The heterodyne two-beam interferometer has been proven to be the optimal solution for laser-Doppler vibrometry (LDV) regarding accuracy and signal robustness. The theoretical resolution limit for a two-beam interferometer of laser class 3R (up to 5 mW visible measurement-light) is in the regime of a few femtometer per square-root Hertz and well suited to study vibrations in microstructures. However, some new applications of radio-frequency microelectromechanical (RF-MEM) resonators, nanostructures, and surface-nano-defect detection require resolutions beyond that limit. The resolution depends only on the photodetector noise and the sensor sensitivity to specimen displacements. The noise is already defined in present systems by the quantum nature of light for a properly designed optical sensor and more light would lead to an inacceptable influence like heating of the tiny specimen. Noise can only be improved by squeezed-light techniques which require a negligible loss of measurement light which is impossible to realize for almost all technical measurement tasks. Thus, improving the sensitivity is the only path which could make attometer laser vibrometry possible. Decreasing the measurement wavelength would increase the sensitivity but would also increase the photon shot noise. In this paper, we discuss an approach to increase the sensitivity by assembling an additional mirror between interferometer and specimen to form an optical cavity. A detailed theoretical analysis of this setup is presented and we derive the resolution limit, discuss the main contributions to the uncertainty budget, and show a first experiment proving the sensitivity and resolution improvement of our approach.

## I. INTRODUCTION

Resolutions of laser-Doppler vibrometers^{1–3} in the attometer regime are desirable to study vibrations at Gigahertz frequencies in structures enabled by the progress in nanotechnology.^{4–6} Attometer resolution is below the theoretical limit for a two-beam interferometer which is in the regime of a few femtometer per square-root Hertz.^{4,7} It has been discussed in a paper at the 10th International Conference on Vibration Measurements by Laser and Noncontact Techniques^{8} that it is theoretically possible to realize laser-Doppler vibrometers with sub-femtometer resolution by using multiple-reflections interferometry. In addition, it has been pointed out that such techniques are successfully applied in gravitational-wave detectors and in cavity opto-mechanics. However, the question if these results can be employed to study vibrations in micro- or nanostructures which do not form an own optical cavity is still unanswered. Especially the effect of the arbitrary reflectivity of a randomly chosen specimen makes it difficult to design a flexible optical sensor for detecting attometer vibration amplitudes.

Multiple-reflections interferometry employs a Fabry-Perot interferometer with well-defined mirror reflectivity.^{9,10} This is obviously impossible to achieve if a sensor for an arbitrary specimen is designed. Thus, a solution for this task is either limited to a fixed specimen reflectivity or would require a mirror with flexible reflectivity to be adaptable to the requirements defined by the object under test. Consequently, the questions to be answered are (1) which applications in nanotechnology guarantee a well-defined surface reflectivity and (2) how to realize a mirror of flexible reflectivity that can be adapted very rapidly. The applications where a well-defined reflectivity is guaranteed are, for example, the detection of defects on hard disks or silicon wafers.^{11–13} Most applications as, for example, MEM-resonators,^{14,15} will require a fast adaption of the reflectivity of the mirror that forms the optical cavity with the surface under test. We name the additional mirror between interferometer and specimen in the following open-optical-resonator (OOR). The reflectivity adaption can, for example, be achieved for a fixed wavelength if the mirror would be a Fabry-Perot-etalon with a moveable distance of 2 highly reflective layers. Shifting the wavelength with respect to a narrow-bandwidth reflective layer is a second possible solution. However, the design of such an adaptable mirror is not addressed in this paper. Here, we concentrate on the physical fundamentals of the proposed solution.^{16}

In this paper, we present the theoretical aspects of a new type of laser-Doppler vibrometer which employs an OOR between a classical two-beam interferometer and the measurement spot on the specimen. First we propose a setup to accomplish attometer vibrometry. Next we derive a theory and a mathematical model to simulate the behavior of such an optical system. With our model, we investigate the behavior and we show how such a system can be designed optimally with respect to achieve the best vibration-amplitude resolution. Our model enables us also to investigate the ultimately achievable resolution and we prove that attometer resolution is possible. We compare our new technique with the existing two-beam-interferometric, laser-Doppler vibrometry. To do so we introduce a gain factor *G _{S}* for the sensitivity gain of the OOR interferometer with respect to the classical two-beam interferometer, an attenuation factor

*G*for the received light power, and a gain factor

_{L}*G*for the resolution. In addition, we employ our model to investigate the influence of the optical-sensitivity amplification with a nonlinear behavior on the uncertainty budget for the vibration-amplitude measurement. We show also the first experiment demonstrating a sensitivity gain factor

_{R}*G*> 1.

_{S}## II. OPTICAL SETUP FOR ATTOMETER LASER VIBROMETRY

The mirror surface of the OOR objective lens forms an optical cavity together with the surface of the vibrating specimen with arbitrary reflectivity. The reflecting OOR surface has a concave spherical shape to allow a coaxial back-reflection of the measurement beam focused on the surface. The OOR is integrated in a microscope-objective lens as it is depicted in Fig. 1. The light field of the interference of the light reflected at the additional mirror and the light reflected at the specimen surface has a detectable phase depending non-linearly on the reflectivity of the OOR, the complex refraction index of the specimen surface, and the distance between the OOR and the specimen surface. The deviation of the phase at a certain operation point with respect to a change in distance defines the phase sensitivity for specimen displacements. The selected working distance is related to the operation point and needs to be stabilized. The laser beam impinging the OOR-objective lens is the measurement beam of a heterodyne interferometer. As it has been shown in Ref. 8, the heterodyne interferometer is the best solution to measure the shot-noise limited phase and the amplitude of the received light wave. Thus, the heterodyne interferometer is employed as means to derive phase and amplitude of the light returning from the optical cavity formed by the OOR and the specimen surface.

The surface of the OOR is the exit face of the microscope objective. The microscope objective has been designed by us such that the wavefront at the concave exit face is already ball shaped and such that the ball-shaped OOR surface does not generate additional refraction. Thus, the OOR surface introduces only partial reflection and transmission to the beam. Assuming a flat specimen surface, the specimen and the OOR form together a semispherical optical resonator. The microscope objective employs a reflection coating and adjustment screws to align the wavefront to coincide with the shape of the exit surface at the position of that surface. We have designed and realized the special OOR objective lens with a focal length of 20 mm (corresponds to 10× magnification for a 200 mm tube length) which we have assembled on the Polytec MSA-500 (prototype system). Figure 2 right shows a photo of the MSA-500 with the OOR 10×-objective. The OOR objective lens has been designed in ZEMAX® such that there is no refraction at the exit face. In addition, a dielectric layer with a reflectivity of *R _{ref}* = 0.88 at 633 nm wavelength has been coated to the exit face of the OOR.

The phase and amplitude of the light returning from the optical cavity depends strongly on the distance *s* of the OOR and the specimen surface. Thus, we have integrated a piezoelectrically driven actuator to adjust the distance between OOR and the specimen. An automated optimization procedure has adjusted the signal level to a minimum to achieve maximum phase gain. The optimization algorithm has been implemented as Basic macro in the Polytec Vibsoft® software. Our solution is capable to keep that distance constant at the optimal sensitivity for approximately 100 s in a quite laboratory environment on an optical table. This simple solution has been sufficient for our first experiments, but the implementation of a robust control loop with a cutoff frequency at a few 100 Hz is a necessary future task to compensate all present disturbances on the vibration-isolated optical table without interruptions. Of course, this would make our setup insensitive against vibrations at frequencies within the bandwidth of the control loop. However, vibrations which require attometer resolutions are rather high (≫1 kHz) while a control loop bandwidth below 100 Hz is very likely sufficient if the setup is placed on a vibration-isolated table.

## III. THEORY OF THE OOR INTERFEROMETER

### A. Sensitivity gain of the OOR interferometer

The electrical field of the light reflected by the optical cavity can be described by the equation

resulting from optical resonator theory.^{9,10} Here $\delta = 4 \pi \u2009 s t \lambda \u2212n\u20092\pi = 4 \pi \u2009 \Delta s t \lambda $ is the phase resulting from one cycle through the resonator, *R _{ref}* is the reflectivity of the OOR mirror,

*R*is the reflectivity of the specimen, and

_{Sample}*E*

_{0}is the amplitude of the impinging electrical wave. Polarization is not considered because losses, due to a depolarizing specimen surface, are considered in

*R*. In addition,

_{Sample}*λ*is the wavelength of the measurement light and

*n*is an integer variable that counts how often the wavelength

*λ*fits into the cavity. The phase of the back-reflected beam results from (1) to

Assuming small displacement amplitudes (≪1 nm) as well as an adjustment of the resonator on the maximum transmission where the phase deviation is maximal and by referencing to the sensitivity of the two-beam interferometer $ \varphi 2 B I = 4 \pi \lambda \Delta s t $, Equation (2) yields

With the sensitivity amplification

for the phase of the reflected light that can be evaluated with a standard heterodyne interferometer and laser-Doppler vibrometer phase decoders. Obviously the reflectivity of the OOR with respect to the specimen reflectivity has a critical influence. The phase gain *G _{S}* at the operation point where the distance is a multiple of the wavelength is not defined for

*R*=

_{ref}*R*. The phase has a point of discontinuity at this point which leads to an undefined sensitivity gain at this condition. Fig. 3 demonstrates the sensitivity gain

_{Sample}*G*for a fixed OOR reflectivity of

_{S}*R*= 0.98. Obviously, a high gain ≫100 can be achieved if the sample reflectivity is only minimally smaller than the OOR reflectivity.

_{ref}### B. Light-power attenuation of the OOR interferometer

The reflected light power at the optical resonance *P _{OORM}* can be calculated with the following equation:

Here, *P*_{0} is the light power of the interferometer impinging the OOR. Obviously, no light is returned to the interferometer if *R _{ref}* =

*R*for

_{Sample}*δ*= 0. The attenuation of the light intensity received by the interferometer can be defined by

with *P _{m}* =

*R*

_{Sample}P_{0}the reflected light power at the sample for the two-beam interferometer. Figure 4 demonstrates the dependence of

*G*on the reflectivity of the sample of an OOR reflectivity of

_{L}*R*= 0.98. It can be seen that the returning light is attenuated if the sample reflectivity is slightly smaller than the OOR reflectivity but the returning light power is magnified for substantially smaller sample reflectivity. Obviously an optimal ratio between

_{ref}*R*and

_{ref}*R*exists.

_{Sample}### C. Resolution gain of the OOR interferometer

Considering the shot noise generated at the interferometer detector

where *P _{I}* is the power of the reference beam of the interferometer and using Equation (3) one can calculate with respect to Equation 3 in Ref. 8

and finally corresponding to Equation 6 in Ref. 8

Equation (10) can be used to define the resolution gain *G _{R}* together with the resolution for the two-beam interferometer $\Delta s= \lambda 2 \pi \epsilon h \nu B P m + P I \eta P m P I $ to

Fig. 5 demonstrates the dependence of *G _{R}* on the reflectivity of the sample for an OOR reflectivity of 0.98. Obviously, the resolution can be improved substantially for a good match of

*R*and

_{ref}*R*. Many micro- and nanosystems are made of silicon. 532 nm has been identified as a well-suited wavelength for a heterodyne-interferometric microscope because a small laser focus is possible as well as a shot-noise-limited detection. Therefore, Fig. 6 demonstrates the situation for silicon with a reflectivity of 0.38 at 532 nm.

_{Sample}Even on plain silicon the resolution can be improved by a factor of more than 1.8. Considering a specimen with a reflectivity of *R _{Sample}* = 0.95 investigated with an OOR reflectivity of

*R*= 0.98 the achievable resolution would result with respect to Equation (10) to (76.6) attometer per square-root Hertz. Thus, attometer resolution is possible with our approach for highly reflecting surfaces. However, even for silicon with

_{OORM}*R*= 0.38 it would be possible with an OOR reflectivity of

_{Sample}*R*= 0.4 to achieve with

_{OORM}*P*= 10 mW reference power and

_{I}*P*

_{0}= 5 mW measurement light power to achieve 823 attometer per square-root Hertz resolution.

## IV. EXPERIMENTS WITH THE OOR INTERFEROMETER

Our first experiment showed clearly that multiple reflections can amplify the phase so that a 100 pm vibration amplitude appears as a 634 pm amplitude in the vibration spectrum. This corresponds to a sensitivity gain of *G _{S}* = 6.34. A reflectivity of 0.835 has been selected for the specimen in this experiment. We have noticed that the angle alignment of the OOR objective with respect to the specimen and the parallel incidence of the laser beam with respect to the optical axis of the OOR objective lens are critical for the maximal achievable gain. However, it is obvious that phase amplification and light-power attenuation is present with correct alignment with our setup. Therefore, the more challenging experimental proof is a resolution gain

*G*> 1. We have excited the piezo actuator at 50 kHz. Fig. 7 shows the LDV measurement of the vibration spectrum at 50 kHz captured with a conventional bright-field microscope objective. Hundred amplitude averages have been applied to achieve a noise level at approximately 550 fm (35 fm/$ Hz $). The measurement demonstrates that the piezo vibrates with an amplitude of approximately 1 pm. Actually we have adjusted the amplitude to 1 pm and made sure that it remains unaltered within <1% during the experiment. The measurement-light power was reduced by an attenuator in the interferometer to approximately 100

_{R}*μ*W.

Fig. 8 shows the single-shot measurement of the piezo-actuator vibration at 50 kHz. The red bars show the spectrum for a measurement with a conventional microscope objective. The noise level is too high to allow measuring the piezo-vibration amplitude without averaging. The measurement with OOR objective was calibrated and corrected with respect to the reference measurement. The noise level achieved with the OOR objective is at 56 fm for 250 Hz (corresponding to 3.5 fm/$ Hz $) without averaging. Thus, a resolution gain of *G _{R}* = 10 was achieved for this measurement. The level of the measurement performed with the OOR-objective is below the quantum shot noise level of the conventional microscope-objective measurement for 100

*μ*W measurement-light power and is marked by a green line at 67 fm (4.2 fm/$ Hz $). In this case 20 complex averages of the spectrum decreases the noise level to 780 attometer for 1 Hz resolution bandwidth. Thus, attometer resolution is feasible for reasonable measurement conditions.

## V. EFFECTS DUE TO MISALIGNMENT

The theory employed so far has been based on the classical Fabry-Perot resonator theory. In reality, the surface has a certain curvature in the measurement spot which will influence the behavior of the OOR system. In most cases, the assumption of a flat surface within the measurement spot is valid and a semispherical resonator can be assumed. The semispherical resonator allows also higher order transverse modes compared to the case above. For example, a mode with two roundtrips is excited if the beam may impinge the OOR objective out of its center. This leads to twice the number of resonances as it is shown in the simulation in Fig. 9. In our first experiments we saw a strong dependency of the returned light power on the alignment due to these effects. The sensitivity is substantially lower if two laser-beam roundtrips in the resonator are adjusted. The reflection can be measured by measuring the analog signal-strength signal of the Polytec laser-Doppler vibrometer OFV-551 employed in the MSA-500 when the focus is changed with the piezo stage integrated in the MSA-500. Figure 10 shows the signal strength measured with an oscilloscope while the piezo stage is driven with a 0.2 Hz triangle signal with an amplitude of 0.2 V. The movement of the OOR visualizes the resonance peaks when the stage is either moved up or down. Thus the adjustment can be optimized and analyzed.

## VI. UNCERTAINTY CONTRIBUTIONS OF THE OOR TECHNIQUE

Accurate measurements are only possible with our proposed system if the OOR objective is perfectly aligned. Every misalignment leads to a change of the sensitivity gain *G _{S}* which directly influences the measurement result. Therefore, the relative error

*E*of

_{RGS}*G*leads directly to an uncertainty contribution for the vibration amplitude measurement

_{S}*E*with

_{RAS}The dependency of the sensitivity on the operation point, on the adjustment of a beam incidence parallel to the specimen-surface normal, on the specimen reflectivity, and on the specimen scatter properties is the major contribution to the measurement uncertainty of the proposed method. Since it will be impossible to know all these influences on a theoretical basis, it will be necessary to determine the sensitivity gain *G _{S}* experimentally for the current measurement situation. The method that we have employed to measure the resonance peaks in the reflected light power can also be evaluated with respect to the phase. Thus, an automated procedure to determine

*G*at a specific measurement point is required if the specimen properties are not absolutely uniform. Further uncertainty contributions may become relevant if a proper operation point control and sensitivity-gain determination are implemented. One obvious problem is the distortion due to the nonlinearity of the displacement-phase response. Figure 11 shows the theoretic response for a vibration amplitude of 1 nm,

_{S}*R*= 0.98, and

_{ref}*R*= 0.95. The sensitivity gain (

_{Sample}*G*= 36.35) has been utilized to correct the amplitude but distortion has reduced the measured amplitude by more than a factor of 2. The error due to distortion is in the range of 1% for an amplitude of 100 pm.

_{S}## VII. CONCLUSIONS AND OUTLOOK

We have proven that it is possible to magnify the sensitivity and to improve the resolution of a two-beam interferometer by employing an OOR lens. In addition, we have demonstrated the theory for the OOR interferometer introduced in this paper. The phase magnification and the resolution improvement have been experimentally verified. We have achieved noise levels below the quantum limit of the two-beam interferometer in our experiments. Our experiments show that attometer resolution is achieved with 20 averages for 1 Hz resolution bandwidth with our current setup. In addition, our theoretical findings show that our approach makes possible in general attometer vibration-amplitude resolution per square-root Hertz for specimen surfaces with reflectivity of 0.38 (corresponds to silicon) or higher without averaging if the alignment is perfect and the interferometric detection is quantum limited. For example, a resolution of <80 attometer per square-root Hertz can be achieved on a specimen with a surface reflectivity of 0.95. Our findings do also reveal that the optical axis of the OOR lens and the impinging laser beam have to be aligned carefully parallel to the surface normal of the specimen surface in the measurement spot. Especially, OOR arrangements with a high sensitivity gain require very accurate alignments. Misalignment leads to intermediate peaks in the reflected light power signal. Thus, the OOR should be aligned to obtain peaks with a maximum reduction of the reflected light power and to avoid intermediate peaks. We have also discussed the major uncertainty contributions. The most critical contribution is the variation of the sensitivity gain in dependence of many different influences because the relative error of the gain equals to the relative error of the determined vibration amplitude.

The most important future goal is the experimental demonstration of attometer resolution per square-root Hertz without averaging on a vibrating microstructure. In addition, we will explore the stabilization of the operation point with a robust electronic control loop.