i + {\raise0.7ex\hbox{${\sqrt {3} }$} \!\mathord{\left/ {\vphantom {{\sqrt {3} } {4}}}\right.}\!\lower0.7ex\hbox{${4}$}}D_B^j)}^2} - ||{{p^i} - {p^j}} ||_2^2} \right)^2},\\ {\left( {0,{{({\raise0.7ex\hbox{${\sqrt {3} }$} \!\mathord{\left/ {\vphantom {{\sqrt {3} } {4}}}\right.}\!\lower0.7ex\hbox{${4}$}}D_S^i + {\raise0.7ex\hbox{${\sqrt {3} }$} \!\mathord{\left/ {\vphantom {{\sqrt {3} } {4}}}\right.}\!\lower0.7ex\hbox{${4}$}}D_B^j)}^2} - ||{{p^i} - {p^j}} ||_2^2} \right)^2},\\ {\left( {0,{{({\raise0.7ex\hbox{${\sqrt {3} }$} \!\mathord{\left/ {\vphantom {{\sqrt {3} } {4}}}\right.}\!\lower0.7ex\hbox{${4}$}}D_S^i + {\raise0.7ex\hbox{${\sqrt {3} }$} \!\mathord{\left/ {\vphantom {{\sqrt {3} } {4}}}\right.}\!\lower0.7ex\hbox{${4}$}}D_S^j)}^2} - ||{{p^i} - {p^j}} ||_2^2} \right)^2} \end{array} \right]} $$

where, D_B is the diameter of large diameter sub-eye lens; D_S is the diameter of small-caliber sub-eye lens; p_i and p_j are the inner points of the i and j regular hexagons; D_R is the total caliber of optical integrator.

\begin{array}{l} {\raise0.7ex\hbox{${{D_S}^i}$} \!\mathord{\left/ {\vphantom {{{D_S}^i} 2}}\right.}\!\lower0.7ex\hbox{$2$}} \le p_1^i \le {D_R}/2 - {\raise0.7ex\hbox{${{D_S}^i}$} \!\mathord{\left/ {\vphantom {{{D_S}^i} 2}}\right.}\!\lower0.7ex\hbox{$2$}}\\ {\raise0.7ex\hbox{${{D_b}^j}$} \!\mathord{\left/ {\vphantom {{{D_b}^j} 2}}\right.}\!\lower0.7ex\hbox{$2$}} \le p_2^j \le {D_R}/2 - {\raise0.7ex\hbox{${{D_b}^j}$} \!\mathord{\left/ {\vphantom {{{D_b}^j} 2}}\right.}\!\lower0.7ex\hbox{$2$}} \end{array}

Without changing the overall aperture of the unequal aperture off-axis optical integrator, the aperture of sub-eye lens is one of the important conditions for the quality of the integrator's uniformity effect. If the aperture of sub-eye lens is too large, the number of integrator channels is small, and the uniformity effect got worse after stacking; on the contrary, if the aperture of sub-eye lens is too small, it will lead to a decrease in the Fresnel number of the integrator, resulting in a stronger diffraction effect and affecting the light intensity distribution. Therefore, when designing the layout, it is necessary to incorporate aperture size limitations.

Since the sub-eye lens of unequal aperture off-axis optical integrator is distributed symmetrically in the ring band, and D_B is greater than D_S, the ring band number of small-aperture sub-eye lens is defined as N₁, and that of large-aperture sub-eye lens is N₂. Based on its geometry, it can be seen that the diameter of sub-eye lens and the number of ring bands should meet Eqs. (14,15):

(14)$${N_1} \ge 2$$

(15)$$2{D_R} = \left( {\left( {({\textrm{2}{N_\textrm{2}} - 1} )\sqrt 3 + \textrm{2}} \right) + \textrm{1}} \right)\left( {({\textrm{2}{N_\textrm{1}} - 1} )\sqrt 3 {D_S}} \right)$$

where, N₁, N₂∈N*; ${D_B} = ({2{N_1} - 1} ){D_S}$.

On the premise of ensuring the connection between the optical integrator and the optical pupil of the front and rear system, the light propagation matrix model at the integrator is established by combining the matrix optical theory. As shown in Eqs. (16,17)

(16)$$\begin{array}{c}\left[ {\begin{array}{c} {{y_\textrm{c}}}\\ {\tan {\alpha_c}} \end{array}} \right] = \left( {\begin{array}{cc} 1&{{l_1}}\\ 0&1 \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ {\frac{{ - 1}}{{{f_2}}}}&1 \end{array}} \right)\left( {\begin{array}{cc} 1&{{l_2}}\\ 0&1 \end{array}} \right)\\ \times \left[ {\left( {\begin{array}{c} {({\textrm{n} + \textrm{1}} ){D_s}}\\ 0 \end{array}} \right) + \left( {\begin{array}{cc} 1&0\\ {\frac{{ - 1}}{{{f_1}}}}&1 \end{array}} \right)\left( {\begin{array}{cc} 1&{{l_1}}\\ 0&1 \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ {\frac{{ - 1}}{{{f_1}}}}&1 \end{array}} \right)\left( {\begin{array}{c} {{y_R}}\\ {\tan {\alpha_R}} \end{array}} \right)} \right] \end{array}$$

(17)$$\begin{array}{c} \left[ {\begin{array}{c} {{y_\textrm{c}}}\\ {\tan {\alpha_c}} \end{array}} \right] = \left( {\begin{array}{cc} 1&{{l_1}}\\ 0&1 \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ {\frac{{ - 1}}{{{f_2}}}}&1 \end{array}} \right)\left( {\begin{array}{cc} 1&{{l_2}}\\ 0&1 \end{array}} \right)\\ \times \left[ {\left( {\begin{array}{c} {{\textrm{N}_\textrm{1}}{D_s} + ({\textrm{n - }{\textrm{N}_\textrm{1}}} ){D_B}}\\ 0 \end{array}} \right) + \left( {\begin{array}{cc} 1&0\\ {\frac{{ - 1}}{{{f_1}}}}&1 \end{array}} \right)\left( {\begin{array}{cc} 1&{{l_1}}\\ 0&1 \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ {\frac{{ - 1}}{{{f_1}}}}&1 \end{array}} \right)\left( {\begin{array}{c} {{y_R}}\\ {\tan {\alpha_R}} \end{array}} \right)} \right] \end{array}$$

where, f₁ is the focal length of the front sub-eye lens group and the back sub-eye lens group of the integrator; f₂ is the focal length of the focusing lens; l₁ is the distance between the front sub-eye lens group and the rear sub-eye lens group, and l₂ is the distance between the rear sub-eye lens group and the focusing lens. y_R, y_C, α_R, α_C are the height and angle of incident light and outgoing light, respectively. n is the number of ring band to which the sub-eye lens belongs, where the central sub-eye lens corresponds to n = 0.

When n + 1 ≤ N₁, take Eq. (16), otherwise, take Eq. (17).

At the same time, the optical integrator should be connected with the aperture of the front and rear system. Taking a solar simulator as an example, the relative aperture D/f of the optical integrator is set to 1/7.391, the optical integral total aperture D_R is set to 45 mm, and the incoming light angle α_R and α_C are set to 7.12°. At this time, the relationship between the number of sub-eye lens and the diameter of small-caliber sub-eye lens can be solved, as shown in Table 1.

Table 1. Relation between the number of sub-eye lens tings and the diameter of sub-eye lens

View Table

It can be seen from the table that the number of lens ring band affects the diameter of small-caliber fly-eye lens, but as shown in Ref. [17], the sub-eye lens system with too small aperture will produce Fresnel diffraction. In order to ensure high uniformity, F_N should be greater than 500 so that diffraction effect cannot affect irradiation uniformity. In order to prevent the decrease of the Fresnel number caused by the excessively small aperture of sub-eye lens, thus affecting the irradiation uniformity of the diffraction light field on the edge of the lens, the integrator's Fresnel number F_N satisfies Eq. (18).

(18)$${F_N} = \frac{{{D_S}D}}{{4\lambda f}}$$

Since the analog spectrum range of the solar simulator is 400 nm to 1100 nm, the maximum wavelength λ=1100 nm can be calculated to obtain D_S greater than 4.46 mm.

The number of sub-eye lens ring band that meets this condition can only be N₁= 1, N₂= 1, that is, the number of small-caliber sub-eye lens ring bands is 1, and the number of large-caliber sub-eye lens ring bands is also 1. When the number of sub-eye lens ring band increases, the diameter of the smallest sub-eye lens will decrease. When the aperture of sub-eye lens drops below 4.46 mm, Fresnel diffraction occurs in the entire sub-eye lens, and the irradiation uniformity decreases, so the number of ring band of sub-eye lens is determined. The optional arrangement of optical integrator under this condition is shown in Fig. 3 (a) and (b).

Fig. 3. Arrangement scheme of unequal aperture optical integrator (a) 1:3 caliber arrangement (b) 1:2 caliber arrangement

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Since the diameter ratio of large-caliber sub-eye lens and small-caliber sub-eye lens in the figure is 1:3 and 1:2 respectively, the arrangement mode shown in Fig. 3(a) is 1:3, and the arrangement mode shown in Fig. 3(b) is 1:2. Since the energy utilization rate of the optical integrator is changed when the arrangement of the optical integrator is changed, the total aperture of the sub-eye lens on the optical integrator is calculated to determine the energy utilization rate of the system.

If the full circle radius is R under the two configurations, then the diameter R₁ of the small sub-eye lens of the 1:3 systems is(R/(3(1 + 3^0.5))), and the diameter R₂ of the small sub-eye lens of the 1:2 system is.(R/6) According to Eqs. (19,20), the sub-eye lens aperture of 1:3 system and 1:2 system is 2.359R² and 2.237R², respectively.

(19)$$7 \times 6 \times \frac{{\sqrt 3 }}{4}R_1^2 + 6 \times 6 \times \frac{{\sqrt 3 }}{4}{({3{R_1}} )^2} = 2.359{R^2}$$

(20)$$7 \times 6 \times \frac{{\sqrt 3 }}{4}R_2^2 + 6 \times 6 \times \frac{{\sqrt 3 }}{4}{({2{R_2}} )^2} = 2.237{R^2}$$

Since the optical aperture of 1:3 arrangement is larger than that of 1:2 arrangement, it can be seen that the energy utilization rate of 1:3 arrangement is higher. Therefore, 1:3 arrangement is selected as the unequal aperture off-axis optical integrator arrangement. At this time, it can be determined that the diameter of the optical integrator small-diameter sub-eye lens is D₁= 5.48 mm, and the diameter of the large-diameter sub-eye lens is D₂= 16.46 mm.

4.3 Determination of optical integrator lens optical axis offset

Based on the principle of geometric optics, the off-axis design changes the light path near the main optical axis of different belt sub-eye lens, so that the projection of the highest energy of belt sub-eye lens coincides with the edge of the effective irradiation surface, and compensates for the uneven irradiation caused by aberrations and the decrease in system energy utilization caused by different aperture designs. The off-axis direction should be close to the principal optic axis because of the distribution of high edge and low center [24] irradiation through the peripheral sub-eye.

Combined with the uniform light principle diagram of unequal aperture off-axis optical integrator shown in Fig. 2, the off-axis values Δh₁ and Δh₂ were determined. First, the sub-eye lens which mapping relationship of the center channel is established, as shown in Eq. (21).

(21)$$\frac{{{R_s}}}{{{l_2} - {R_c}\left( {1 - \cos \left( {ar\sin \frac{{{R_s}}}{{Rc}}} \right)} \right)}} = \frac{{{D_1}}}{{2{l_1}}}$$

where, R_s is the image height of the image surface; the diameter of the sub-eye lens of the center channel is the same as that of the sub-eye lens of the first ring band, D₁= 5.48 mm; R_c is the radius of curvature of the focusing mirror; l₁ and l₂ are the distances between the front and rear fly-eye lens arrays and the distances between the unequal aperture off-axis optical integrator and the focusing mirror, respectively.

The image height R_s and the off-axis quantity Δh_i (i = 1.2) of sub-eye lens with different girdles should satisfy Eq. (22).

(22)$$\begin{array}{c} {n_2}\sin (90^\circ{-} \arccos (\frac{{{\raise0.7ex\hbox{${3{D_1}}$} \!\mathord{\left/ {\vphantom {{3{D_1}} 2}}\right.}\!\lower0.7ex\hbox{$2$}} + ({{l_1} - \Delta {h_{iOUT}}} )\tan ({{\beta_i}} )- {R_S}}}{{{f_2} + {n_2}{d_2} - \Delta {h_i}}})\\ - \arccos \left( {\frac{{{\raise0.7ex\hbox{${3{D_1}}$} \!\mathord{\left/ {\vphantom {{3{D_1}} 2}}\right.}\!\lower0.7ex\hbox{$2$}} + ({{l_1} - \Delta {h_{iOUT}}} )\tan ({{\beta_i}} )}}{{{R_C}}}} \right)\\ = \sin \left( {\arctan ({{\beta_i}} )- \arccos \left( {\frac{{{\raise0.7ex\hbox{${3{D_1}}$} \!\mathord{\left/ {\vphantom {{3{D_1}} 2}}\right.}\!\lower0.7ex\hbox{$2$}} + ({{l_1} - \Delta {h_{iOUT}}} )\tan ({{\beta_i}} )}}{{{R_C}}}} \right)} \right) \end{array}$$

where, n₁ and n₂ are unequal aperture off-axis optical integrator and refractive index of focusing mirror, respectively. Δh_i is the off-axis amount of sub-eye lens in the i ring band. Δh_iOUT is the axial height difference between the incident point of sub-eye lens and the main point of sub-eye lens on the focusing mirror. f₁ and f₂ are the focal length of sub-eye lens and focusing mirror respectively. ${\beta _i}$ is the incidence angle of the sub-eye lens of the i ring band, ${\beta _i} = \frac{{{\raise0.7ex\hbox{${{D_i}}$} \!\mathord{\left/ {\vphantom {{{D_i}} 2}}\right.}\!\lower0.7ex\hbox{$2$}} - \Delta {h_i}}}{{{f_1}}}$.

The off-axis quantity of Δh_iOUT and the fly-eye lens of the i band shall meet Eq. (23).

(23)$$\Delta h_{iOUT}^2 = R_C^2 - \left( {\frac{{3{D_2}}}{2} + {{({{l_1} - \Delta {h_{iOUT}}} )}^2}\tan ({\beta_i})} \right)$$

At the same time, the sub-eye lens aperture, focal length, and relative position should meet the following relations

(24)$$\left\{ {\begin{array}{{c}} {{f_1} = \frac{{{n_1}{r_1}}}{{1 - {n_1}}}}\\ {{l_1} = {f_1} - {n_1}{d_1}} \end{array}} \right.$$

where, d₁ is the thickness of the optical integrator's sub-eye lens; r₁ is the curvature radius of the first ring band sub-eye lens; based on processing experience, usually the thickness d₁= 7 mm [4]; D₁/f₁ is 1/7.391 of the system relative aperture; n₁ is the optical integrator refractive index, because the optical integrator material is JGS3 [23] heat-resistant quartz glass, n₁= 1.4586.

The specific parameters of unequal aperture off-axis optical integrator are determined to accord sections 4.2 and 4.3 as follows: sub-eye lens diameter D₁= 5.48 mm for the first ring band, sub-eye lens diameter D₂= 16.46 mm for the second ring band, sub-eye lens focal length f₁= 40.42 mm, and the distance between the front and rear fly-eye lens groups l₁= 30.21 mm. The off-axis quantity of sub-eye lens in the first ring band Δh₁= 0.1 mm, and the off-axis quantity of sub-eye lens in the second ring band Δh₂= 0.2 mm.

5. Simulation and verification of the principle of light uniformity in unequal aperture off-axis optical integrator

The Light Tools is used to simulate the unequal aperture off-axis optical integrator. The number of ray tracing strips is 1 × 10⁷[22], the intensity curve of the light source is directly imported from the xenon lamp toe-cutting file, the material of the ellipsoidal condenser, additional mirror and collimator is K9, the material of the optical integrator is heat-resistant quartz glass JGS3, and the stop is indium steel 4J36, and the simulation system is shown in Fig. 4.

Fig. 4. Simulation modeling diagram

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In the Fig. 4, the color of the light represents the path of the light, and different colors of the light through different paths. Generally, irradiation non-uniformity is used to measure irradiation uniformity of solar simulator. The specific calculation equation is as follows [21]

(25)$$\varepsilon ={\pm} \frac{{{E_{\max }} - {E_{\min }}}}{{{E_{\max }} + {E_{\min }}}} \times 100\%$$

where, ε represents irradiation irregularity; E_max represents the maximum irradiance in the irradiation surface; E_min indicates the minimum irradiance in the irradiation plane.

Analyze the distribution of irradiation surfaces at the focal plane of optical integrator, the picture plane of optical integrator, and effective irradiated surface to verify the correctness of the design principle of unequal aperture off-axis optical integrator, and the rationality of the selection of sub-eye lens aperture ratio for annular distribution the scientificity of the off-axis quantity of sub-eye lens with annular distribution and the authenticity of the improvement of irradiation uniformity by the unequal aperture off-axis optical integrator. Considering the engineering practice and Fresnel diffraction problem, the traditional integrator chooses 37 channels. Considering the engineering practice and Fresnel diffraction problem, the traditional integrator chooses 37 channels.

1) Irradiation distribution at focal plane of optical integrator of unequal aperture and traditional optical integrator, as shown in Fig. 5. Compared with traditional optical integrator, the edge channel aberration of unequal aperture off-axis optical integrator has been corrected to a certain extent, while the irradiation distribution within each sub-eye lens has changed.
2) In the picture plane of optical integrator, the irradiation non-uniformity in the ф 26 mm irradiation plane is 26.02%, 20.73%, 16.09%, 14.87%. It can be seen from Fig. 6 (a), (b) and (c), compared with the traditional optical integrator, the irradiation uniformity of the unequal aperture system has increased to a certain extent. It can be seen from Fig. 6 (c) and (d) that the irradiation uniformity is further improved after the off-axis design is carried out.
3) Keep the off-axis amount of the second ring band sub-eye lens 0.2 mm unchanged, and the off-axis amount of the first ring band sub-eye lens from 0-0.2 mm. The off-axis amount of sub-eye lens in the first ring band remains 0.1 mm unchanged, while the off-axis amount of sub-eye lens in the second ring band ranges from 0-0.4 mm for simulation. The axial irradiation distribution on the irradiation surface is shown in Fig. 7. Through simulation analysis can be seen it can be seen that when the off-axis amount of sub-eye lens in the first ring band is 0.1 mm and the off-axis amount of sub-eye lens in the second ring band is 0.2 mm, the irradiation uniformity of the system is the best. The scientificity of off-axis selection of ring band sub-eye lens is verified.
4) As shown in Fig. 8, the radiation distribution on the effective irradiation surface ф 300 mm irradiation non-uniformity is 0.53%, 1.63%, 1.11%, 2.28% successively. It can be found that under the combined effect of different aperture and off-axis design, the irradiation irregularity index of the system changes from failing to meet the A-level solar simulation standard to exceeding the A-level solar simulator standard, and the irradiance only decreases by 16.5%. Reference [18] points out that only 10 kW xenon lamp can provide 2087.5 W/m² irradiation within ф 300 mm irradiation surface under traditional optical integrator, and can still reach 1700 W/m² after 16.5% reduction to meet the index requirement of a solar constant (1353 W/m²).

Fig. 5. Irradiation distribution at the picture plane of Optical integrator unequal aperture off-axis optical integrator(b) tradition optical integrator

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Fig. 6. Picture plane of Optical integrator Radiation distribution in the ф26 mm irradiation plane tradition optical integrator(b) 1:2 system unbiased axis optical integrator (c) 1:3 system unbiased axis optical integrator(d) unequal aperture off-axis optical integrator

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Fig. 7. Irradiation distribution on the image plane under different off-axis quantities change Δh₁ (b) change Δh₂

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Fig. 8. Irradiation distribution at the effective irradiated surface of different systems Unequal aperture off-axis optical integrator(b) 1:3 variable aperture optical integrator 1:2 variable aperture optical integrator(d) 37 channel optical integrator

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In conclusion, the unequal aperture off-axis optical integrator designed in this paper can improve the irradiation uniformity of the solar simulator without greatly reducing the irradiance.

6. Summary

The unequal aperture off-axis optical integrator design method is proposed to improve the optical homogenizing ability. Firstly, the unequal aperture off-axis optical integrator structure is designed based on the scalar diffraction theory to analyze the factors affecting the optical homogenization ability of the optical integrator. Then, the relationship between sub-eye lens aperture and arrangement is explored in combination with Lagrange invariance principle and semi-definite programming theory. Finally, the optimum off-axis amount of sub-eye lens with different ring band is determined from the perspective of geometric optics by using aberration theory and following the principle of edge light, so as to improve the evenness of optical integrator. The design results are verified by simulation analysis. The simulation results show that: in the picture plane of optical integrator, the irradiation non-uniformity in the ф 26 mm irradiation plane is 14.87%, which is better than 26.02% in the traditional optical integrator. At the same time, at the effective irradiated surface, the irradiation non-uniformity of 0.53% within the ф 300 mm reaches the irradiation standard of the class A + solar simulator, and the irradiance only decreases by 16.5% compared with the traditional optical integrator, which still meets the index requirement of a solar constant. The goal of improving the evenness of optical integrator is realized without greatly affecting irradiance and without introducing aspherical design.

Funding

111 Project (D21009); National Natural Science Foundation of China (616003061); Jilin Province Innovation and Entrepreneurship talent funding project (2023QN13); Jilin Scientific and Technological Development Program (20210201034GX).

Acknowledgments

The authors would like to thank the support of doctoral research project of Changchun University of Science and Technology.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Band number		Diameter Ds(mm)
N₁	N₂	Diameter Ds(mm)
1	1	5.48
1	2	3.29
2	1	3.36
2	2	2.02
3	1	2.42

Design and analysis of unequal aperture off-axis optical integrator for solar simulators

Abstract

1. Introduction

2. Analysis of factors affecting the uniformity ability of optical integrator

3. Composition and the working principle of unequal aperture off-axis optical integrator

4. Unequal aperture off-axis optical integrator design

4.1 Principle of irradiation uniformity by unequal aperture off-axis optical integrator

4.2 Sub-eye lens aperture value and arrangement method

4.3 Determination of optical integrator lens optical axis offset

5. Simulation and verification of the principle of light uniformity in unequal aperture off-axis optical integrator

6. Summary

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (26)

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