{
    "cells": [
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# Pickups"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Pickups define relationships between the components of a lens. For instance, in a singlet lens, the radii of curvature can be constrained to be equal in magnitude but opposite in sign."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 1,
            "metadata": {},
            "outputs": [],
            "source": [
                "import numpy as np\n",
                "\n",
                "from optiland import optic, optimization"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Define a starting lens:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 2,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "image/png": 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",
                        "text/plain": [
                            "<Figure size 1000x400 with 1 Axes>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "lens = optic.Optic()\n",
                "\n",
                "# add surfaces\n",
                "lens.surfaces.add(index=0, radius=np.inf, thickness=np.inf)\n",
                "lens.surfaces.add(index=1, radius=50, thickness=3, material=\"SK16\", is_stop=True)\n",
                "lens.surfaces.add(index=2, radius=-50, thickness=30)\n",
                "lens.surfaces.add(index=3)\n",
                "\n",
                "# set aperture\n",
                "lens.set_aperture(aperture_type=\"EPD\", value=10)\n",
                "\n",
                "# set fields\n",
                "lens.fields.set_type(field_type=\"angle\")\n",
                "lens.fields.add(y=0)\n",
                "\n",
                "# set wavelengths\n",
                "lens.wavelengths.add(value=0.48)\n",
                "lens.wavelengths.add(value=0.55, is_primary=True)\n",
                "lens.wavelengths.add(value=0.65)\n",
                "\n",
                "lens.draw()"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Define pickups:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": [
                "lens.pickups.add(\n",
                "    source_surface_idx=1,\n",
                "    attr_type=\"radius\",\n",
                "    target_surface_idx=2,\n",
                "    scale=-1,\n",
                "    offset=0,\n",
                ")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Note that pickups can also act on generic surface attributes using the string parameter \u0007ttr_type. When doing so, you can use [i] to stand in for the dynamic surface index that is resolved at runtime so the target does not accidentally overwrite the source attribute if you provide a hardcoded index.\n\n",
                "As an example, if you wanted to pickup the geometry coefficients, we can provide [i] as the index for the generic path:\n\n",
                "`python\n",
                "lens.pickups.add(\n",
                "    source_surface_idx=1,\n",
                "    attr_type='surface_group.surfaces[i].geometry.coefficients',\n",
                "    target_surface_idx=2\n",
                ")\n",
                "`\n\n",
                "Note: We omit executing this here since our current lens geometry does not have coefficients."
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Define optimization problem:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 4,
            "metadata": {},
            "outputs": [],
            "source": [
                "problem = optimization.OptimizationProblem()"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Add operands (targets for optimization):"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": [
                "\"\"\"\n",
                "Add a wavefront error operand for all wavelengths.\n",
                "\n",
                "Use Gaussian quadrature distribution for the rays (see distribution documentation for\n",
                "more information).\n",
                "\"\"\"\n",
                "\n",
                "for wave in lens.wavelengths.get_wavelengths():\n",
                "    input_data = {\n",
                "        \"optic\": lens,\n",
                "        \"Hx\": 0,\n",
                "        \"Hy\": 0,\n",
                "        \"num_rays\": 3,\n",
                "        \"wavelength\": wave,\n",
                "        \"distribution\": \"gaussian_quad\",\n",
                "    }\n",
                "    problem.add_operand(\n",
                "        operand_type=\"OPD_difference\",\n",
                "        target=0,\n",
                "        weight=1,\n",
                "        input_data=input_data,\n",
                "    )"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Define variables - let first radius of curvature vary (the second surface will match this value, but with opposite sign):"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 6,
            "metadata": {},
            "outputs": [],
            "source": [
                "problem.add_variable(lens, \"radius\", surface_number=1)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Check initial merit function value and system properties:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 7,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502   Merit Function Value \u2502   Improvement (%) \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502                3879.67 \u2502                 0 \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n",
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502 Operand Type   \u2502   Target \u2502   Weight \u2502   Value \u2502   Delta \u2502   Contribution (%) \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502 OPD difference \u2502        0 \u2502        1 \u2502 40.0587 \u2502 40.0587 \u2502            41.3618 \u2502\n",
                        "\u2502  1 \u2502 OPD difference \u2502        0 \u2502        1 \u2502 36.0072 \u2502 36.0072 \u2502            33.4184 \u2502\n",
                        "\u2502  2 \u2502 OPD difference \u2502        0 \u2502        1 \u2502 31.2801 \u2502 31.2801 \u2502            25.2198 \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n",
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502 Variable Type   \u2502   Surface \u2502   Value \u2502 Min. Bound   \u2502 Max. Bound   \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502 radius          \u2502         1 \u2502      50 \u2502              \u2502              \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n"
                    ]
                }
            ],
            "source": [
                "problem.info()"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Define optimizer:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 8,
            "metadata": {},
            "outputs": [],
            "source": [
                "optimizer = optimization.OptimizerGeneric(problem)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Run optimization:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 9,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "  message: CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH\n",
                            "  success: True\n",
                            "   status: 0\n",
                            "      fun: 7.40021876171174\n",
                            "        x: [-6.131e-01]\n",
                            "      nit: 9\n",
                            "      jac: [ 2.496e+02]\n",
                            "     nfev: 58\n",
                            "     njev: 29\n",
                            " hess_inv: <1x1 LbfgsInvHessProduct with dtype=float64>"
                        ]
                    },
                    "execution_count": 9,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "optimizer.optimize()"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Print merit function value and system properties after optimization:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 10,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502   Merit Function Value \u2502   Improvement (%) \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502                7.40022 \u2502           99.8093 \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n",
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502 Operand Type   \u2502   Target \u2502   Weight \u2502    Value \u2502    Delta \u2502   Contribution (%) \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502 OPD difference \u2502        0 \u2502        1 \u2502 1.81071  \u2502 1.81071  \u2502            44.3049 \u2502\n",
                        "\u2502  1 \u2502 OPD difference \u2502        0 \u2502        1 \u2502 0.899126 \u2502 0.899126 \u2502            10.9244 \u2502\n",
                        "\u2502  2 \u2502 OPD difference \u2502        0 \u2502        1 \u2502 1.8202   \u2502 1.8202   \u2502            44.7707 \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n",
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502 Variable Type   \u2502   Surface \u2502   Value \u2502 Min. Bound   \u2502 Max. Bound   \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502 radius          \u2502         1 \u2502 38.6889 \u2502              \u2502              \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n"
                    ]
                }
            ],
            "source": [
                "problem.info()"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Draw final lens:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 11,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "image/png": 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",
                        "text/plain": [
                            "<Figure size 1000x400 with 1 Axes>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "lens.draw(num_rays=5)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Confirm that the radii of curvature are equal and opposite:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 12,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "\u2552\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2564\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2555\n",
                        "\u2502    \u2502 Type            \u2502   Radius \u2502   Thickness \u2502 Material   \u2502   Conic \u2502   Semi-aperture \u2502\n",
                        "\u255e\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u256a\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2561\n",
                        "\u2502  0 \u2502 Planar          \u2502 inf      \u2502         inf \u2502 Air        \u2502       0 \u2502        5        \u2502\n",
                        "\u2502  1 \u2502 Stop - Standard \u2502  38.6889 \u2502           3 \u2502 SK16       \u2502       0 \u2502        5        \u2502\n",
                        "\u2502  2 \u2502 Standard        \u2502 -38.6889 \u2502          30 \u2502 Air        \u2502       0 \u2502        4.85123  \u2502\n",
                        "\u2502  3 \u2502 Planar          \u2502 inf      \u2502         nan \u2502 Air        \u2502       0 \u2502        0.095248 \u2502\n",
                        "\u2558\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2567\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u255b\n"
                    ]
                }
            ],
            "source": [
                "lens.info()"
            ]
        }
    ],
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