感谢收件箱上的人们帮助我。

1. 我不需要完整的球体，只有一半的球体包含与横截面相同的能量。

2. 我可以使用我的点网格并想象它们之间的一个小区域。我设置一个点并根据最近的邻居创建一个正方形。

   2.1 在球坐标系中，面积元素为。为了将此表达式转换为离散情况，我们使用和以及是最近邻的平均值$dA = r^2sin(\theta)d\theta d\phi$$d\theta = \theta[i+1]-\theta[i]$$d\phi= \phi[i+1]-\phi[i]$$\theta$$\theta = (\theta[i+1]+\theta[i])/2$

   2.2 为简单起见，截面积可以只是一个正方形网格点

3. 此过程仅适用于远场，也就是说，如果与高斯腰部相比，球体的半径应该很大（我在示例中使用了错误的值选择）

对于所有感兴趣的人，这里有完整的工作代码：

function E_gaussian(x,y,z, k₀, w₀, E₀)
    first_term = exp(im*k₀*z)/(1 + 2*im*z/(k₀*w₀^2))
    second_term = exp(  - ((x^2 + y^2)/w₀^2)*(1/(1 + 2*im*z/(k₀*w₀^2)))  )
    E = E₀*first_term*second_term
    return E
end

function createGaussianSection(nPoints, w₀ )
    x_range = range(-w₀, stop=w₀, length=nPoints)
    y_range = range(-w₀, stop=w₀, length=nPoints)
    screen = zeros(nPoints, nPoints, 3)
    screen[:,:,1] = [x for x in x_range, y in y_range]
    screen[:,:,2] = [y for x in x_range, y in y_range]
    screen[:,:,3] .= 0
    return screen, x_range, y_range
end



function createSphere(Radius, θNintervals, φNintervals, w₀, k₀)
    θ_range = range(0, stop=π/2, length=θNintervals)
    φ_range = range(0, stop=2π, length=φNintervals)

    xₛ = Radius.*[cos(φ)*sin(θ) for θ in θ_range, φ in φ_range]
    yₛ = Radius.*[sin(φ)*sin(θ) for θ in θ_range, φ in φ_range]
    zₛ = Radius.*[cos(θ) for θ in θ_range, φ in φ_range]

    screen = zeros(size(xₛ,1),size(xₛ,2),3)
    screen[:,:,1] = xₛ
    screen[:,:,2] = yₛ
    screen[:,:,3] = zₛ

    return screen, θ_range, φ_range
end


function getE(screen, k₀,  w₀, E₀)
    nΘ = size(screen,1)
    nΦ = size(screen,2)
    E_sphere = zeros(Complex{Float64},nΘ, nΦ )

    for θ in 1:nΘ, φ in 1:nΦ
        xₛ = screen[θ,φ,1]
        yₛ = screen[θ,φ,2]
        zₛ = screen[θ,φ,3]
        E_sphere[θ,φ] = E_gaussian(xₛ, yₛ, zₛ, k₀,  w₀, E₀)
    end
    return E_sphere
end

k₀ =  1
w₀ = 30/k₀
E₀ = 1


screenGaussin, x_range, y_range = createGaussianSection(150, 2*w₀)
E_section = getE(screenGaussin, k₀,  w₀, E₀)
Intensity_section = real.( conj.(E_section).*E_section)

I_section = 0
for x = 1:(length(x_range)-1), y = 1:(length(y_range)-1)
    global I_section
    E_mean = (E_section[x, y] + E_section[x+1, y] + E_section[x, y+1]+E_section[x+1, y+1])/4
    ΔA = (x_range[x+1]-x_range[x])*(y_range[y+1]-y_range[y])
    I_section += real(conj(E_mean)*E_mean)*ΔA
end
println("Total Power Section: ", round(I_section,digits=3))


Radius = 1000/k₀
θNintervals = φNintervals = 200
screen, θ_range, φ_range = createSphere(Radius, θNintervals, φNintervals, w₀, k₀)
E_sphere = getE(screen, k₀,  w₀, E₀)
Intensity_sphere = real.( conj.(E_sphere).*E_sphere)

I_sphere = 0
for θ = 1:(θNintervals-1), φ = 1:(φNintervals-1)
    global I_sphere
    E_medio = (E_sphere[θ, φ] + E_sphere[θ+1, φ] + E_sphere[θ, φ+1]+E_sphere[θ+1, φ+1])/4
    ΔA = (Radius^2)*sin(   (θ_range[θ]+θ_range[θ+1])/2  )*(θ_range[θ+1]-θ_range[θ])*(φ_range[φ+1]-φ_range[φ])
    I_sphere += real(conj(E_medio)*E_medio)*ΔA
end
println("Total Power Sphere: ", round(I_sphere,digits=3))

println("Relation:",round(I_section/I_sphere,digits=3))

结果：

Total Power Section: 1413.028
Total Power Sphere: 1413.567
Relation:1.0