Single-Shot Dual-Wavelength Holography

• Single-shot dual-wavelength off-axis digital holography combines dual-wavelength interferometry with off-axis holography to measure thicker structures in a single camera exposure.
• The approach employs Fourier-domain separation or direct phase difference recovery to reconstruct complex fields and extend the unambiguous measurement range.
• Experimental validations on step targets and micro-structures demonstrate robust quantitative phase retrieval with trade-offs in phase noise and dynamic range.

Single-shot dual-wavelength off-axis digital holography is an interferometric imaging modality in which two wavelength channels are encoded in a single camera exposure by off-axis holographic carriers and then used for complex-field reconstruction or synthetic-wavelength phase retrieval. In its conventional form, the method combines the single-frame character of off-axis digital holography with dual-wavelength interferometry, so that structures far thicker than a single optical wavelength can be measured without sequential acquisition. The defining operational features are simultaneous illumination at two wavelengths, a single recorded interferogram, Fourier-domain separation or direct phase-difference recovery, and quantitative phase or height reconstruction over an extended unambiguous range (Turko et al., 2019, Amani et al., 29 Jul 2025).

1. Physical basis and defining criteria

The off-axis component follows the standard holographic intensity model

$$I(x,y)=|U_s+U_r|^2=|U_s|^2+|U_r|^2+U_sU_r^*+U_s^*U_r,$$

in which a small tilt of the reference wave shifts the cross-correlation terms away from the DC term in spatial-frequency space. After a 2D Fourier transform, one cross term can be windowed and inverse transformed to recover the complex sample field; its argument yields the wrapped phase and its modulus the amplitude (Turko et al., 2019).

The dual-wavelength component introduces two phases, one at each wavelength. For reflective samples, the measured phase at wavelength $\lambda_i$ is

$$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$

so any optical thickness larger than a single wavelength produces wrapping. Combining two wavelengths generates the synthetic wavelength

$$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$

and the phase difference obeys

$$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$

Because $\Lambda\gg\lambda_1,\lambda_2$ when the wavelengths are close, the unambiguous height range is correspondingly extended (Turko et al., 2019, Amani et al., 29 Jul 2025).

The single-shot condition is stricter than mere simultaneous dual-wavelength illumination. It requires that both interferometric channels be encoded into one sensor frame, with no sequential switching, no phase stepping, and no temporal delay between wavelength-specific phase maps. In the canonical implementation, this is achieved by optical multiplexing of two off-axis holograms onto one monochrome camera; in a later variant, the two wavelength contributions deliberately overlap in the Fourier domain and the synthetic phase is recovered directly from the combined sideband magnitude (Turko et al., 2019, Amani et al., 29 Jul 2025).

2. Optical architectures and multiplexing strategies

A representative conventional architecture uses a reflectance microscope as the base imaging system and attaches an external dual-wavelength off-axis holographic module at the image plane. Illumination is provided by a supercontinuum source and an acousto-optical tunable filter that generate two simultaneous narrow spectral bands, with reported pairs $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$ and $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$, each with approximately $5.4\,\mathrm{nm}$ bandwidth. A 4-f relay projects the sample image onto the module input plane. Inside the module, a beam splitter forms sample and reference arms, a $30\,\mu\mathrm{m}$ pinhole spatially filters the reference arm, and a dichroic mirror separates the two wavelength channels. Slightly tilted mirrors then return wavelength-specific reference beams with orthogonal off-axis directions, so that one wavelength produces fringes along $\lambda_i$0 and the other along $\lambda_i$1. The two interferograms are thereby optically multiplexed onto a single monochrome camera in one exposure (Turko et al., 2019).

This external module is self-referenced and near common-path in the sense that both sample and reference beams originate from the sample image and share most of the optical path. The reference arm is produced by low-spatial-frequency filtering of the sample-derived field rather than by an independent external reference. The reported design is therefore mechanically stable, compatible with existing microscopes, and explicitly intended to avoid motion artefacts associated with sequential two-wavelength acquisition (Turko et al., 2019).

A distinct architecture was introduced for direct phase-difference retrieval in an off-axis Michelson interferometer illuminated by a sodium-vapor lamp. Here the two wavelengths are the sodium D-lines, $\lambda_i$2 and $\lambda_i$3, selected by a $\lambda_i$4 bandpass filter. Because both wavelengths propagate through exactly the same optical components and geometry in each arm, the setup behaves like a single-wavelength arrangement from an alignment standpoint. The fixed mirror is slightly tilted to create the off-axis carrier, but the two wavelength sidebands are intentionally allowed to overlap in Fourier space; a single combined +1 order is then processed, rather than two separately resolved wavelength channels (Amani et al., 29 Jul 2025).

The broader multiplexing principle is not limited to two channels. “Six-pack off-axis holography” shows that six off-axis holograms can be compressed into one multiplexed hologram by assigning six different fringe orientations so that the corresponding cross-correlation regions do not overlap in the Fourier plane. The paper explicitly identifies wavelength multiplexing as one of the intended applications. In that framework, a single-shot dual-wavelength system is a simpler two-channel case of a more general spatial-bandwidth packing problem (Rubin et al., 2019).

3. Reconstruction workflows

In orthogonal-carrier dual-wavelength off-axis holography, numerical reconstruction proceeds by Fourier-domain order separation. A single multiplexed hologram is recorded, its 2D Fourier transform is computed, and the spatial spectrum shows a central DC term together with two cross-correlation lobes along $\lambda_i$5 for one wavelength and two along $\lambda_i$6 for the other. One cross term per wavelength is cropped, inverse transformed, and converted into wrapped phase maps $\lambda_i$7 and $\lambda_i$8. The synthetic-wavelength phase is then obtained by subtraction,

$\lambda_i$9

followed, where needed, by a local correction that adds $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$0 when the difference is negative in order to remove residual jumps caused by wrap-index mismatch. The final height map is recovered through $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$1 (Turko et al., 2019).

This workflow is designed to bypass the failure modes of conventional 2D phase unwrapping on discontinuous profiles. The 2019 reflectance implementation explicitly notes that traditional 2D unwrapping fails for sharp discontinuities such as a step of many wavelengths, whereas dual-wavelength combination yields a clean synthetic-wavelength height map for such objects (Turko et al., 2019).

A later reconstruction paradigm dispenses with separate per-wavelength phase recovery. After Fourier filtering of the combined +1 order, the retained complex sideband is written as

$$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$2

Under the equal-amplitude assumption $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$3 and $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$4, this becomes a mean-phase term modulated by $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$5, so that the sideband magnitude satisfies

$$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$6

The phase difference is then recovered directly as

$$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$7

The method requires a calibration of $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$8 from a background hologram, in which a known zero-phase-difference region is established and a single-wavelength reconstruction at $$\phi_i(x,y)=\frac{4\pi}{\lambda_i}h(x,y)\pmod{2\pi},$$9 is used to infer the corresponding synthetic-wavelength phase map (Amani et al., 29 Jul 2025).

The direct method changes the usual interpretation of dual-wavelength off-axis holography. In conventional approaches, the two wavelengths must typically be spatially separated in the Fourier plane and individually reconstructed before subtraction. In the overlapping-order method, no such wavelength separation is required. The synthetic phase is retrieved from amplitude modulation of a single combined sideband, which is what the paper identifies as “direct phase difference reconstruction” (Amani et al., 29 Jul 2025).

4. Experimental validation and quantitative performance

The external-module reflectance system was validated on a commercial step-height standard with nominal height $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$0 and on circular copper pillars of nominal height $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$1 and diameter $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$2. For the step target, the wavelengths were $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$3 and $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$4, giving $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$5. A histogram of the reconstructed height distribution, fitted with two Gaussians for the top and bottom levels, yielded a step height of $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$6. The reported average accuracy relative to a white-light interferometer reference was between $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$7 and $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$8, and the repeatability over 20 repeated measurements was below $$\Lambda=\frac{\lambda_1\lambda_2}{|\lambda_1-\lambda_2|},$$9 (Turko et al., 2019).

For the $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$0 copper pillars, the wavelengths were $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$1 and $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$2, corresponding to a synthetic wavelength of approximately $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$3. Because the nominal pillar height exceeded both the synthetic wavelength and the depth of field, the reconstruction was shifted by adding four multiples of $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$4, or approximately $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$5, so that the final estimate lay within the unambiguous interval around the expected height. Over 26 pillars, the average reconstructed height was $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$6, the standard deviation was $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$7, and the average deviation from nominal was $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$8 (Turko et al., 2019).

The direct phase-difference Michelson method was validated first on an air wedge and then on large-step objects. With $$\Delta\phi_\Lambda(x,y)=\phi_1(x,y)-\phi_2(x,y)=\frac{4\pi}{\Lambda}h(x,y), \qquad h(x,y)=\frac{\Delta\phi_\Lambda(x,y)\Lambda}{4\pi}.$$9 and $\Lambda\gg\lambda_1,\lambda_2$0, the synthetic wavelength was $\Lambda\gg\lambda_1,\lambda_2$1. In the air-wedge experiment, the single-wavelength unwrapped phase reached about $\Lambda\gg\lambda_1,\lambda_2$2, while the directly reconstructed dual-wavelength synthetic phase remained around $\Lambda\gg\lambda_1,\lambda_2$3, so no spatial phase unwrapping was required. The 1D height profiles from single-wavelength unwrapping and from the direct dual-wavelength method matched closely (Amani et al., 29 Jul 2025).

The same method was then used to measure nominal $\Lambda\gg\lambda_1,\lambda_2$4 step increments and a glass plate of nominal thickness $\Lambda\gg\lambda_1,\lambda_2$5. For four effective height conditions, $\Lambda\gg\lambda_1,\lambda_2$6, $\Lambda\gg\lambda_1,\lambda_2$7, $\Lambda\gg\lambda_1,\lambda_2$8, and $\Lambda\gg\lambda_1,\lambda_2$9, the retrieved step differences were $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$0, $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$1, and $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$2. For the glass plate, with refractive index $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$3 and $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$4, the reconstructed thickness was $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$5, which lay within the stated nominal tolerance (Amani et al., 29 Jul 2025).

Taken together, these measurements establish two experimentally distinct operating regimes: orthogonally multiplexed off-axis dual-wavelength holography for calibrated micro-topography at roughly $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$6 to $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$7 scale, and overlapping-sideband direct phase-difference holography for synthetic wavelengths approaching $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$8 and object heights extending to approximately $$\lambda_1=580\,\mathrm{nm}, \lambda_2=597\,\mathrm{nm}$$9 (Turko et al., 2019, Amani et al., 29 Jul 2025).

5. Noise, ambiguity, and design trade-offs

The principal design trade-off is set by the synthetic wavelength. Smaller $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$0 produces a larger $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$1 and hence a wider unambiguous range, but it also amplifies phase noise. In the 2019 reflectance implementation, the synthetic-wavelength phase map was reported to have noise approximately 35 times higher than the single-wavelength phase maps, with maximum spatial noise up to about $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$2. The method therefore extends range at the cost of increased noise sensitivity (Turko et al., 2019).

Carrier placement in the Fourier plane introduces a second trade-off. In orthogonal multiplexing, the two wavelength channels must be sufficiently separated from the DC term and from each other to permit robust filtering. The general packing problem is formalized by six-pack holography, which shows that six non-overlapping cross-correlation regions can occupy the Fourier plane without loss of magnification or resolution and reports cross-term occupancies of about $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$3 for a single off-axis hologram, $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$4 for two orthogonal-carrier holograms, $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$5 for four holograms, and $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$6 for six holograms. For dual-wavelength systems, this implies that spatial-bandwidth consumption is usually not the limiting factor; greater separation can be retained for robustness rather than packing density (Rubin et al., 2019).

Dynamic-range sharing is an intrinsic consequence of optical multiplexing onto a single sensor. The 2019 dual-wavelength reflectance module states that sharing the monochrome camera dynamic range had negligible impact on measurement accuracy for the reported experiments. Six-pack holography notes the same issue more generally for optically multiplexed holograms and reports no visible degradation for phase objects, with a mean square error in phase of $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$7 in its own six-channel quantization test. A plausible implication is that two-channel wavelength multiplexing is less demanding than higher-order multiplexing in this respect (Turko et al., 2019, Rubin et al., 2019).

At the detector-noise level, off-axis heterodyne holography is fundamentally limited by shot noise on the reference beam. The cited analysis shows that, for a weak signal, the equivalent noise on the signal beam is one photoelectron per pixel for the whole sequence of images used to build the digital hologram. The same source connects this result explicitly to single-shot dual-wavelength off-axis holography by treating each wavelength channel independently, provided each local oscillator is strong and the +1 orders are well separated (Joud et al., 2012).

A common misconception is that dual-wavelength off-axis holography always requires two separately resolvable wavelength sidebands. The overlapping-order sodium-lamp method contradicts that assumption by showing that very closely spaced wavelengths can be used without Fourier-domain wavelength separation, precisely because the desired observable is the phase difference rather than the two individual phase maps (Amani et al., 29 Jul 2025).

6. Related variants and adjacent research directions

Single-shot dual-wavelength off-axis holography has been extended conceptually beyond classical synthetic-wavelength profilometry. In imaging with undetected photons, a nonlinear Michelson interferometer based on spontaneous parametric down-conversion uses a detected visible field to encode the transmission and phase of an object placed only in an infrared idler arm. One implementation uses a $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$8 pump, $$\lambda_1=580\,\mathrm{nm}, \lambda_2=605\,\mathrm{nm}$$9 detected signal photons, and $5.4\,\mathrm{nm}$0 idler photons; a single off-axis interferogram recorded in the visible is processed by FFT–filter–IFFT to recover the object field at the infrared wavelength. The paper reports transmission-image signal-to-noise ratio $5.4\,\mathrm{nm}$1 at $5.4\,\mathrm{nm}$2 frames per second and dynamic-scene imaging at $5.4\,\mathrm{nm}$3 frames per second (Pearce et al., 2024).

A related quantum imaging with undetected light experiment employs a hybrid induced-coherence interferometer with $5.4\,\mathrm{nm}$4, $5.4\,\mathrm{nm}$5, and $5.4\,\mathrm{nm}$6. Here the object is probed at the idler wavelength, the hologram is recorded at the signal wavelength, and off-axis Fourier filtering recovers amplitude and phase from a single shot in a wide-field configuration. The reported field of view is $5.4\,\mathrm{nm}$7 diameter at the camera, and the measured engraved-feature heights include $5.4\,\mathrm{nm}$8 and $5.4\,\mathrm{nm}$9 for two phase objects (León-Torres et al., 2024).

These quantum implementations differ from conventional dual-wavelength synthetic-wavelength interferometry because the two wavelengths are not both detected and no synthetic wavelength is formed. Nevertheless, the cited works explicitly describe them as intrinsically dual-wavelength in the sense that probing and recording occur at distinct wavelengths while the single-shot, off-axis reconstruction logic remains the same (Pearce et al., 2024, León-Torres et al., 2024).

A further adjacent development is polarization-multiplexed second-harmonic generation holography. Although it is not dual-wavelength, it demonstrates a closely analogous two-channel strategy in which a Wollaston prism creates two off-axis reference beams with orthogonal polarizations and non-parallel propagation directions. From one hologram, two second-harmonic fields corresponding to orthogonal polarizations are separated in the angular spectrum and back-propagated independently. The paper presents this as a single-shot 3D mapping method for collagen and reports recovery of two polarization-resolved second-harmonic fields from one measurement (Goldmann et al., 8 Mar 2026). The multiplexing logic is directly relevant to dual-wavelength design because it shows how two independent channels can be assigned distinct carriers, separated in the spatial-frequency domain, and reconstructed with wavelength- or channel-specific propagation kernels.

Multi-wavelength in-line holography provides another adjacent perspective. A three-wavelength Gerchberg–Saxton method using $30\,\mu\mathrm{m}$0, $30\,\mu\mathrm{m}$1, and $30\,\mu\mathrm{m}$2 holograms recorded on an RGB CCD exploits the phase-scaling relation

$30\,\mu\mathrm{m}$3

under weak dispersion and uses combined similarity scores for detector-plane intensity consistency and object-plane phase covariance to locate the object plane automatically. The cited work is not off-axis, but it shows that multi-wavelength constraints can be used for axial localization and phase retrieval under noisy conditions. This suggests a natural algorithmic complement to dual-wavelength off-axis systems, where complex fields are already available from Fourier filtering and can then be subjected to wavelength-consistency refinement (Zhang et al., 2018).

In its mature form, the field therefore encompasses at least three distinct but connected interpretations of the term: classical dual-wavelength synthetic-wavelength profilometry with orthogonal carrier multiplexing, direct phase-difference recovery from overlapping dual-wavelength sidebands, and broader single-frame off-axis schemes in which probing and recording wavelengths differ but the reconstruction remains holographic and single-shot (Turko et al., 2019, Amani et al., 29 Jul 2025, Pearce et al., 2024, León-Torres et al., 2024).