How to use mgrid to interpolate between a rectangle and a circle

Question

I am trying to create a 3D surface that has a 1/4 rectangle for the exterior and 1/4 circle for the interior. I had help before to create the 3D surface with an ellipse as an exterior but I cannot do this for a rectangle for some reason. I have done the math by hand which makes sense, but my code does not. I would greatly appreciate any help with this.

import numpy as np
import pyvista as pv

# parameters for the waveguide
# diameter of the inner circle
waveguide_throat = 30
# axes of the outer ellipse
ellipse_x = 250
ellipse_y = 170
# shape parameters for the z profile
depth_factor = 4
angle_factor = 40
# number of grid points in radial and angular direction
array_length = 100

phase_plug = 0
phase_plug_dia = 20
plug_offset = 5
dome_dia = 28
# theta is angle where x and y intersect
theta = np.arctan(ellipse_x / ellipse_y)
# chi is for x direction and lhi is for y direction
chi = np.linspace(0, theta, 100)
lhi = np.linspace(theta, np.pi/2, 100)

# mgrid to create structured grid
r, phi = np.mgrid[0:1:array_length*1j, 0:np.pi/2:array_length*1j]

# Rectangle exterior, circle interior 
x = (ellipse_y * np.tan(chi)) * r + ((waveguide_throat / 2 * (1 - r)) * np.cos(phi))
y = (ellipse_x / np.tan(lhi)) * r + ((waveguide_throat / 2 * (1 - r)) * np.sin(phi))


# compute z profile
angle_factor = angle_factor / 10000
z = (ellipse_x / 2 * r / angle_factor) ** (1 / depth_factor)

plotter = pv.Plotter()

waveguide_mesh = pv.StructuredGrid(x, y, z)
plotter.add_mesh(waveguide_mesh)
plotter.show()