Imaging by Refraction

Refraction changes more than the direction of an individual ray. A group of rays leaving one object point can converge at a new location or appear to originate from one. That location is its image. A single interface can therefore shift the apparent position of a feature or provide focusing power before a complete lens is introduced.

Apparent depth at a flat interface

A feature viewed through an index boundary may appear displaced along the viewing direction. For observation close to the surface normal, its apparent depth is:

d_{\mathrm{apparent}}=d_{\mathrm{actual}}\frac{n_2}{n_1}

Here, $n_1$ is the index surrounding the object and $n_2$ is the index on the observer’s side. Both depths are positive distances from the interface. A feature 18 mm beneath a surface in an index-1.50 medium appears 12 mm deep when viewed from air with index 1.00.

Derivation: apparent depth from Snell's law

Take a ray from an object at depth $d_{\mathrm{actual}}$ . It reaches the flat interface a horizontal distance $x$ from the point directly above the object. Its incident and transmitted angles, $\theta_1$ and $\theta_2$ , are measured from the surface normal.

Extend the transmitted ray backward until it meets the normal through the object. That intersection defines its apparent depth. The two right triangles give:

\tan\theta_1=\frac{x}{d_{\mathrm{actual}}},\qquad \tan\theta_2=\frac{x}{d_{\mathrm{apparent}}}

Snell’s law connects the angles:

n_1\sin\theta_1=n_2\sin\theta_2

For rays close to the normal, $\sin\theta\approx\tan\theta$ . Substitute the triangle relationships:

n_1\frac{x}{d_{\mathrm{actual}}}\approx n_2\frac{x}{d_{\mathrm{apparent}}}

Cancel the common nonzero $x$ and rearrange:

d_{\mathrm{apparent}}\approx d_{\mathrm{actual}}\frac{n_2}{n_1}

The axial ray alone does not locate the image; nearby rays establish this limiting depth. Their backward extensions approximately share a virtual image point. At larger angles the intercept depends on the ray angle, so the simple depth ratio no longer describes a single exact image.

$Rays leaving a submerged feature refract at a flat boundary; their backward extensions locate a shallower apparent image.$

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The apparent image is virtual: the transmitted rays do not physically converge there. This depth relation is useful when estimating focus through a transparent boundary, but it assumes near-normal viewing through one interface. An oblique view or an additional window surface requires a more complete ray model.

Image formation at a curved surface

A spherical interface has a different normal at each ray intersection. In the paraxial approximation, its image location follows:

\frac{n_1}{s}+\frac{n_2}{s'}=\frac{n_2-n_1}{R}

The real object distance $s$ is positive on the incident side. The image distance $s'$ is positive for a real image in the transmitted medium and negative for a virtual image on the incident side. The radius $R$ is positive when the center of curvature lies in the transmitted medium. All distances use the same units and are measured from the surface vertex.

$A spherical refracting surface redirects paraxial rays toward a real image in the second medium; object distance, image distance, and radius use explicit reference points.$

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For $n_1=1.00$ , $n_2=1.50$ , $R=30$ mm, and $s=120$ mm, the image forms at $s'=180$ mm. Moving the object to $s=60$ mm produces collimated output instead of a finite image.

Separate image position from image scale

Across different indices, transverse magnification includes the index ratio:

m=\frac{h'}{h}=-\frac{n_1s'}{n_2s}

The first example gives $m=-1$ : equal image size with inverted orientation. These results describe rays near the optical axis and a single refracting surface. Larger ray angles introduce departures from the approximation, and any subsequent exit surface changes the final image. Use the estimates to establish geometry before evaluating a practical lens or inspection window.

Imaging by Refraction

Apparent depth at a flat interface

Image formation at a curved surface

Separate image position from image scale

Spherical refracting surface

Apparent depth

Connect interface geometry to measurement geometry.