Optics 032 · Imaging, instruments, and visual systems

Binocular Prism Lab

Open two orthogonal Porro prisms, trace finite Snell–Fresnel paths and their delivered pupil, then compare the polarization-resolved diffraction of bare and ideally compensated roof pupils.

Porro rotation / bare roof retardance

\Phi_P,\Delta_{sp}

$0.50$

Smallest total-reflection margin

\min(\theta_i)-\theta_c

$50\%$

Delivered power / roof on-axis peak

T,S_0

$0.00\pi$

Model regime $\text{valid model regime}$

Porro is a finite geometric module with a real square entrance port; field cards are equal-scale orientation diagrams. Roof is a separate axial, in-glass 90-degree module with an oversized circular pupil and ideal retardance compensation, not a complete binocular or a fabricated coating design.

Physics tutorial

Audit the physics of Binocular Prism Lab

BackgroundA positive telescope objective supplies an inverted field. Image erection is determined by the actual reflection operator $H_{\hat{\mathbf n}}=I-2\hat{\mathbf n}\hat{\mathbf n}^{\mathsf T}$ . Each Porro contains two perpendicular reflecting facets. An orthogonal pair reverses both transverse coordinates while restoring the forward propagation direction.

Why it mattersHow do four reflections erect a field, and why can a lossless roof still degrade its image?

Start with the essentials

Focus question: How do four reflections erect a field, and why can a lossless roof still degrade its image?
One-sentence intuition: A roof splits a pupil by reflection order, not into independent photographs. For unpolarized light, the image intensity is $I(\mathbf q)=\frac12\left\|J_+A_+(\mathbf q)+J_-A_-(\mathbf q)\right\|_F^2$ . The two complex vector fields must interfere before averaging the independent input polarizations. Unit reflection power does not guarantee a diffraction-limited point image.

Core mathematical model

Four reflections, two coordinate flips

H_4H_3H_2H_1=\operatorname{diag}(1,-1,-1)

The first coordinate is the forward optical axis. The other two are the transverse world coordinates. The displayed objective-inverted field therefore becomes upright after the second prism. All field cards retain the same symbolic scale; they are not photographs or a full image relay.

Critical angle and actual finite surfaces

\theta_c=\arcsin(1/n_g),\qquad \hat{\mathbf k}_r=H_{\hat{\mathbf n}}\hat{\mathbf k}_i

Angles are measured from the face normal. The traced ray must intersect the finite face before reflection can occur. Below the critical angle a refracted branch leaks into air; the remaining reflected branch follows Fresnel amplitudes. The square entrance stop and blackened extrusion ends are actual absorbing hardware.

Lossless reflection still changes phase

r_s=\frac{\cos\theta-i\gamma}{\cos\theta+i\gamma},\quad r_p=\frac{\eta^2\cos\theta-i\gamma}{\eta^2\cos\theta+i\gamma},\quad \gamma=\sqrt{\sin^2\theta-\eta^2},\quad \eta=1/n_g

Both amplitudes have unit modulus in total internal reflection, but unequal phases. In a roof, the two pupil halves encounter the non-coplanar incidence frames in opposite orders. The code transforms the full electric-field vector at each facet rather than assigning a fictitious scalar path difference.

Two-half Jones pupil diffraction

A_\pm(\mathbf q)=\frac1\pi\iint_{P_\pm}e^{-i\mathbf q\cdot\boldsymbol\rho}\,\mathrm d^2\boldsymbol\rho,\qquad S_0=I(\mathbf 0)

The roof model uses a uniform circular pupil at 550 nm. A semi-disk quadrature and analytic disk transform generate the point image; no decorative blur is used. All PSFs use the same 100 arcsecond field and unit ideal-peak reference, with square-root brightness. Full correction is an ideal lossless retardance equalizer, not a specified multilayer or thickness.

Common difficulties

Total reflection guarantees a sharp image

Typical misconceptionIf both roof faces reflect all power, phase correction cannot improve resolution.

Better mental modelPower and complex field are different. Opposite reflection orders alter the polarization-resolved pupil and redistribute image energy. The roof reference excludes glass entry/exit losses; Porro direct throughput includes them. Neither should be mistaken for a complete binocular transmission rating.

Run the experiment

01
Scene 1: Double Porro erection
Open Double Porro. Follow the four physical reflection vertices and compare the asymmetric field before, between and after the prisms. Orbit to see that the two folds are perpendicular.
What to observe: The double Porro has parallel input and output directions but a lateral displacement and a 180-degree field rotation.
02
Scene 2: Finite pupil & TIR leakage
Open Finite pupil. Raise the field angle or lower the index; inspect pink leakage and the critical-angle margin. Increase pupil size or shrink Porro width to expose actual stop and end-face clipping.
What to observe: Fresnel losses and geometric clipping are separate: delivered power and accepted pupil area are reported independently.
03
Scene 3: Roof pupil diffraction
Open Roof. Move ideal correction from zero to one. Compare bare, selected and ideal circular-pupil point images at equal brightness. Index sets bare Fresnel phase; pupil diameter changes the diffraction angular scale.
What to observe: The roof module turns through 90 degrees; it isolates the phase problem and does not pretend to be a whole straight-through Schmidt–Pechan instrument.