The surface shape of an object can be measured by illuminating it with coherent light of wavelength $\lambda$ and analyzing the interference pattern formed by the superposition of the wave reflected from the surface and a reference wave. For accurate measurements, the surface must be “smooth.” Specifically, the interference pattern should be free of speckles, and any abrupt changes in the surface shape must be smaller than $\lambda /2$. Interferometry has emerged as a powerful technique for determining the shape of optically flat surfaces, typically achieving resolutions of $\lambda /100$ or better. In addition to its high resolution, interferometry offers a significant advantage in acquisition speed compared with tactile surface mapping methods.

However, a vast class of functionally rough surfaces cannot be classified as “smooth.” These surfaces are of great importance because they are widely encountered in industrial and technological applications, including machined, cast, forged, welded, and 3D-printed components. The introduction of multiwavelength interferometry has enabled the interferometric determination of their shapes.1 This approach has been studied for over half a century.2^–5 It uses coherent light at two wavelengths, ${\lambda }_1$ and ${\lambda }_2$. By evaluating and subtracting the resulting phase maps, speckle patterns are suppressed, producing a phase map equivalent to one obtained with coherent light at the synthetic wavelength $\mathrm{\Lambda }=c/|{\nu }_2-{\nu }_1|$, where $c$ is the speed of light and the optical frequencies are given by ${\nu }_{\mathrm{1,2}}=c/{\lambda }_{\mathrm{1,2}}$. The synthetic wavelength extends the unambiguous measurement range $\mathrm{\Lambda }/2$ by orders of magnitude compared with single-wavelength interferometry, albeit with a proportional decrease in resolution.

This trade-off is addressed by employing multiple synthetic wavelengths through hierarchical unwrapping of phase maps.6 The longest synthetic wavelength is chosen to be at least twice the largest deformation of the surface, ensuring an adequate measurement range, whereas the shortest wavelength determines the resolution. Selecting intermediate wavelengths among these extremes is crucial for accurate phase unwrapping, especially for surfaces with complex deformations or noisy data.7^,8 Too few wavelengths risk phase ambiguity, whereas too many increase measurement time and complexity without significant accuracy gains.

The significant potential of multiwavelength interferometry lies in its ability to flexibly adjust synthetic wavelengths across the desired range, providing a balance between resolution and unambiguous measurement range. This adaptability, in principle, makes it an invaluable tool for characterizing a wide variety of surfaces in industrial and technological applications. However, despite decades of research, its full potential remains untapped. In recent review articles, it is pointed out that current systems are unique to special measuring tasks; however, they must be flexible. One main contributing factor is the absence of suitable light sources capable of reliably and rapidly altering the synthetic wavelength across several orders of magnitude.9^,10

The importance of the light source is highlighted by the following equations that describe the reconstruction and uncertainty of a height distribution $h(x,y)$ based on an interferometrically determined phase map $\mathrm{\Phi }(x,y)$ corresponding to a synthetic wavelength $\mathrm{\Lambda }$. $$h(x,y)=[m(x,y)+\frac{\mathrm{\Phi }(x,y)}{2\pi }]\frac{\mathrm{\Lambda }}{2},\frac{\mathrm{\Delta }h}{h}=\frac{\mathrm{\Delta }\mathrm{\Lambda }}{\mathrm{\Lambda }}+\frac{\mathrm{\Delta }\mathrm{\Phi }}{2\pi m+\mathrm{\Phi }}+\frac{\mathrm{\Delta }m}{m+2\pi /\mathrm{\Phi }}.$$(1)

The uncertainty in the synthetic wavelength is determined by the properties of the light source, whereas phase uncertainty arises from the interferometric setup and the data evaluation process. Accurate determination of the integer order $m$ is critical as it depends on both the synthetic wavelength and the measured phase. Consequently, the overall measurement uncertainty can be attributed to three main sources: the light source, the interferometric system, and the data processing. Uncertainties related to the refractive index—such as those induced by environmental fluctuations—have been neglected. Although all components of the measurement system contribute to the overall uncertainty, the light source plays a particularly crucial role. It not only defines the synthetic wavelength, which directly affects the absolute scale of the measurement, but also determines the spectral purity and switching speed—key parameters for achieving high precision and flexibility in dynamic applications. In the following, we therefore concentrate on the light source, which represents the core innovation of this work.

Various methods exist for generating synthetic wavelengths. They can be grouped into four categories sketched in Fig. 1. A commonly used approach relies on multiple free-running lasers [Fig. 1(a)], each operating at its specific optical frequency.11^–13 A switching mechanism ensures that interferograms are recorded sequentially at each optical frequency. The synthetic wavelengths are then derived from the corresponding difference frequencies. In this method, the lasers are selected to produce synthetic wavelengths tailored to a particular surface to be measured, which inherently limits flexibility. In addition, accurate determination requires measuring the optical frequencies of the free-running lasers, their individual frequencies, or their differences. The uncertainty in these optical frequencies restricts the practical synthetic wavelengths of such devices to millimeter-scale values or smaller. Despite its limitations, this approach has successfully found a way out of the lab into industrial applications.14

To address this limitation, a tunable laser can be used instead of multiple lasers [Fig. 1(b)].15^–23 In this approach, the synthetic wavelength can be adjusted within a range determined by two factors: the tuning range of the laser, which sets the minimum value, and the tuning step size, which determines the maximum value. For mode-hop-free tunable lasers, the step size is limited by the uncertainty in the optical frequency, enabling synthetic wavelengths to extend to meter scales. Although this method offers a high degree of flexibility, reliably tuning the laser across a broad range requires precise frequency tracking and a control loop. Despite its versatility, the approach is limited by relatively slow tuning speeds. This limitation can be addressed by implementing a continuous frequency sweep synchronized to a stable clock, which ensures that interferograms are acquired at well-defined frequencies.24 Although this method enables switching among different synthetic wavelengths on the timescale of tens of milliseconds, it lacks flexibility in selecting arbitrary synthetic wavelengths. Furthermore, the frequency sweep has to be calibrated by a reference system. More recently, an alternative approach has been demonstrated that achieves impressive sub-millisecond switching time. It is based on injection locking a diode laser to a frequency comb,25 where the laser current is tuned to shift the emission frequency among individual comb lines. However, this method also suffers from limited versatility as the available frequencies are restricted to the discrete spacing of the comb.

All of the approaches mentioned above share a common feature: at some stage, they require a wavelength reference such as a wavemeter, an optical spectrum analyzer, or a frequency comb. A third category of synthetic-wavelength generators avoids this requirement using a frequency comb not as a reference, but as the primary light source.26^,27 In this scheme, a tunable filter sequentially selects individual comb lines [see Fig. 1(c)]. As the mode spacing of the comb is precisely known, no external wavelength reference is needed. However, this method only allows discrete frequency steps, reducing flexibility. In addition, the switching speed is relatively slow.

All of the approaches discussed above are summarized in Table S1 in the Supplementary Material, where they are compared in terms of synthetic wavelength range, tuning flexibility, and switching speed. The comparison clearly shows that none of the existing methods combines wide-range, flexible synthetic wavelength tuning over several orders of magnitude with sub-100 ms switching time—without relying on an external wavelength reference.

To overcome these limitations, we present a synthetic-wavelength generator based on single-sideband modulation that combines high flexibility, fast switching, and independence from external wavelength references. Our approach enables dynamic tuning over several orders of magnitude in synthetic wavelength with switching time below 30 ms—using only commercially available components. These capabilities are key prerequisites for dynamically reconfigurable multiwavelength interferometry.

The approach is based on the following idea: single-frequency light at the optical frequency ${\nu }_0$ is converted by a frequency shifter, which is driven at the radio frequency $f$, to produce light at the new frequency ${\nu }_0+f$ [Fig. 1(d)]. Here, the synthetic wavelength is given by $c/f$. Thus, no additional monitoring of the optical frequency is necessary. The application of radio-frequency-controlled frequency shifters for multiwavelength interferometry was demonstrated with acousto-optic modulators.28 However, these devices do not operate in a wide radio frequency range. Thus, the flexibility was very limited. In this contribution, we use an electro-optic single-sideband modulator as a frequency shifter. These devices can be operated from 0.1 GHz up to the 100 GHz range.29^–31 Thus, synthetic wavelengths can be adjusted to any value between millimeters and meters.

In our proof-of-concept study, we demonstrate a generator for synthetic wavelengths based on single-sideband modulation. This device is characterized by spectral purity, the flexibility of the synthetic wavelengths, and tuning speed. We show its application for multiwavelength holography. Two different samples—one machine-milled part made of metal and another made out of plastic via injection molding—are investigated with different cascades of synthetic wavelengths.

The generation of synthetic wavelengths with a radio-frequency-driven modulator is based on the following idealizations. The output spectrum contains only the shifted optical frequency component at ${\nu }_0+f$, where ${\nu }_0$ is the original optical frequency and $f$ is the radio frequency. No additional spectral components, such as sidebands or noise, are present. The synthetic wavelength is precisely given by $\mathrm{\Lambda }=c/f$, whereas $f$ is variable over a wide frequency range. In the following, we evaluate how closely electro-optic single-sideband modulation aligns with these idealizations.

We start by examining the spectral purity of the output spectrum. Single-sideband modulation relies on the interferometric combination of four phase modulators, each producing multiple sidebands. By precisely tuning the phase relationships among the outputs of these modulators, the power of the sideband at the desired frequency ${\nu }_0+f$ is enhanced to exceed that of all other spectral components by several orders of magnitude. Theoretical limits of spectral purity in single-sideband modulation have been analyzed in several previous works.32^,33 For completeness, we provide a summary of the underlying principles in the Supplementary Material (Figs. S1–S3). Commercially available devices achieve side-mode suppression ratios exceeding 20 dB.

Strictly following the idealized relation $\mathrm{\Lambda }=c/f$ indicates that the required radio-frequency bandwidth is given by $B=f_{\max }-f_{\min }=c/{\mathrm{\Lambda }}_{\min }-c/{\mathrm{\Lambda }}_{\max }$. As our goal is to vary the synthetic wavelength over orders of magnitude, we can assume ${\mathrm{\Lambda }}_{\max }\gg {\mathrm{\Lambda }}_{\min }$ and consequently $B\approx c/{\mathrm{\Lambda }}_{\min }$. If we want to determine phase maps between 10 and 1250 mm synthetic wavelengths in four steps, we need the radio frequencies of 0, 0.24, 1.2, 6, and 30 GHz, i.e., a bandwidth close to 30 GHz. However, the same synthetic wavelengths can be generated using 0, 24, 28.8, 29.76, and 30 GHz as visualized in Fig. 2(a). Thus, if we determine the synthetic wavelength from $\mathrm{\Lambda }=c/|f_2-f_1|$, the required radio-frequency bandwidth can be significantly decreased.

However, this is still an idealization that holds true only if the laser frequency ${\nu }_0$ remains constant. In practice, however, lasers exhibit short-term fluctuations, defined by their linewidth, and long-term drifts, as shown in Fig. 2(b). To determine a phase map at a specific synthetic wavelength, the radio frequency is set to $f_1$ for a duration $T_{\mathrm{m}1}$ to capture the first set of interferograms. The frequency is then switched to $f_2$, acquiring a time $T_{\mathrm{s}}$, followed by capturing the second set of interferograms over $T_{\mathrm{m}2}$. The total time $T_{\mathrm{m}1}+T_{\mathrm{s}}+T_{\mathrm{m}2}$ must be much shorter than the typical timescale of long-term drift to minimize synthetic-wavelength uncertainty, which is then determined only by the laser’s linewidth. Using a single-frequency laser with a megahertz-level linewidth, one can reliably determine synthetic wavelengths up to meters, provided the total duration for capturing and switching remains on the order of seconds.

Figure 3 shows the setup of the synthetic-wavelength generator. Near-infrared light at 1560 nm wavelength with less than 1 MHz linewidth is provided by a fiber laser (Koheras, BasiK E15) and shifted in frequency by a single-sideband modulator (Exail, MXIQER LN30). The modulator is driven by two radio-frequency signals at the frequency $f$, one of them phase-shifted by $\pi /2$ with respect to the other one. They are delivered by the combination of a tunable signal generator (Anritsu, MG3696A), a hybrid coupler (Sigatek, SQ20509), and two amplifiers (Exail, DR-VE-10-MO). Across all measurements carried out in this work, a microwave-signal power of –4 dBm is maintained. This leads to an estimated modulation index of 0.7. Furthermore, three direct current (DC) voltages are fed into the modulator to optimize the side-mode suppression.

The interferometer (Fraunhofer IPM, HoloTop NX NIR) available for this study requires visible light with a wavelength of around 780 nm. Thus, 90% of the near-infrared light is converted into the visible spectral range using an optical amplifier (IPG, EAD-3-C-PM) and a frequency doubler (HC Photonics, PMC2307050016). The rest of the near-infrared light is used for characterization.

In principle, it would have been possible to use a laser emitting directly at a wavelength of 780 nm in combination with an electro-optic modulator designed for this spectral range,34 which would simplify the experimental setup. However, our approach offers several key advantages. First, single-sideband modulators designed for the telecommunications band are readily available as commercial off-the-shelf components and offer high performance. Second, by employing second-harmonic generation (SHG), the frequency shift is effectively doubled: a shift of 10 GHz in the near-infrared corresponds to a 20 GHz shift in the visible. This allows access to significantly smaller synthetic wavelengths, which are essential for high-resolution measurements. Moreover, the quadratic dependence of SHG output power on the input power improves the spectral purity of the frequency-doubled light. Specifically, a sideband suppression of 20 dB in the near-infrared translates to a suppression of 40 dB in the visible.35

To be useful for multiwavelength interferometry, the output power of the synthetic-wavelength generator should be in the milliwatt range at 780 nm wavelength. As discussed above, it needs to provide spectrally pure light. The values for $\mathrm{\Lambda }$ should be adjustable to arbitrary values over 2 orders of magnitude with switching time far below 1 s.

The output power at 780 nm wavelength is measured to be 10 mW when the pump laser emits 15 mW at 1560 nm wavelength and the erbium-doped amplifier is set to 200 mW output power.

The spectral purity is determined with an optical spectrum analyzer (OSA, AD6730). Figure 4(a) shows an exemplary near-infrared output spectrum when the modulator is driven at 10 GHz radio frequency. The side-mode suppression is better than 20 dB. As the conversion efficiency of frequency doubling scales quadratically with input power, we assume that the side-mode suppression in the visible spectral range is better than 40 dB, which is in line with recently published data.35Figure 4(b) shows the long-term stability of the normalized power of the main spectral components in the near-infrared. With constant DC bias voltages, sideband suppression remains better than 20 dB for $<2\mathrm{h}$. This level of stability was sufficient for all subsequent proof-of-concept experiments, which were conducted under these conditions.

Over longer timescales, however, a gradual degradation is observed, leading to reduced suppression performance. For applications outside the laboratory, where long-term stability is critical, active stabilization becomes essential. As demonstrated in the Supplementary Material (Fig. S4), the use of a commercially available locking system—capable of dynamically adjusting the DC bias voltages36—suppression of all unwanted spectral components below the $-24\mathrm{dB}$ level relative to the target sideband.

To demonstrate the flexibility of the synthetic-wavelength generator, we vary the radio frequency $f$ over a period of 100 s in such a way that the function $f(t)$ resembles the contour of the remains of the “Hochburg Emmendingen” near Freiburg, Germany. Simultaneously, we record the near-infrared output wavelength with a wavemeter (HighFinesse, WS7-60) and determine the corresponding frequency shift from the initial laser frequency. Figure 5 shows the near-infrared frequency shift as well as the doubled value that is expected for the visible output. We do not observe any significant difference between the target values given by $f(t)$ and the measured frequency shifts. We have reliably generated frequency shifts in the 100 MHz region up to 20 GHz, which is given by the maximum frequency of the hybrid coupler. This result shows that we can vary the synthetic wavelength to arbitrary values between 15 mm and $<1\mathrm{m}$. The 1 s long zoom in Fig. 5(b) shows that the switching takes less than $30\text{m}\mathrm{s}$, which is limited by the measurement period of the wavemeter. Figure S5 in the Supplementary Material shows that this would not be possible with a tunable diode laser.

The performance of the synthetic-wavelength generator regarding output power, spectral purity, flexibility, and switching time meets all abovementioned requirements for multiwavelength interferometry. Furthermore, we do not observe any significant changes in its performance over several hours without any active stabilization.

However, these values do not represent the limit of this approach. Erbium-doped amplifiers are capable of delivering output powers in the Watt range, which is 10 times higher than the power levels used in our experiment. At such power levels, the frequency-doubling stage could produce visible light with an output of $\sim 100\mathrm{mW}$. In addition, we are witnessing significant advancements in the performance of electro-optic devices, driven by the adoption of thin-film lithium niobate.37 This technology enables single-sideband modulators to operate at radio frequencies exceeding 100 GHz. Furthermore, by changing the DC voltages at the modulator, the output of the single-sideband modulator can be switched from $\nu +f$ to $\nu -f$. Combining both strategies, synthetic wavelengths below 1 mm are achievable. Regarding switching speed, the limit is set by the radio-frequency signal generator. Here, values below $15\mu \mathrm{s}$ are typical,38 i.e., several orders of magnitude below the limit of the wavemeter used in our experiment.

To showcase the applicability of the synthetic-wavelength generator demonstrated above, we connect the visible light output to a commercially available holographic measurement sensor (Fraunhofer IPM, HoloTop NX NIR), as illustrated in Fig. 6. This sensor incorporates a Mach–Zehnder–type interferometer. To determine the phase maps, a three-step phase-shifting method is employed.39^,40 Consequently, three interferograms are recorded for each radio frequency $f$, requiring $\sim 100\text{ms}$. This duration is determined by the motion of a piezo actuator used for phase shifting and the camera’s exposure time. For a series of four synthetic wavelengths, as shown in Fig. 2(a), all necessary interferograms are captured in less than a second. The accompanying software (Fraunhofer IPM, Holo software) handles phase reconstruction, hierarchical unwrapping, and numerical propagation. Additional details about the sensor head can be found in Ref. 41.

To show the flexibility of the synthetic-wavelength generator, we use two different samples of different sizes and materials. Our first sample is a machine-milled metallic component featuring several stepped surfaces with height differences ranging from 1 to 120 mm [see Fig. 7(a)]. To resolve these height variations unambiguously, the maximum synthetic wavelength must exceed 240 mm. Given a minimum synthetic wavelength of 15 mm, we selected synthetic wavelengths of 15, 75, and 375 mm—spanning two factor-of-five steps.

The acquisition of 12 interferograms, each with $\sim 9$ million pixels, required just 0.7 s. Figure 7(b) shows the surface reconstruction using only $\mathrm{\Lambda }=375\mathrm{mm}$. Including the 75 mm synthetic wavelength refines the reconstruction, as seen in Fig. 7(c), whereas Fig. 7(d) displays the final result using all three synthetic wavelengths. The complete numerical evaluation took 0.6 s.

In this final reconstruction, the standard deviation of the measured height values is below 100th of the smallest synthetic wavelength. Moreover, the mean values of the individual surface levels agree well with their nominal heights. Details of the statistical analysis, including histograms and a repeatability study, are provided in the Supplementary Material (Figs. S6 and S7).

A distinctive feature of our synthetic-wavelength generator is the ability to tune the spectral purity by adjusting the DC bias voltages. This allows us to experimentally assess how the level of sideband suppression influences the accuracy of height measurements. To investigate this effect, we use a single synthetic wavelength of 15 mm and focus on the 1 mm step of the machine-milled sample. First, the DC voltages are set such that all unwanted spectral components are suppressed by more than 20 dB relative to the target sideband [blue spectrum in Fig. 8(a)]. We then vary the carrier extinction ratio (CER)—i.e., the suppression of the carrier at frequency ${\nu }_0$—from 4 to 17 dB while keeping all other undesired components below the $-20\mathrm{dB}$ level [orange spectrum in Fig. 8(a)]. For each CER value, we determine the corresponding step height. As shown in Fig. 8(b), reliable height values are obtained for CER values exceeding 10 dB.

Next, we vary the suppression of the lower sideband at ${\nu }_0-f$, referred to as the sideband extinction ratio (SER), between 3 and 18 dB, again keeping all other unwanted components below $-20\mathrm{dB}$. For each SER value, the step height is measured. Figure 8(b) shows that reliable results are only obtained when the SER exceeds 15 dB. These results suggest that the system is relatively tolerant of a reduced carrier extinction ratio. However, strong suppression of the sideband with the opposite sign to the desired frequency shift is critical for accurate height determination.

The second sample is composed of multiple commercially available toy-building-blocks, made of plastic via injection molding [see Fig. 9(a)]. The total height of the object is $\sim 45\mathrm{mm}$. Furthermore, we want to maintain the smallest wavelength of 15 mm for the highest resolution. To minimize the number of interferograms, we chose 15 and 150 mm, i.e., only one factor-of-ten step. Figure 9(b) shows the surface reconstruction using only $\mathrm{\Lambda }=150\mathrm{mm}$, whereas Fig. 9(c) displays the final result using both synthetic wavelengths. Here, the total reconstruction time including capturing all interferograms and numerical evaluation was reduced to 1.0 s. Also here, the determined height values nicely agree with the expected ones.

Our experiments show that the synthetic-wavelength generator described above can be reliably applied for multiwavelength interferometry. The total reconstruction time is on the level of a second for three synthetic wavelengths. This duration can be further decreased. Applying spatial phase shifting42 rather than temporal phase shifting reduces the capture time to the time that it takes to expose the camera chip and to save the data. Commercially available devices provide this on the level below 10 ms. Thus, with improved hardware, data capture could be faster by a factor of 10.

We have proposed and demonstrated a synthetic-wavelength generator for multiwavelength interferometry based on electro-optical single-sideband modulation. It provides a variation of the synthetic wavelength to arbitrary values between 15 and 1500 mm with switching time below 30 ms. This combination of flexibility and speed paves the way for dynamically reconfigurable interferometric measurements. The actual values for the synthetic wavelength can be specifically optimized to any reflecting object without changing the hardware of the light source. We have demonstrated that surfaces with deformations larger than 100 mm can be reliably reconstructed with uncertainties of the level of 0.1 mm.

As discussed, the performance of the light source can be significantly enhanced in terms of output power, minimum achievable synthetic wavelength, and data acquisition time. Such synthetic-wavelength generators open the door to reconstructing large-area surface deformations on the meter scale with sub-$10\mu \mathrm{m}$ resolution. Moreover, with acquisition time below 10 ms, even dynamic or transient surface changes can be captured.

The concept is based on the integration of a laser source with a single-sideband modulator, an erbium-doped fiber amplifier, and a frequency doubler. Notably, all required components are readily available as commercial off-the-shelf products and are known for their robustness, making the system well-suited for use in industrial environments.

Furthermore, the latter three components have already been implemented using thin-film lithium niobate.43^–45 As a result, they can be integrated into a single chip, significantly reducing the size of the optical setup and enhancing its robustness.

We believe that the capabilities offered by this concept will greatly expand the range of applications for multiwavelength interferometry. It will not only benefit fundamental scientific research but, thanks to its robust components, also enable applications beyond the laboratory environment.

Acknowledgment. This work was supported by the German Federal Ministry of Education and Research, Research Program Quantum Systems (Grant No. 13N16774).

Biographies of the authors are not available.