1.  Introduction

In a previous article, we explored holography from a signal processing perspective: an object wave and a reference wave interfere on a film, and the recorded interference fringes — a modulated carrier — can later be demodulated by the same reference wave to reconstruct the full 3D wavefront, phase included.

But there exists a second, radically different kind of “hologram” that requires no laser, no interference, and not even a camera: the scratch hologram (also called abrasion hologram), popularized by William Beaty in the 1990s (#ref1). Take a piece of clear plastic, scratch a set of circular arcs on it with a compass, hold it under a point light source (the sun works perfectly), and a ghostly 3D image floats above — or sinks below — the plate.

Naturally, one asks: is this really a hologram? In this article, we answer the question in three steps:

1. Distill the mathematical essence of (wave) holography: what is the minimal structure a 2D surface must have in order to “project” a 3D point?
2. Re-derive the same structure from geometric optics, showing step by step why a circular scratch works, and visualizing the local principle with an interactive demo.
3. Unify the two pictures in one equation, connecting the geometric-optics notation of scratch holography with the signal-processing notation of the previous article.

2.  The Mathematical Essence of Holography

Recall the setup of the previous article. The object wave and the reference wave on the film are denoted as

$$[U_o(x,y)\exp(j\Phi(x,y))]\,e^{j\omega t}, \qquad [U_r\exp(j2\pi f_{xr}x)]\,e^{j\omega t},$$

and the film linearly records the interference intensity, $T(x,y) = \beta I(x,y)$. Expanding $|U_o e^{j\Phi} + U_r e^{j2\pi f_{xr}x}|^2$, the information-bearing part of the hologram is the cross term

$$T_{\text{info}}(x,y) \;=\; 2\beta\, U_o(x,y)\,U_r \cos\big(2\pi f_{xr}x - \Phi(x,y)\big),$$

a sinusoidal fringe pattern: the object phase $\Phi(x,y)$ is encoded as the local phase of a carrier of spatial frequency $f_{xr}$.

Now let us distill this formula step by step, stripping away everything non-essential.

Step 1: a hologram is locally just a grating. Zoom into any point $(x,y)$ of the hologram: the fringe pattern there looks like a plane wave. Define the local fringe vector

$$\vec{G}(x,y) \;=\; \nabla\big(\Phi(x,y) - 2\pi f_{xr}x\big).$$

The fringes run perpendicular to $\vec{G}$, with local spacing $\Lambda = 2\pi/|\vec{G}|$. The whole hologram is nothing but a smoothly varying field of little gratings.

Step 2: reconstruction is the grating equation. When the hologram is illuminated by the reference wave, each local grating redirects it by the grating equation: the wavevector component tangential to the plate changes by exactly the fringe vector,

$$\vec{k}_{out}^{\parallel} \;=\; \vec{k}_{in}^{\parallel} + \vec{G}(x,y).$$

Since $\vec{G}$ was recorded as the difference between the reference and object wavevectors, adding it back returns a wave whose local direction is that of the original object wave — this is Part 2 of the reconstruction formula in the previous article, the virtual image.

Step 3: the key is a conservation law. A fringe pattern is translation-invariant along its own tangent $\hat{t}(x,y)$, so $\vec{G}\cdot\hat{t} = 0$ by construction. Therefore the redirection always obeys

$$\big(\vec{k}_{out}^{\parallel} - \vec{k}_{in}^{\parallel}\big)\cdot\hat{t} \;=\; 0 .$$

Physics only allows a locally 1D structure to change the wavevector component perpendicular to itself; the component along the structure is conserved. This is the true mathematical essence of holography: a hologram is a 2D surface patterned with curves that are everywhere perpendicular to the desired wavevector-change field.

Step 4: what it takes to project a point. Suppose we want the plate to project a virtual point $O = (X,Y,z_0)$. At every plate point $P=(x,y,0)$, the outgoing wave must look as if it diverges from $O$, i.e. its phase must be

$$\Phi(x,y) \;=\; \frac{2\pi}{\lambda}\sqrt{(x-X)^2+(y-Y)^2+z_0^2} \;\approx\; \frac{\pi}{\lambda z_0}\big[(x-X)^2+(y-Y)^2\big] + \text{const},$$

where the approximation is the Fresnel (paraxial) one. The level sets of $\Phi$ — the fringes — are concentric circles centered at $(X,Y)$, with radii $r_n \approx \sqrt{2n\lambda z_0}$: the famous zone plate. The depth $z_0$ is encoded in the fringe spacing; the lateral position $(X,Y)$ in the fringe centers.

So, to make a hologram of a 3D scene, one “only” needs to draw, for every scene point, a family of concentric circular curves. Optical holography draws them by interference. But nothing in the conservation law above says the curves must be drawn by interference — any physical mechanism that redirects light while preserving the wavevector component along the curve will do.

3.  Scratch Holography: the Geometric-Optics Twin

Geometric optics provides exactly one everyday element with this property: a scratch. A scratch on a plastic plate is a groove with (roughly) V-shaped cross-section. Locally it is translation-invariant along its tangent $\hat{t}$, so it behaves like a cylindrical mirror whose axis is $\hat{t}$. Reflection from a cylindrical mirror obeys:

• Axis conservation: the direction component along the groove is unchanged. Writing $\hat{d}_i$ for the incident ray direction and $\hat{d}_o$ for the outgoing ray direction (both unit vectors, i.e. $\vec{k}/k$ in the $\lambda \to 0$ limit),

$$\hat{d}_o\cdot\hat{t} \;=\; \hat{d}_i\cdot\hat{t} \qquad\Longleftrightarrow\qquad \big(\hat{d}_o-\hat{d}_i\big)\cdot\hat{t} \;=\; 0 .$$

• Mirror law in the normal plane: in the plane perpendicular to $\hat{t}$, the groove facet reflects like an ordinary mirror — the facet normal bisects the incident and reflected rays.

Compare the first property with Step 3 above: it is exactly the same conservation law. A scratch is a geometric-optics fringe. This is the key that makes scratch holography possible.

3.1.  Where does the highlight appear?

Now scratch a circular arc of radius $R$ centered at $C$ on the plate ($z=0$), illuminate it with a point light $S$, and look at it from eye position $E$. The scratch does not shine uniformly: the eye sees a single bright highlight at the point $P$ on the arc where the reflection law can be satisfied, i.e. where

$$\big(\hat{d}_o(P)-\hat{d}_i(P)\big)\cdot\hat{t}(P) = 0, \qquad \hat{d}_i = \frac{P-S}{|P-S|},\quad \hat{d}_o = \frac{E-P}{|E-P|}.$$

Since for a circle $\hat{t} \perp (P-C)$, the condition says: the radius $C\to P$ must be parallel to the plate-projection of $\hat{d}_o-\hat{d}_i$. Move your head, and the highlight slides along the arc — this sliding is the motion parallax that makes scratch holograms feel so three-dimensional.

3.2.  Two eyes turn the sliding highlight into depth

Your two eyes $E_1$ and $E_2$ sit at different positions, so they see the highlight at two different points $P_1$ and $P_2$ on the same arc. Your brain back-projects the two sightlines and finds their intersection $O$: a luminous point hanging in space. Let us compute where it is.

Put the circle center at $C=(0,0,0)$, the two eyes at $E_{1,2}=(\mp i,\rho,h)$ — inter-pupillary half-distance $i$, horizontal distance $\rho$, height $h$ — and take the light near the zenith, $\hat{d}_i \approx (0,0,-1)$, so that the highlight condition reduces to “$C\to P$ parallel to the in-plane direction of the eye”. For $\rho \gg R$, each eye has two solutions:

$$P_{1,2}^{\text{far}} \approx \big(\pm\tfrac{iR}{\rho},\,-R,\,0\big), \qquad P_{1,2}^{\text{near}} \approx \big(\mp\tfrac{iR}{\rho},\,+R,\,0\big),$$

on the far side and the near side of the circle relative to the viewer. Both are valid roots of the highlight condition; which one you see is selected by the illumination direction and the groove facets.

Far-side highlight. The sightline from $E_1=(-i,\rho,h)$ through $P_1=(iR/\rho,-R,0)$ is $(x(s),z(s)) = \big(-i+si(1+R/\rho),\; h(1-s)\big)$. By symmetry the two sightlines meet at $x=0$, i.e. at $s=\rho/(\rho+R) < 1$ — between the eyes and the plate:

$$z_0 \;=\; +\,\frac{hR}{\rho+R} \;\xrightarrow{\;\rho\gg R\;}\; +\,\frac{h}{\rho}R .$$

The image floats above the plate (crossed disparity): this is the virtual image.

Near-side highlight. The same computation gives $s = \rho/(\rho-R) > 1$ — the sightlines meet behind the plate:

$$z_0 \;=\; -\,\frac{hR}{\rho-R} \;\xrightarrow{\;\rho\gg R\;}\; -\,\frac{h}{\rho}R .$$

The image sinks below the plate (uncrossed disparity): this is the conjugate image. A scratch hologram always has both, exactly like the Part 2 (virtual) and Part 3 (conjugate) terms of wave holography — which one appears depends on how you illuminate and hold the plate.

Three takeaways:

• Depth is proportional to the radius of curvature. For a fixed viewing geometry, $z_0 \propto R$. This is the design rule: to place a point deeper, scratch with a larger compass radius. Beaty’s practical recipe — compass centered at the point’s projection, radius proportional to the desired depth — is exactly this formula (#ref1).
• A 3D scene is a sum of arcs. Every scene point gets its own arc(s); a scratch hologram of a wireframe object is just a stack of circular arcs with different centers and radii.
• The depth cue is geometric, not wave-based. It works through binocular triangulation (and motion parallax), so unlike a true hologram it needs two eyes to deliver full depth — we will come back to this honest caveat below.

4.  Interactive Demo

The demo below visualizes the local principle derived above. Left: top view of the plate — one circular scratch, a point light, two eyes, the two highlights $P_1$, $P_2$, and the sightlines that triangulate the image point $O$. Right top: side view showing how binocular triangulation places $O$ above or below the plate. Right bottom: the groove cross-section at $P_1$ perpendicular to the tangent — axis conservation $\hat{d}_o\cdot\hat{t}=\hat{d}_i\cdot\hat{t}$ holds globally, while in this plane the facet acts as an ordinary mirror with equal angles.

Things to try:

• Drag the arc radius $R$: the triangulated depth $z_0$ tracks $hR/(\rho+R)$ — curvature is depth.
• Switch to “Conjugate image”: the highlight jumps to the near side of the circle and the image sinks below the plate — the scratch-hologram twin image.
• Drag the light azimuth: the highlights slide along the arc while the image point barely moves — the sliding highlight is motion parallax.
• Zone-plate overlay: concentric circles centered at the triangulated point $O$ — the fringes that a wave hologram of the same point would record. Note that the scratch is everywhere tangent to these fringes: scratch and zone plate are curves of the same family.

5.  One Equation for Both Pictures

We can now place the two “holograms” side by side. On the plate, define the redirection field

$$\Delta\vec{k}^{\parallel}(x,y) \;\equiv\; \vec{k}_{out}^{\parallel}(x,y) - \vec{k}_{in}^{\parallel}(x,y),$$

the wavevector change the plate must produce at each point so that light appears to come from the desired scene. Then:

• Wave holography records fringes whose local fringe vector equals the redirection field, $\vec{G} = \Delta\vec{k}^{\parallel}$ — recording by interference guarantees this automatically, since $\Phi$ is the object-wave phase. Fringes are the level sets of the recorded phase, hence $\vec{G}\cdot\hat{t}=0$.
• Scratch holography engraves grooves whose tangent satisfies $(\hat{d}_o-\hat{d}_i)\cdot\hat{t}=0$; multiplying by $k=2\pi/\lambda$, this is again $\Delta\vec{k}^{\parallel}\cdot\hat{t}=0$.

Both patterns are therefore integral curves of the same geometric condition:

$$\boxed{\;\hat{t}(x,y)\cdot\Delta\vec{k}^{\parallel}(x,y) \;=\; 0\;}$$

The recorded/drawn curves are everywhere perpendicular to the desired wavevector change. A scratch hologram of a point is quite literally a sparse, mechanically engraved sampling of the zone plate of that point — this is what the zone-plate overlay in the demo shows.

The difference lies in what each technology stores about $\Delta\vec{k}^{\parallel}$:

The last difference in the middle rows is the deep one. Wave holography stores both the orientation and the magnitude of $\Delta\vec{k}^{\parallel}$ — orientation in the fringe direction, magnitude in the fringe spacing — and therefore can reconstruct the full wavefront for a single eye. Scratch holography stores only the orientation and throws the magnitude away; the missing information is supplied by your brain, which triangulates the two highlights $P_1, P_2$ and computes the depth that a zone plate would have diffracted. Strictly speaking, a scratch hologram is a stereogram built from holographic fringe geometry — “hologram” with quotes (#ref3).

Yet the organizing principle is one and the same: a 2D surface patterned with curves perpendicular to a desired redirection field encodes a 3D scene (#ref4).

6.  The Philosophical Insight

In the previous article, we concluded that holography teaches us how a 2D surface can fully encode a 3D world, and that most of that information lives in the phase that our eyes never record. Scratch holography adds a humble but beautiful twist: the encoding does not even need waves. A compass and a plastic plate suffice, because the phase gradient $\nabla\Phi$ — the mathematical soul of the hologram — has a purely geometric shadow: the direction in which light must bend.

What changes is only who does the computation. In wave holography, the reference wave performs the demodulation physically and instantaneously; in scratch holography, the demodulation is performed neurally, by two eyes and a brain solving a triangulation that nature has been doing for half a billion years. The mathematics on the plate is the same.