feat[adjoint]: Add conductivity gradient for CustomMedium

[yaugenst-flex]

Summary

This PR introduces the capability to compute gradients with respect to the conductivity of a CustomMedium.

Implementation Details

• Generalizes the _derivative_field_cmp helper method with a new component parameter that can be set to 'real', 'imag', or 'complex'.
• Implements the Vector-Jacobian Product (VJP) for conductivity (component='imag') by scaling the imaginary part of the complex permittivity derivative by 1 / (omega * epsilon_0).
• Updates the _compute_derivatives dispatcher to handle the new "conductivity" parameter path and invoke the generalized helper method.
• Adds test_custom_medium_conductivity_only_gradient to validate the gradient calculation for a CustomMedium with variable conductivity and constant permittivity.
• Adds a test case structure, custom_med_with_conductivity, that contains both variable permittivity and conductivity.

Greptile Summary

Implements conductivity gradient calculations for CustomMedium by extending the permittivity gradient framework to handle real, imaginary, and complex components, enabling material conductivity optimization.

• Added component parameter to _derivative_field_cmp to support 'real', 'imag', and 'complex' gradient types
• Implemented conductivity Vector-Jacobian Product (VJP) by scaling imaginary permittivity derivative by 1/(omega * epsilon_0)
• Added new test case custom_med_with_conductivity to validate combined permittivity and conductivity gradients
• Refactored test code in test_autograd.py to improve maintainability with shared coordinate dictionaries

[greptile-apps]

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[github-actions]

Diff Coverage

Diff: origin/develop...HEAD, staged and unstaged changes

• tidy3d/components/medium.py (96.0%): Missing lines 3013

Summary

• Total: 25 lines
• Missing: 1 line
• Coverage: 96%

tidy3d/components/medium.py

  3009             # for complex permittivity in eps_dataset, return the full complex derivative
  3010             E_der_dim_interp = E_der_dim_interp_complex
  3011         elif component == "imag":
  3012             # pure imaginary component (no conductivity conversion)
! 3013             E_der_dim_interp = E_der_dim_interp_complex.imag
  3014         else:
  3015             E_der_dim_interp = E_der_dim_interp_complex.real
  3016 
  3017         E_der_dim_interp = E_der_dim_interp.sum("f")

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[greptile-apps]

LGTM

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[tylerflex]

Looks good to me!

a really minor comment is that I notice that we sum("f") the E_der_dim_interp in each of the handlings of component. could be slightly more robust if w just sum that after constructing E_der_dim_interp.

[yaugenst-flex]

a really minor comment is that I notice that we sum("f") the E_der_dim_interp in each of the handlings of component. could be slightly more robust if w just sum that after constructing E_der_dim_interp.

good point, fixed

[groberts-flex]

looks good! not sure this would be in the scope of the PR or not but would love to have a numerical test for conductivity as well. I'm happy to take this on at some point, think I could modify the field and/or mode numerical tests we have to check conductivity (right now, they use custom medium for permittivity, but should be quick change to also test conductivity.

[daquinteroflex]

Hello! Is the plan to merge this by 2.9.0 ? We can add the .1 tag possbly if not?

[groberts-flex]

overall looks good, couple of small comments.

did you want me to send you the fixed up numerical test to include here as well? or do you want to save that for a separate PR?