TMM

Backend lightsmith-tmm (method tmm, module lightsmith.backends.tmm). The transfer-matrix method for layer stacks. It handles coherent and incoherent layers, is exact, differentiable, and runs on the CPU or a GPU.

Method

• Coherent layers. A stable Airy recursion of r from the substrate upwards. It is equivalent to 2 × 2 transfer matrices, but every phase factor is exp(i k_z d) with Im k_z ≥ 0, so thick absorbing or evanescent layers never overflow. The branch of k_z handles both absorption and total internal reflection.
• Incoherent layers (coherent=False). An intensity cascade over coherent groups (Katsidis & Siapkas 2002).
• Amplitude conventions. Amplitudes follow Byrnes' tmm package. For s polarisation, r and t are ratios of E_y. For p polarisation, r is the ratio of tangential H, so r_p = −r_s at normal incidence, and t is the ratio of the full E field. Amplitudes are omitted when any layer is incoherent.
• Gradients. lightsmith.backends.tmm.tmm_tensors(problem, device, thicknesses=, eps=) is differentiable with respect to thicknesses and permittivities.
• Thickness optimisation. optimize_thicknesses(problem, objective, ...) runs Adam over layer thicknesses. It needs PyTorch.

Options

options["tmm"] or options["lightsmith-tmm"].

Option	Default	Meaning
precision	"auto"	"auto" means double, because interference needs the digits

Validation

All tests are in tests/test_tmm.py.

Test	Reference	Tolerance
test_fresnel_single_interface (0–89°)	Fresnel equations	amplitudes 1e-13
test_brewster_angle	R_p = 0	< 1e-28
test_total_internal_reflection	R = 1, evanescent Fresnel amplitudes	1e-14, 1e-13
test_frustrated_tir_and_energy_conservation: random 12-layer stack	R + T = 1 (lossless); R + T + A = 1	1e-12; 1e-14
test_quarter_wave_ar_coating	R = 0 at λ₀; single-film Airy formula	< 1e-28; 1e-14
test_bragg_mirror_stop_band: 10 pairs	analytic admittance at the centre	1e-12
test_incoherent_thick_slab_matches_closed_form: 1 mm slab	closed form	1e-10
test_incoherent_equals_phase_averaged_coherent	coherent result averaged over thickness	1e-6
test_tmm_matches_rcwa_for_unpatterned_stack	RCWA	R, T, A and amplitudes 1e-12
test_gradients_match_finite_differences	central differences	1e-6 relative
test_optimize_ar_coating_finds_quarter_wave	λ/(4n)	0.5 nm
test_tmm_gpu_matches_cpu (gpu)	CPU	1e-12

Limits

• Layer stacks only. For patterned layers, use RCWA.
• With an absorbing superstrate, R and T are normalised to the incident Poynting flux at the first interface, and the result carries a warning saying so.