<Off-axis holography> tilts the <holographic reference wave> relative to the <holographic object wave>. Its <interference fringes> acquire a carrier <spatial frequency>; the two cross terms occupy different <spatial frequency> regions when the carrier is sufficiently large compared with the object's <signal bandwidth>. This permits filtering a single image term.

Gabor's lecture credits <Emmett Leith> and <Juris Upatnieks> with the later skew-reference method using helium–neon <laser> illumination, separating reconstructed images angularly.<Holography, 1948-1971 by Dennis Gabor>[Gabor, Nobel lecture, Fig. 8 and surrounding account of the 1962–1963 laser work]{ref}

For an original periodic calculation, let $x$ run over a unit-length <interval> and choose
$$
O(x)=0.2e^{2\pi i3x}+0.1e^{-2\pi i5x},\qquad R(x)=e^{2\pi i20x}.
$$
All <spatial frequencies> here are cycles per unit length; the data are a chosen field at the recording plane, not measurements or a physical object profile. Direct expansion gives
$$
I(x)=1.05+0.04\cos(2\pi8x)+0.4\cos(2\pi17x)+0.2\cos(2\pi25x).
$$
Consequently its <Fourier series> coefficients are $1.05$ at zero, $0.02$ at $\pm8$, $0.2$ at $\pm17$ and $0.1$ at $\pm25$. The $R^*O$ component lies at $-17,-25$; the $RO^*$ component lies at $17,25$; the background lies at $0,\pm8$. Filtering the negative-frequency cross term and multiplying by $R$ recovers $O$, including its relative <wave phase>.

\Image[/holography-off-axis-cross-term-separation.png]
{title=Original periodic <off-axis holography> calculation: separated cross terms and recovered <complex amplitude>}

The figure samples $N=1024$ equally spaced positions with the endpoint excluded, computes a normalized <discrete Fourier transform> using a <Fast Fourier transform>, retains bins $-30\leq m\leq-12$, then applies the <inverse discrete Fourier transform> and multiplies by $R$. The recovered real and imaginary parts coincide with the chosen $O$ to numerical precision. This exact periodic example avoids leakage by placing every frequency on an integer bin; finite nonperiodic records need allowance for broadening and an <aperture>.

For a continuous object field with <Fourier transform> confined to $|f|\leq B$, the background $|O|^2$ occupies $|f|\leq2B$. A carrier $f_c>3B$ is a sufficient separation condition for the sideband centred at $-f_c$, and $f_c+B$ must remain below the <Nyquist spatial frequency>. Here $B=5$, $f_c=20$ and the <Nyquist spatial frequency> is $512$. This sufficient bound describes a whole bandwidth, rather than exploiting the gaps in this particular two-component field.

The same example also makes optical playback explicit. Choose $t_0=0.3$ and $\beta=0.01$. Since $|O|\leq0.3$, the bounds $0.49\leq I\leq1.69$ imply $0.3049\leq t\leq0.3169$, and $|\beta I|/t_0\leq0.0564$. The affine response is stipulated; its weak-modulation interpretation is approximate. In $Rt$, the desired field $0.01O$ has frequencies $3,-5$; the conjugate term $0.01R^2O^*$ has frequencies $37,45$; the background has frequencies $20,12,28$. Thus the same carrier that separates the recorded cross terms separates the playback components. Reillumination restores a field through <diffraction>; computational filtering extracts the corresponding encoded field through <Fourier analysis>.


 Off-axis holography (source code)