Astronomy · How new tools extend what we can know

Improving contrast and angular resolution

Separate spatial resolution from bright-source suppression, then derive and check an ideal two-output interference model with phase and amplitude mismatch.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

Back to the chapter

After reading, you can

Distinguish angular response from contrast and calibrated detection performance.
Derive dark-output leakage for a lossless two-mode combiner.
Compare five exact model cases while preserving normalization, power conservation and observing limits.

Separating sources and revealing faint light are distinct tasks

Angular resolution concerns the distinguishability of source directions through an observing response. Contrast concerns a relative signal or residual brightness, with its normalization stated. A faint companion near a bright star can be difficult even when its separation is large enough for the instrument to transmit spatial information about it.

For a uniform ideal circular pupil, monochromatic scalar diffraction gives the familiar small-angle scale 1.22λ/D. Rayleigh uses that scale in a particular criterion for two incoherent point sources. Source structure, relative brightness, noise, aberrations and estimation method influence the actual inference; one criterion is not a universal information limit for every experiment.

Suppression methods reduce unwanted bright-source light in a region or output used for science. NASA’s Roman coronagraph description combines masks and active wavefront control for that purpose. Its spatial response and companion throughput are different properties from a mere increase in image magnification.

The following original interference model isolates one aspect of suppression. It is not an optical prescription for Roman or HWO, an angular contrast curve, a forecast or a companion-detection calculation. Its deliberately ideal conditions make the roles of phase, amplitude and normalization calculable.

Combine fields before calculating output power

Assume one ideal on-axis point source, monochromatic mutually coherent scalar beams with the same polarization and spatial mode, and an ideal lossless two-mode combiner. Exclude finite stellar size, chromaticity, spatial wavefront errors, backgrounds, detector noise and the response to an off-axis companion.

Choose normalized complex inputs e1 = 1 and e2 = (1 + ε) exp(iδ), with i² = −1. Here ε is fractional field-amplitude mismatch, not fractional intensity mismatch, and 1 + ε is positive. The phase δ is in radians relative to the stipulated ideal cancellation configuration.

Define complementary output fields Ebright = (e1 + e2)/√2 and Edark = (e1 − e2)/√2. Power is the squared modulus of each normalized field in one common arbitrary power unit. These definitions include the factor needed for a unitary, power-conserving transformation.

The total input power is Ptotal = |e1|² + |e2|² = 1 + (1 + ε)². Direct expansion shows Pdark + Pbright = Ptotal. Destructive interference at the dark output redistributes coherent power into the complementary output; it does not destroy the total energy.

Dark over total is not dark over bright

Expand the dark output: Pdark = [1 + (1 + ε)² − 2(1 + ε) cos δ]/2. Using 1 − cos δ = 2 sin²(δ/2), this becomes [ε² + 4(1 + ε) sin²(δ/2)]/2.

Define f = Pdark/Ptotal. The exact fraction is f = [ε² + 4(1 + ε) sin²(δ/2)]/{2[1 + (1 + ε)²]}. With equal amplitudes, ε = 0, it reduces to sin²(δ/2). At perfect phase, δ = 0, a nonzero amplitude mismatch retains f = ε²/{2[1 + (1 + ε)²]}.

A different normalization is the dark-to-bright ratio N = Pdark/Pbright = f/(1 − f), when Pbright is nonzero. The historical NExScI nuller description uses a null-to-peak leakage ratio. It must not be silently equated with the dark-to-total fraction plotted here.

For very small leakage the numerical values can be close, but their denominators remain different. Neither f nor N is a detection probability. No model of an astronomical companion’s angular transmission or its detection statistics has been supplied.

Correcting phase leaves an amplitude-mismatch floor

Case A has ε = 0 and δ = 0: Ptotal = 2, Pdark = 0 and Pbright = 2. Case B keeps equal amplitudes but sets δ = 0.02 rad, giving Pdark ≈ 0.000199993333 and f ≈ 0.000099996667. Its N is approximately 0.000100006667.

Case C instead has ε = 0.02 with δ = 0. Its total power is 1 + 1.02² = 2.0404, dark power is 0.02²/2 = 0.0002 and bright power is 2.0402. The retained fraction is f = 1/10202 ≈ 0.000098019996, despite exact phase cancellation.

Case D combines ε = 0.02 and δ = 0.02 rad. Its dark power is approximately 0.000403993200 and f ≈ 0.000197997059. Case E improves only phase to 0.01 rad, reducing dark power to approximately 0.000250999575 and f to approximately 0.000123014887.

In D and E, total power remains 2.0404 and the bright output holds the complement. Phase improvement helps, but approaching δ = 0 with ε fixed at 0.02 approaches C’s positive amplitude-only leakage rather than A’s perfect null. All five rows use the same power unit and definitions; displayed rounding does not alter the exact conservation identity.

Five original ideal-model cases. Both fractions are dimensionless, δ is in radians and ε is field-amplitude mismatch. Values are rounded to nine decimal places for comparison with the schematic.
Case	ε	δ (rad)	f = dark/total	N = dark/bright
A	0	0	0.000000000	0.000000000
B	0	0.02	0.000099997	0.000100007
C	0.02	0	0.000098020	0.000098030
D	0.02	0.02	0.000197997	0.000198036
E	0.02	0.01	0.000123015	0.000123030

Phase correction and an amplitude-mismatch floor

Scroll sideways to see the full diagram. Its explanation is below.

This is an original analytic teaching schematic, not observations, an instrument forecast, an optical layout or an on-sky companion contrast curve. It assumes one ideal on-axis point source, monochromatic mutually coherent co-polarized scalar beams in one spatial mode and a lossless unitary two-mode combiner. Finite stellar size, wavelength spread, spatial aberrations, backgrounds, detector noise, throughput and off-axis companion response are excluded. The normalized complex input fields are e1=1 and e2=(1+epsilon) exp(i delta), where i²=−1. Epsilon is fractional field-amplitude mismatch, not intensity mismatch; delta is phase in radians relative to the stipulated ideal dark-port cancellation configuration. The output fields are E_bright=(e1+e2)/sqrt(2) and E_dark=(e1−e2)/sqrt(2). Power is the squared modulus in one arbitrary common power unit. All rows use that same unit and the same fraction conventions. The exact input and combined output power is P_total=1+(1+epsilon)²=P_dark+P_bright. The plotted fraction is f=P_dark/(P_dark+P_bright)=[epsilon²+4(1+epsilon)sin²(delta/2)]/{2[1+(1+epsilon)²]}. Its numerator is not the power lost from the two-output system; the complementary output retains the rest. The single shared linear plot has delta from 0 to 0.04 rad and f from 0 to 0.0006, both starting at zero. The solid blue curve with circle case markers uses epsilon=0. The dashed orange curve with diamond case markers uses epsilon=0.02. Both curves use 801 exact analytic sample evaluations at delta=i/20000 rad for i=0 through 800, joined by straight segments. Case markers A–E are placed at their exact model coordinates without displacement. At zero phase, A has f=0; C has f=1/10202, so improving phase alone cannot remove the fixed amplitude mismatch. Historical null leakage N uses a different denominator: N=P_dark/P_bright=f/(1−f). The two fractions are close at small leakage but are not equal. The vertical axis is f, not N, not a host/companion contrast and not an angular-separation response. The displayed tables round powers and fractions to nine decimal places; the exact expressions and more precise numerical evaluations below define the five rows. Five cases: A: epsilon=0, delta=0 rad, P_input=2.000000000000000, P_dark=0.000000000000000, P_bright=2.000000000000000 common power units, f=0.000000000000000, N=0.000000000000000. B: epsilon=0, delta=0.02 rad, P_input=2.000000000000000, P_dark=0.0001999933334222216, P_bright=1.999800006666578 common power units, f=0.00009999666671111079, N=0.0001000066670444641. C: epsilon=0.02, delta=0 rad, P_input=2.040400000000000, P_dark=0.0002000000000000000, P_bright=2.040200000000000 common power units, f=0.00009801999607920016, N=0.00009802960494069210. D: epsilon=0.02, delta=0.02 rad, P_input=2.040400000000000, P_dark=0.0004039932000906660, P_bright=2.039996006799909 common power units, f=0.0001979970594445531, N=0.0001980362700436851. E: epsilon=0.02, delta=0.01 rad, P_input=2.040400000000000, P_dark=0.0002509995750014167, P_bright=2.040149000424999 common power units, f=0.0001230148867875988, N=0.0001230300213117420.

A scalar null does not establish a planet detection

A real system must account for wavelength spread, polarization and spatial response, imperfect alignment, finite stellar extent, stability and backgrounds. A real target may leak light even when the ideal point-source cancellation is good. Calibration must distinguish instrumental contributions from the source-dependent response.

The NExScI account treats null observations and calibrator-based instrumental correction as separate stages. That historical description supplies context, not current performance or a finite-star numerical prediction for this toy. The component calculation here does not replace an end-to-end observing model.

Residual starlight can also have correlated spatial and temporal structure. Mawet and colleagues emphasize the small number of comparison resolution elements close to a bright source. A familiar Gaussian significance label cannot automatically be assigned the same false-positive probability under every such condition.

To claim improved companion detectability, specify angular separation, band, target properties, throughput, calibrated residuals and the decision and recovery procedure. Optical suppression, resolution, calibration and inference work together. The next lesson considers another source of capability: coordinating observing windows and messages across a network.

Worked examples

Calculate perfect-phase leakage with unequal amplitudes

What is f when ε = 0.02 and δ = 0 in this model?

Compute Ptotal = 1 + 1.02² = 2.0404.
At zero phase, Pdark = ε²/2 = 0.0002.
Divide 0.0002 by 2.0404.

f = 1/10202 ≈ 0.000098019996, with Pbright = 2.0402 in the same arbitrary power unit.

Convert the dark fraction to a null ratio

What is N for that same amplitude-only case?

Use N = Pdark/Pbright rather than Pdark/Ptotal.
Divide 0.0002 by 2.0402.
Equivalently evaluate f/(1 − f).

N ≈ 0.000098029605, close to f but not equal to it.

Try the reasoning

Where does power go when the ideal dark output is suppressed?
Show the answer and reasoning
Into the complementary bright output.
The stipulated unitary model conserves the sum of output powers.
Does ε = 0.02 mean a 2% intensity mismatch?
Show the answer and reasoning
No.
It specifies field amplitude 1.02 relative to 1; the corresponding input power is 1.02².
Can improving phase alone remove the floor at fixed ε = 0.02?
Show the answer and reasoning
No.
The zero-phase amplitude mismatch leaves Pdark = ε²/2.
Is the plotted f an on-sky companion contrast curve as a function of angular separation?
Show the answer and reasoning
No.
The toy contains no angular companion response, throughput or detection model.

What to carry forward

Angular response, suppression and calibrated detection performance are different properties.
Interference combines fields, and a correctly normalized lossless model conserves total power.
Phase and amplitude errors affect leakage, whose denominator and excluded observing effects must remain explicit.

References

Teaching resources and research supporting this lesson’s explanations and examples.