01. Units, photon budget, optics and receiver algebra

This executable notebook independently derives and checks the design calculations. It is written for inspection as well as execution: the equations are in the markdown cells, the numerical implementation is in modules/design_review.py, and the detailed derivations/source locators are in methods/design_and_diagnostics.md.

Scope: the parameters are explicit hypothetical design inputs. A checked calculation does not turn a toy Gaussian into an N2 spectrum, a dark-only NEP into a device specification, or a photon fluence into a source luminosity function. Run all cells from a kernel with NumPy, Matplotlib and IPython. No network requests or installations are performed. Original result tables are comparison targets, not copied calculation outputs.

from pathlib import Path
import sys, os, json, math, hashlib
import numpy as np
from io import BytesIO
from IPython.display import display, Image, Markdown
candidates = [Path.cwd(), *Path.cwd().parents]
ROOT = next(p for p in candidates if (p/'data/design_results.json').exists() and (p/'modules/design_review.py').exists())
if (ROOT/'runtime/deps').is_dir(): sys.path.insert(0, str(ROOT/'runtime/deps'))
sys.path.insert(0, str(ROOT))
os.environ.setdefault('MPLCONFIGDIR', str(ROOT/'checks/design_mplconfig'))
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from modules import design_review as v
from modules.common import table
plt.rcParams.update({'figure.figsize': (8.2, 4.3), 'font.size': 10, 'axes.grid': True, 'grid.alpha': .22})
def show(fig):
    b=BytesIO(); fig.savefig(b, format='png', dpi=125, bbox_inches='tight')
    display(Image(data=b.getvalue())); plt.close(fig)
design = v.load('design_results.json')
analytic = v.load('analytic_results.json')
print('Independent module:', ROOT/'modules/design_review.py')
print('SHA256:', hashlib.sha256((ROOT/'modules/design_review.py').read_bytes()).hexdigest())

Independent module: D:\Photometer_research_20261022\verification_20261022_r1\modules\design_review.py
SHA256: af4c3056a61716d48c5eb8eec3f1abb67be7dee123d78c6bbc7c6f6cc37322ca

D01. Energy and reference planes

For a vacuum wavelength in metres, $E_\gamma=hc/\lambda$ (joules). A received energy fluence in J m$^{-2}$ divided by this energy gives photon fluence in photons m$^{-2}$. The nominal wavelength is retained from the legacy model; this is not an air-to-vacuum spectral-registration correction.

Keep four planes separate: emitted source; atmospheric output; entrance aperture; and detected counts. A spacecraft fluence already includes real cloud/air delivery. Never multiply it by atmospheric transmission a second time.

table([{'wavelength_nm': w, 'photon_energy_J': v.photon_energy(w*1e-9),
        'photons_per_J': 1/v.photon_energy(w*1e-9)} for w in [337.1,777.4]])
print('One Rayleigh:', v.RAYLEIGH_PHOTONS, 'photons m^-2 sr^-1 s^-1, integrated over the feature')
print('At nominal 337.1 nm:',v.RAYLEIGH_PHOTONS*v.photon_energy(337.1e-9),'W m^-2 sr^-1 per R')

One Rayleigh: 795774715.4594767 photons m^-2 sr^-1 s^-1, integrated over the feature
At nominal 337.1 nm: 4.689301058286399e-10 W m^-2 sr^-1 per R

D02-D03. Ordinary and projected solid angle

For a circular cone, $\Omega=2\pi(1-\cos\theta)$ while $\Omega_p=\int\cos\vartheta\,d\Omega=\pi\sin^2\theta$. A flat aperture viewing uniform radiance uses $A\Omega_p$.

MMIA's quoted field is a square diagonal, not a circular cone diameter. The independent square integral below shows why this distinction affects background and etendue. Steradians are retained in the bookkeeping even though they are dimensionless in SI.

rows=[]
for full in [30.,60.,80.,90.,180.]:
    ordinary,projected=v.circular_angles(math.radians(full/2))
    rows.append({'field':'circular','full_or_diagonal_deg':full,'Omega_sr':ordinary,'projected_Omega_sr':projected})
ordinary,projected=v.square_angles(math.radians(40))
rows.append({'field':'square diagonal','full_or_diagonal_deg':80.,'Omega_sr':ordinary,'projected_Omega_sr':projected})
table(rows)
angles=np.linspace(1,120,150)
fig,ax=plt.subplots()
ax.plot(angles,[v.circular_angles(math.radians(a/2))[0] for a in angles],label='ordinary solid angle')
ax.plot(angles,[v.circular_angles(math.radians(a/2))[1] for a in angles],label='projected angular integral')
ax.scatter([80],[projected],label='square 80° diagonal: projected',color='black')
ax.set(xlabel='Full circular field (degrees)',ylabel='Angular integral (sr)',title='Projection matters for a wide field');ax.legend();show(fig)

D04-D05. A worked received-fluence and background conversion

At nadir, $S=\Phi_{ap}A\eta$. For uniform spectral energy radiance and a flat response, $b_{sky}=L_\lambda\Delta\lambda A\Omega_p\eta/E_\gamma$ and $B=(b_{sky}+d)T$.

The following example assumes a 25 mm entrance, 60 degree circular full field, 20% end-to-end probability, a 4 nm rectangular/equivalent band, and 50 primary dark detections per second. The value $L_\lambda=1.77\times10^{-9}$ is a reference continuum scale, not a polar cloud background measurement. No false-alarm or detection-power criterion is imposed here.

area=v.aperture_area(.025);eta=.2;L=1.77e-9
rate=v.continuum_rate(L,4.,area,math.pi/6,eta,337.1e-9)
table([{'window_us':T*1e6,'sky_primary_s':rate,'dark_primary_s':50.,
    'background_mean_count':(rate+50)*T,'ideal_known_background_RMS_count':math.sqrt((rate+50)*T)}
    for T in [30e-6,80e-6,200e-6,.003]])
print('Input received photon fluence of 1e8 photons/m² ->',v.counts_from_received_fluence(1e8,area,eta),'expected detected photons')
print('Required aperture photons for an explicitly chosen 100 detected photons:',100/eta)
print('That count choice is not a validated detection threshold.')

Input received photon fluence of 1e8 photons/m² -> 9817.477042468105 expected detected photons
Required aperture photons for an explicitly chosen 100 detected photons: 500.0
That count choice is not a validated detection threshold.

D06. Etendue is a necessary constraint

In uniform air-to-air geometrical optics, $G=A\pi\sin^2\theta$. A detector hemisphere gives $A_{det}\ge A\sin^2\theta$; a filter internal half-cone $\beta$ requires $A_{filter}\ge G/(\pi\sin^2\beta)$.

This does not specify a lens, detector numerical aperture, vignetting or a usable 1U layout. The MMIA 25 mm photocathode is not a 25 mm UV entrance. The proposed larger entrance is a new phase-space design.

rows=[]
for full in [30.,60.,80.,90.]:
    g,ad,dd,df=v.etendue_bounds(.025,math.radians(full/2),math.radians(10))
    rows.append({'full_circular_deg':full,'G_m2_sr':g,'detector_min_mm2':ad*1e6,
                 'detector_min_diameter_mm':dd*1e3,'filter_min_diameter_mm':df*1e3})
table(rows)
beta=np.linspace(4,25,100);fig,ax=plt.subplots()
for full in [60,90]:
    ax.plot(beta,[v.etendue_bounds(.025,math.radians(full/2),math.radians(b))[3]*1e3 for b in beta],label=f'{full}° full circular field')
ax.set(xlabel='Allowed internal filter half-cone (degrees)',ylabel='Necessary clear filter diameter (mm)',title='25 mm entrance: ideal air-to-air bounds');ax.legend();show(fig)

D07-D09. Filter walk and a hypothetical Gaussian overlap

The effective-index approximation is $\lambda_c=\lambda_0\sqrt{1-\sin^2\theta/n_{eff}^2}$. Here $n_{eff}=1.8$ is an assumption. A normalized Gaussian photon source and unit-peak Gaussian filter have capture

C={\sigma_f\over\sqrt{\sigma_s^2+\sigma_f^2}}\exp[-\delta^2/(2(\sigma_s^2+\sigma_f^2))],\qquad W_{eq}=\sqrt{2\pi}\sigma_f.

The capture formula is independently compared with 48 direct Gaussian-product quadratures. The relative $C/\sqrt{W_{eq}}$ diagnostic holds the photon-background density per wavelength fixed. It is not a reconstructed N2 band or a flight filter optimum. A real cone, polarization, temperature and source population must be folded with measured transmission.

fig,ax=plt.subplots();angles=np.linspace(0,30,121);ss=v.sigma_from_fwhm(1e-9)
for width in [1,2,4,8]:
    sf=v.sigma_from_fwhm(width*1e-9)
    cap=[v.gaussian_overlap(ss,sf,v.shifted_center(337.1e-9,math.radians(a),1.8)-337.1e-9) for a in angles]
    ax.plot(angles,cap,label=f'filter FWHM {width} nm')
ax.set(xlabel='Single-ray incidence (degrees)',ylabel='Unit-peak photon capture fraction',title='Hypothetical 1 nm Gaussian source; effective-index toy');ax.legend();show(fig)
print('4 nm Gaussian FWHM equivalent width:',v.gaussian_equivalent_band(v.sigma_from_fwhm(4e-9))*1e9,'nm')
q=[]
for sw in [.5,1.,2.]:
 for fw in [1.,2.,4.,8.]:
  for angle in [0.,5.,10.,20.]:
   ss=v.sigma_from_fwhm(sw*1e-9);sf=v.sigma_from_fwhm(fw*1e-9);delta=v.shifted_center(337.1e-9,math.radians(angle),1.8)-337.1e-9
   q.append(abs(v.gaussian_overlap_integral(ss,sf,delta)-v.gaussian_overlap(ss,sf,delta)))
print('Independent quadratures:',len(q),'maximum absolute capture disagreement:',max(q))

4 nm Gaussian FWHM equivalent width: 4.257868077724905 nm
Independent quadratures: 48 maximum absolute capture disagreement: 8.881784197001252e-16

D10-D12. NEP, noise bandwidth and leak rejection

For one-sided PSD, $\sigma_P^2=\int_0^\infty NEP^2(f)|H(f)|^2df$. A first-order electrical low-pass has ENBW $\pi f_c/2$; an exposure average has ENBW $1/(2T)$. These Hz widths are unrelated to filter widths in nanometres.

The ideal dark-primary model gives $NEP_{dark}=E_\gamma\sqrt{2Fd}/\eta$. The following values use assumed $d$, $F$ and $\eta$, not published candidate device specifications. Amplifier noise, sky, radiation, correlations and readout are absent.

Uniform out-of-band blocking requires $OD\ge\log_{10}(R/f)$ for the actual detector-weighted unblocked out/in ratio $R$ and allowed fraction $f$. R is not known from an OD certificate alone.

table([{'wavelength_nm':w,'eta_assumed':.2,'dark_primary_s_assumed':d,'F_assumed':F,
        'dark_only_NEP_W_sqrtHz':v.dark_nep(w*1e-9,.2,d,F)}
        for w in [337.1,777.4] for d in [1.,50.,1000.] for F in [1.,1.5]],max_rows=12)
table([{'electrical_fc_Hz':fc,'one_sided_ENBW_Hz':v.electrical_enbw_first_order(fc)} for fc in [1e4,5e4]])
print('3 ms boxcar ENBW:',v.boxcar_enbw(.003),'Hz')
table([{'unblocked_detector_weighted_ratio':r,'allowed_leak_fraction':.01,'uniform_OD_required':v.blocking_od(r,.01)} for r in [1e2,1e5,1e9]])

3 ms boxcar ENBW: 166.66666666666666 Hz

D13-D14. A pulse plateau, sampling cadence and integration phase

Analog high-pass droop obeys $\tau\ge -T/\ln(1-\delta)$. This step response is different from AC shaping of individual counted photon pulses.

The deterministic sampling example averages a unit ramp over exact exposures, assigns each sample to its midpoint, and interpolates the 10%/90% crossings. At 10 kS/s, samples are 100 microseconds apart; at 50 kS/s, 20 microseconds. A 50 kHz analog signal cannot be assumed unaliased at a 50 kS/s baseband cadence. Noise and unknown source onset make the real problem harder than this calculation.

table([{'plateau_ms':3.,'allowed_droop':f,'tau_min_s':v.hp_droop_timeconstant(.003,f)[0],
      'fc_max_Hz':v.hp_droop_timeconstant(.003,f)[1]} for f in [.01,.05,.1]])
phases=np.linspace(0,.99,100);fig,ax=plt.subplots()
for fs in [1e4,5e4,1e5]:
 ax.plot(phases,[v.sampled_rise(30e-6,fs,p)*1e6 for p in phases],label=f'{fs/1000:g}kS/s')
ax.axhline(30,color='black',ls='--',lw=1,label='true30 µs')
ax.set(xlabel='Onset phase within one exposure',ylabel='Interpolated 10-90 rise (µs)',title='Noiseless exact exposure averages; phase bias is estimator-specific');ax.legend();show(fig)

Full-table independent verification

This cell checks every straight-algebra output in all 1,639 rows, all constants, the 48 selected numerical Gaussian integrations and unit/limiting cases. It does not reuse original code functions. It raises if any declared comparison fails. The JSON comparison receipts also retain hashes, field-appropriate tolerances and explicit scope. Tiny SI quantities have no permissive generic absolute floor. Separate in-memory fault controls must reject wrong photon energies, NEP and capture values; reproduce them with python checks/design_negative_controls.py. Notebook execution uses write=False to avoid overwriting the separate verification receipts.

result,recomputed=v.check_design(write=False)
assert result['status']=='PASS',result['failures']
table([{'table':k,'rows_checked':n} for k,n in result['rows_checked'].items()])
print('Status:',result['status'],'scalar comparisons:',result['comparisons'],'independent integrals:',result['independent_Gaussian_quadratures'])
table([{'hand_or_unit_limit':k,'passes':ok} for k,ok in result['unit_and_limiting_checks'].items()],max_rows=30)

Status: PASS scalar comparisons: 3679 independent integrals: 48

Primary references and limits

• BIPM SI defining constants: exact values of c, h, e and k in the official table.
• MMIA design: instrument optics/photometer architecture, field convention, entrance versus detector sizes; not a prescription for our 25 mm entrance.
• Analog Devices MT-002, pp1,4-8 and figures1,4-7: baseband sampling and anti-aliasing.
• ET Enterprises PMT output circuits: DC/AC output architectures; our single-pole and count-noise examples are explicit derivations.
• NIST wavelength conventions: air/vacuum refractivity conventions; not physical filter-vacuum shift.

The full source applicability, equation derivations, units, original-file/function map and residual measured inputs are in methods/design_and_diagnostics.md and checks/design_method_map.json. No published device NEP, lifetime damage, calibrated band capture or useful rate is established by these assumed scenarios.