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# Spherical Gradient-Index Lens
## The Lens
A spherical gradient-index (GRIN) lens is made of concentric shells, each with a slightly different refractive index. The refractive index is lowest at the center and increases toward the outer edge, following a logarithmic profile. The total variation is very small on the order of the difference between air and vacuum.
Because light bends toward regions of higher refractive index, this gradient causes rays to curve outward as they travel away from the center.
## How It Looks from Outside
When an observer outside the lens looks at an object inside it, the light from that object has traveled outward through increasing refractive index. The rays curve outward away from the center so they arrive at the observer from a direction that points back toward a position closer to the center than the object actually is.
The lens compresses everything inside it toward the center. Objects appear both closer to the center and smaller than they really are. The effect is strongest near the center, where the refractive index gradient is steepest, and weakest near the edge.
## How It Looks from Inside
An observer sitting at the center of the lens sees the opposite effect. Incoming light travels inward through decreasing refractive index, and the gradient bends the rays inward concentrating them toward the observer. The lens acts as a converging lens for incoming light, making the outside world appear magnified and spread out.
Because this magnification applies uniformly to everything the observer sees, it is difficult to detect from the inside. There is no internal reference frame to compare against.
## The Asymmetry
The same lens has two different faces. It diverges outgoing light (shrinking the interior when viewed from outside) and converges incoming light (magnifying the exterior when viewed from inside). This asymmetry is central to how two lenses interact.
## Parallax from Inside the Lens
An observer inside the lens, orbiting its center, can make parallax measurements of outside objects. As the observer moves along the orbit, the apparent angular shift of an external object gives its parallax distance.
However, the lens compresses all positions toward the center including the observer's. An orbit with true radius *R* corresponds to a compressed effective radius *R'* as seen from outside. The parallax angles the observer measures correspond to this smaller baseline *R'*, not the true *R*. The observer believes they are moving a distance *R*, but the effective baseline that determines parallax is shorter.
The result is that all parallax distances are systematically overestimated everything outside appears more distant than it really is. This effect is uniform and cannot be distinguished from the objects actually being farther away, unless the observer has an independent distance measurement that does not rely on light passing through the lens.
## Two Lenses
Place two identical lenses, A and B, some distance apart. An observer sits inside Lens A, near its center, and looks toward Lens B.
### Objects inside Lens B
Light from an object inside Lens B first exits Lens B. During this exit, Lens B compresses the object's apparent position toward B's center the object looks closer to B's center and smaller than it is.
This light then crosses the gap and enters Lens A. Lens A bends the incoming light inward, but it does this to all incoming light equally both the light from Lens B's center and the light from the object. The relative separation between them is unchanged. Lens A's effect is a uniform magnification that does not undo Lens B's compression.
The observer inside A sees the object compressed toward B's center, just as any external observer would.
### Background objects behind Lens B
Light from distant objects behind Lens B passes through all of Lens B before reaching the observer. Lens B refracts this background light, compressing it toward B's center. When the observer shifts position, Lens B and the background both appear to move, but the background's apparent motion partially tracks Lens B because the light has been refracted through it. The parallax between Lens B and the background is reduced.
### Why the observer cannot detect Lens A
The observer is inside Lens A but outside Lens B. Lens B's full compression effect is visible because the observer sees light that has traversed the entire lens. Lens A's effect, on the other hand, is applied uniformly to the observer's entire field of view every direction is magnified by the same amount. With no unaffected reference to compare against, the observer has no straightforward way to notice Lens A's influence.