دفترچهٔ فصل ۵: تمرکز نمونه‌برداری‌شده و فاز هندسی

from pathlib import Path
from dataclasses import replace,asdict
import sys,json,csv,hashlib,platform,html
import numpy as np
import scipy,matplotlib
import matplotlib.pyplot as plt
from IPython.display import display,HTML
ROOT=next((p for p in [Path.cwd(),*list(Path.cwd().parents)[:3]] if (p/'simulations/metasurface_synthesis.py').is_file()),None)
if ROOT is None:raise FileNotFoundError('Extract the Chapter 5 source bundle, preserving simulations/, data/, and notebooks/.')
sys.path.insert(0,str(ROOT/'simulations'))
from unit_cell_te import Cell,solve_cell,sample_ey,phase_error_deg
from metasurface_synthesis import solve_supercell,staircase_coefficients,select_widths,angular_spectrum_1d,rotated_jones,line_fwhm,SOLVER_VERSION
DATA=ROOT/'data/chapter-05';DATA.mkdir(parents=True,exist_ok=True)
def table(headers,rows):
    def escape(x):return html.escape(f'{x:.7g}' if isinstance(x,(float,np.floating)) else str(x))
    display(HTML('<table><thead><tr>'+''.join('<th>'+escape(x)+'</th>' for x in headers)+'</tr></thead><tbody>'+''.join('<tr>'+''.join('<td>'+escape(x)+'</td>' for x in row)+'</tr>' for row in rows)+'</tbody></table>'))
plt.rcParams.update({'figure.dpi':125,'font.size':10,'axes.spines.top':False,'axes.spines.right':False})
print('Python',platform.python_version(),'NumPy',np.__version__,'SciPy',scipy.__version__,'Matplotlib',matplotlib.__version__)

# Editable scalar aperture parameters.
WAVELENGTH_UM=1.55;OUTPUT_INDEX=1.5;FOCAL_DISTANCE_UM=30.
PITCH_UM=.65;N_APERTURE_CELLS=48;APERTURE_UM=PITCH_UM*N_APERTURE_CELLS
DX_UM=.10;N_FFT=2048;FOCUS_WINDOW_HALF_UM=1.;FOCAL_COMPARE_HALF_UM=4.
if not isinstance(N_APERTURE_CELLS,(int,np.integer)) or N_APERTURE_CELLS<2:
    raise ValueError('Use at least two whole aperture cells.')
if not all(np.isfinite(v) and v>0 for v in [WAVELENGTH_UM,OUTPUT_INDEX,FOCAL_DISTANCE_UM,PITCH_UM]):
    raise ValueError('Wavelength, output index, focal distance, and pitch must be finite and positive.')
REFERENCE=Cell(period_um=PITCH_UM,wavelength_um=WAVELENGTH_UM,n_substrate=OUTPUT_INDEX)
REFERENCE.validate()

def require_matching_library(parameters, reference, phase_references, label):
    """Reject reuse outside the saved TE library's experiment and face references."""
    expected=asdict(reference)
    keys=('period_um','wavelength_um','n_ridge','n_gap','n_incident','n_substrate','angle_deg')
    mismatches=[]
    for key in keys:
        actual=parameters.get(key) if isinstance(parameters,dict) else None
        if actual is None or not np.isfinite(actual) or not np.isclose(actual,expected[key],rtol=1e-12,atol=1e-12):
            mismatches.append(f'{key}: saved {actual!r}, requested {expected[key]!r}')
    if phase_references.get('r')!='z=0' or phase_references.get('t')!='z=height':
        mismatches.append('reflection/transmission must be referenced to z=0/z=height')
    if mismatches:
        raise ValueError(f"{label} is incompatible with this experiment: "+'; '.join(mismatches)+
                         '. Generate a matching library before changing its pitch, wavelength, materials, or incidence.')

data=json.loads((DATA/'beam-steering.json').read_text(encoding='utf-8'))
library_metadata=data['metadata']
require_matching_library(library_metadata.get('library_parameters',{}),REFERENCE,
                         library_metadata.get('library_phase_references',{}),'Saved Chapter 5 library')
if (library_metadata.get('time_convention')!='exp(-i omega t)' or
        library_metadata.get('spatial_convention')!='exp(i k.r)'):
    raise ValueError('Saved Chapter 5 library uses incompatible phasor conventions.')
rows=data['extended_library']
print('Aperture width',APERTURE_UM,'um; nominal line-focus distance',FOCAL_DISTANCE_UM,'um')

k=2*np.pi*OUTPUT_INDEX/WAVELENGTH_UM
cell_centers=(np.arange(N_APERTURE_CELLS)+.5)*PITCH_UM-APERTURE_UM/2
phase_cells=-k*(np.sqrt(cell_centers**2+FOCAL_DISTANCE_UM**2)-FOCAL_DISTANCE_UM)
choices=[]
for offset in np.linspace(-np.pi,np.pi,361,endpoint=False):
    selected,error=select_widths(rows,phase_cells,.8,float(offset))
    choices.append((float(np.mean(error**2)),float(offset),selected))
score,phase_offset,selected=min(choices,key=lambda v:v[0])
local_t=np.array([complex(r['t_real'],r['t_imag']) for r in selected])
def validate_focus_grid(dx,n):
    if not np.isfinite(dx) or dx<=0 or not isinstance(n,(int,np.integer)) or n<16 or n%2:
        raise ValueError('Use positive dx and an even integer FFT length of at least 16.')
    half_window=n*dx/2
    if dx>=WAVELENGTH_UM/(2*OUTPUT_INDEX):
        raise ValueError('The grid must resolve the propagating angular-spectrum band.')
    if APERTURE_UM/2+4*dx>=half_window:
        raise ValueError('The physical aperture must fit inside every padded FFT window.')
    if not 0<FOCUS_WINDOW_HALF_UM<FOCAL_COMPARE_HALF_UM<half_window-dx:
        raise ValueError('The fixed integration and comparison regions must fit in every grid without extrapolation.')

def aperture_fields(dx,n):
    validate_focus_grid(dx,n)
    x=(np.arange(n)-n//2)*dx
    gate=abs(x)<APERTURE_UM/2
    bins=np.clip(np.floor((x+APERTURE_UM/2)/PITCH_UM).astype(int),0,N_APERTURE_CELLS-1)
    continuous=np.where(gate,np.exp(-1j*k*(np.sqrt(x*x+FOCAL_DISTANCE_UM**2)-FOCAL_DISTANCE_UM)),0)
    sampled=np.where(gate,np.exp(1j*phase_cells[bins]),0)
    local=np.where(gate,local_t[bins],0)
    return x,dict(continuous=continuous,sampled=sampled,local_library=local)

def fixed_window_integral(x,intensity,half_width):
    """Trapezoidal integral with interpolated endpoints at the same physical +/- width."""
    if -half_width<x[0] or half_width>x[-1]:
        raise ValueError('Requested physical integration interval lies outside the FFT grid.')
    interior=(x>-half_width)&(x<half_width)
    coordinates=np.r_[-half_width,x[interior],half_width]
    values=np.r_[np.interp(-half_width,x,intensity),intensity[interior],np.interp(half_width,x,intensity)]
    return float(np.sum((values[1:]+values[:-1])*np.diff(coordinates)/2))

def focus_metrics(x,field):
    intensity=np.abs(field)**2
    peak=int(np.argmax(intensity))
    fwhm=float(line_fwhm(x,intensity))
    if not np.all(np.isfinite(intensity)) or intensity[peak]<=0 or not np.isfinite(fwhm):
        raise ValueError('Finite, nonzero focus and resolved half-maximum crossings are required.')
    if abs(x[peak])>=FOCAL_COMPARE_HALF_UM or fwhm>=2*FOCAL_COMPARE_HALF_UM:
        raise ValueError('The focal comparison interval does not contain the intended central focus.')
    window_integral=fixed_window_integral(x,intensity,FOCUS_WINDOW_HALF_UM)
    total_integral=float((x[1]-x[0])*np.sum(intensity))  # Periodic discrete norm, not a power-flux claim.
    return dict(fwhm_um=fwhm,peak_intensity=float(intensity[peak]),
                relative_peak=float(intensity[peak]/ideal_peak),
                fixed_window_line_intensity=window_integral,
                line_intensity_per_aperture=float(window_integral/APERTURE_UM),
                total_line_intensity=total_integral,
                line_intensity_fraction=float(window_integral/total_integral))

x,apertures=aperture_fields(DX_UM,N_FFT)
focus={name:angular_spectrum_1d(field,DX_UM,WAVELENGTH_UM,FOCAL_DISTANCE_UM,OUTPUT_INDEX) for name,field in apertures.items()}
# Freeze ONE peak normalization for every aperture model, window, and grid.
ideal_peak=float(np.max(abs(focus['continuous'])**2))
metrics=[dict(model=name,**focus_metrics(x,field)) for name,field in focus.items()]
table(['Scalar aperture model','FWHM (um)','Peak / fixed ideal peak','Fixed-window line intensity / aperture width','Fixed-window / total modeled line intensity'],
      [[r[key] for key in ['model','fwhm_um','relative_peak','line_intensity_per_aperture','line_intensity_fraction']] for r in metrics])
print('All fixed-window integrals use the same physical +/-',FOCUS_WINDOW_HALF_UM,'um boundaries, with interpolated endpoints.')
fig,axes=plt.subplots(1,2,figsize=(11,4),layout='constrained')
for name,field in apertures.items():axes[0].plot(x,np.angle(field),label=name)
axes[0].set(xlim=(-APERTURE_UM/2,APERTURE_UM/2),xlabel='x (um)',ylabel='Wrapped aperture phase (rad)',title='Continuous, sampled, and library fields')
for name,field in focus.items():axes[1].plot(x,abs(field)**2/ideal_peak,label=name)
axes[1].set(xlim=(-4,4),xlabel='x at z=f (um)',ylabel='Squared field / ideal peak',title='Scalar line-focus profile')
for ax in axes:ax.grid(alpha=.2);ax.legend()
plt.show()

# Window changes at fixed dx; grid changes at fixed physical window.
# Every case regenerates all THREE aperture fields with the same selected cells.
study_specs=[('window',[(DX_UM/2,N_FFT),(DX_UM/2,2*N_FFT),(DX_UM/2,4*N_FFT)]),
             ('grid',[(DX_UM,2*N_FFT),(DX_UM/2,4*N_FFT),(DX_UM/4,8*N_FFT),(DX_UM/8,16*N_FFT)])]
comparison_x=np.linspace(-FOCAL_COMPARE_HALF_UM,FOCAL_COMPARE_HALF_UM,1601)
propagation_cache={}
def scalar_case(dx,n):
    key=(float(dx),int(n))
    if key not in propagation_cache:
        xr,fields=aperture_fields(dx,n)
        if comparison_x[0]<xr[0] or comparison_x[-1]>xr[-1]:
            raise ValueError('Do not extrapolate a focal profile beyond its physical FFT domain.')
        cases={}
        for name,field in fields.items():
            propagated=angular_spectrum_1d(field,dx,WAVELENGTH_UM,FOCAL_DISTANCE_UM,OUTPUT_INDEX)
            cases[name]=dict(metrics=focus_metrics(xr,propagated),
                             profile=np.interp(comparison_x,xr,np.abs(propagated)**2))
        propagation_cache[key]=cases
    return propagation_cache[key]

refinements=[];refinement_checks=[]
refinement_limits={
    'window':dict(profile_difference_over_fixed_peak=.02,relative_fwhm_change=.02,
                  peak_change_over_fixed_peak=.02,fixed_window_integral_change_over_aperture=.01,
                  line_concentration_change=.02),
    'grid':dict(profile_difference_over_fixed_peak=.03,relative_fwhm_change=.02,
                peak_change_over_fixed_peak=.03,fixed_window_integral_change_over_aperture=.02,
                line_concentration_change=.02)}
for study,specs in study_specs:
    fine_dx,fine_n=specs[-1]
    reference_cases=scalar_case(fine_dx,fine_n)
    for dx,n in specs:
        cases=scalar_case(dx,n)
        for name,case in cases.items():
            error=float(np.max(np.abs(case['profile']-reference_cases[name]['profile']))/ideal_peak)
            refinements.append(dict(study=study,model=name,dx_um=float(dx),fft_samples=int(n),
                                    window_um=float(dx*n),**case['metrics'],
                                    roi_peak_normalized_difference=error))
    coarse_dx,coarse_n=specs[-2]
    previous_cases=scalar_case(coarse_dx,coarse_n)
    for name,fine in reference_cases.items():
        previous=previous_cases[name];a=previous['metrics'];b=fine['metrics']
        changes=dict(profile_difference_over_fixed_peak=float(np.max(np.abs(previous['profile']-fine['profile']))/ideal_peak),
                     relative_fwhm_change=float(abs(a['fwhm_um']-b['fwhm_um'])/b['fwhm_um']),
                     peak_change_over_fixed_peak=float(abs(a['peak_intensity']-b['peak_intensity'])/ideal_peak),
                     fixed_window_integral_change_over_aperture=float(abs(a['line_intensity_per_aperture']-b['line_intensity_per_aperture'])),
                     line_concentration_change=float(abs(a['line_intensity_fraction']-b['line_intensity_fraction'])))
        passed=all(value<refinement_limits[study][key] for key,value in changes.items())
        refinement_checks.append(dict(study=study,model=name,
            compared_dx_um=[float(coarse_dx),float(fine_dx)],
            compared_windows_um=[float(coarse_dx*coarse_n),float(fine_dx*fine_n)],
            **changes,limits=refinement_limits[study],passed=bool(passed)))

table(['Study','Aperture','dx (um)','Window (um)','FWHM (um)','Peak / fixed ideal peak','Fixed-window integral / aperture','Profile difference / fixed ideal peak'],
      [[r[key] for key in ['study','model','dx_um','window_um','fwhm_um','relative_peak','line_intensity_per_aperture','roi_peak_normalized_difference']] for r in refinements])
table(['Last independent refinement','Aperture','Relative FWHM change','Peak change / fixed peak','Window-integral change / aperture','Concentration change','Pass'],
      [[r[key] for key in ['study','model','relative_fwhm_change','peak_change_over_fixed_peak','fixed_window_integral_change_over_aperture','line_concentration_change','passed']] for r in refinement_checks])
assert {(r['study'],r['model']) for r in refinement_checks}=={(s,m) for s in ['window','grid'] for m in apertures}
assert all(r['passed'] for r in refinement_checks), 'A scalar aperture refinement failed; inspect its recorded changes and retain the failed evidence.'
print('Separate window/grid checks passed for continuous, sampled, and local-library apertures under the recorded demonstration tolerances.')
mode=3;plane=np.exp(2j*np.pi*mode*np.arange(N_FFT)/N_FFT)
kx_mode=2*np.pi*mode/(N_FFT*DX_UM)
exact=plane*np.exp(1j*np.sqrt(k*k-kx_mode*kx_mode)*FOCAL_DISTANCE_UM)
propagated=angular_spectrum_1d(plane,DX_UM,WAVELENGTH_UM,FOCAL_DISTANCE_UM,OUTPUT_INDEX)
assert np.max(abs(propagated-exact))<1e-10
spectrum=np.zeros(N_FFT,complex);spectrum[[0,2,7,-3]]=[1,2j,.4,1-.2j]
test=np.fft.ifft(spectrum);after=angular_spectrum_1d(test,DX_UM,WAVELENGTH_UM,FOCAL_DISTANCE_UM,OUTPUT_INDEX)
assert abs(np.linalg.norm(after)-np.linalg.norm(test))<1e-12
z_scan=np.linspace(20,40,81)
axial=np.array([abs(angular_spectrum_1d(apertures['continuous'],DX_UM,WAVELENGTH_UM,float(z),OUTPUT_INDEX)[N_FFT//2])**2 for z in z_scan])
print('Nominal focus:',FOCAL_DISTANCE_UM,'um; sampled on-axis maximum:',z_scan[np.argmax(axial)],'um (finite-aperture focal shift is possible).')
fig,ax=plt.subplots(figsize=(7,3.4),layout='constrained');ax.plot(z_scan,axial/ideal_peak);ax.axvline(FOCAL_DISTANCE_UM,color='gray',ls='--');ax.set(xlabel='z (um)',ylabel='On-axis squared field / ideal peak at f',title='Scalar finite-aperture axial response');ax.grid(alpha=.2);plt.show()

angles=np.linspace(0,np.pi,181);retardances=np.linspace(0,np.pi,181)
phase_curves=[];max_jones_error=0.
for sigma in [1,-1]:
    incident=np.array([1,1j*sigma])/np.sqrt(2);converted=np.array([1,-1j*sigma])/np.sqrt(2)
    amplitude=[]
    for theta in angles:
        J=rotated_jones(1.,-1.,float(theta));a=np.vdot(converted,J@incident)
        max_jones_error=max(max_jones_error,float(abs(a-np.exp(2j*sigma*theta))),float(np.linalg.norm(J.conj().T@J-np.eye(2))))
        amplitude.append(a)
    phase_curves.append(np.unwrap(np.angle(amplitude)))
converted_power=[]
incident=np.array([1,1j])/np.sqrt(2);opposite=np.array([1,-1j])/np.sqrt(2)
for delta in retardances:
    J=rotated_jones(np.exp(.5j*delta),np.exp(-.5j*delta),.37)
    converted_power.append(abs(np.vdot(opposite,J@incident))**2)
assert max_jones_error<1e-12
assert max(abs(np.array(converted_power)-np.sin(retardances/2)**2))<1e-12
fig,axes=plt.subplots(1,2,figsize=(10,3.6),layout='constrained')
for sigma,phase in zip([1,-1],phase_curves):axes[0].plot(np.rad2deg(angles),np.rad2deg(phase),label=f'sigma={sigma:+d}')
axes[0].set(xlabel='Element rotation (deg)',ylabel='Unwrapped converted phase (deg)',title='Ideal half-wave geometric phase');axes[0].legend()
axes[1].plot(np.rad2deg(retardances),converted_power);axes[1].set(xlabel='Retardance (deg)',ylabel='Converted fraction',title='Retardance controls conversion')
for ax in axes:ax.grid(alpha=.2)
plt.show()

summary=dict(model='1D scalar angular spectrum plus separate analytical Jones matrices',aperture_um=APERTURE_UM,focal_distance_um=FOCAL_DISTANCE_UM,wavelength_um=WAVELENGTH_UM,output_index=OUTPUT_INDEX,dx_um=DX_UM,fft_samples=N_FFT,window_um=DX_UM*N_FFT,focus_window_half_um=FOCUS_WINDOW_HALF_UM,metrics=metrics,refinements=refinements,refinement_checks=refinement_checks,focus_comparison_half_um=FOCAL_COMPARE_HALF_UM,metric_normalization=dict(peak='one frozen baseline continuous-aperture peak',fixed_window='same physical endpoints integrated with interpolation',line_intensity='fixed-window integral divided by physical aperture width',concentration='fixed-window integral divided by each model total line intensity',scope='scalar squared-field metrics, not electromagnetic focusing efficiency'),axial_peak_um=float(z_scan[np.argmax(axial)]),max_jones_error=max_jones_error,library_phase_rms_deg=float(np.rad2deg(np.sqrt(score))),source_sha256=hashlib.sha256((ROOT/'simulations/metasurface_synthesis.py').read_bytes()).hexdigest(),limitations=['Cylindrical scalar focus, not a full-wave metalens','Library aperture uses local periodicity','Jones matrices are ideal analytical elements, not the TE ridge','No physical focusing efficiency is reported'])
(DATA/'wavefront-models.json').write_text(json.dumps(summary,indent=2)+'\n',encoding='utf-8')
print('Saved wavefront-models.json; model checks passed.')