Interference Undulations in Coupled-Cavity Lasers and Their Applications for Optical Measurements

3.3.2 Interference Undulations in Coupled-Cavity Lasers and Their Applications for Optical Measurements

3.3.2.1 Characterization of Coupled-Cavity Lasers

Coupled-cavity lasers with a simple configuration consisting of a Fabry–Perot laser diode and an external mirror (see Figure 3.21) have been the subject of intense investigation because of their attractive oscillation performance, which is readily controlled by external mirrors [Morikawa et al., 1971; Voumard et al., 1977; Lang and Kobayashi, 1980; Fleming and Mooradian, 1981; Acket et al., 1984]. Substitution of the external cavity with an effective mirror having a complex reflectivity is also available to characterize the FIGURE 3.19 Near-field pattern of tapered laser diode.

The wavelength is in the 1.3-µm range.

FIGURE 3.20 Scheme of tracking based on sampled servo control.

FIGURE 3.21 Schematics of laser with light feedback from external mirror.

wobbling mark

on-tracking off-tracking

on-tracking

off-tracking

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performance, although the external cavity is relatively long compared to optically switched lasers. The reflectivity is represented considering multiple reflections in the external cavity as

(3.10) with the following values:

: oscillation angular frequency : light speed in a vacuum

: external-cavity length

: reflectivity of laser facet facing external cavity : reflectivity of external mirror

The oscillation frequency is pulled into one of the eigenmodes of the laser diode owing to the strong optical gain; hence, it is represented as

(3.11) with the following values represented:

: longitudinal-mode number : length of laser diode : refractive index of laser diode

: optical angular frequency of an eigenmode

We can readily derive the interference undulation with a period of half the wavelength from the representation of the effective reflectivity, assuming a single-mode oscillation with no mode-hopping.

However, the undulation actually exhibits various increased spatial frequencies such as λ/4, λ/6, λ/8, etc., according to the external-cavity length (see Figure 3.22) [Katagiri and Hara, 1994]. Such behavior of the coupled-cavity lasers is explained by a mode selection rule according to which one of the eigenmodes with maximum reflectivity, corresponding to the minimum threshold, is selected as the oscillation mode.

The substitution of the external cavity with the effective reflectivity is insufficient for discussing asymmetric sawtooth undulations. We must take into account the time-dependent field component of

FIGURE 3.22 Interference undulations by light feedback from external mirror located several millimeters away from the laser facet.

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the light in the external-cavity laser. Assuming the stationary condition, this consideration results in the oscillation frequency change being dependent on the external-cavity length as given by

(3.12)

where η is the coupling efficiency of the optical feedback from the external mirror to the laser diode and L0is the cavity length of the laser diode. Assuming that the equation gives a unique solution for Ω, the frequency change Ω−ωm corresponds to the phase change in the interference undulations at a period of half the wavelength. This phase change generates the sawtooth undulation curves [Spano et al., 1984;

Olesen et al., 1986].

3.3.2.2 Displacement Detection Principle [Katagiri and Itaoh, 1998]

The measurement of a small displacement based on the coherence of a single-mode laser light has features of both high resolution and sensitivity; hence, it has been widely performed in various fields. However, conventional measurement schemes need stable, narrow-linewidth light sources consisting of relatively large and expensive optical components; thus, they are inadequate for general use. The coupled-cavity lasers in a simple configuration exhibiting the interference performance are generally expected to be used for this sort of displacement measurement. The problem is how to remove the oscillation instability at the same time as the displacement of the external cavity is induced for the measurement.

The paradoxical problem is eliminated by using a coupled-cavity laser stabilized by a mechanical negative-feedback loop circuit. In the loop the position of the laser diode is controlled along the axial direction by using a high-resolution actuator to cancel the transient displacement of the external mirror while the light output is monitored (see Figure 3.23). Consider the state of the laser on the sawtooth undulation curve with linear portions in every interval of half the wavelength. The initial state is defined at the halfway point on the curve, where the light output corresponds to P0. A small temporal displace-ment ∆h of the external mirror is detected by a photodiode as a differential signal P − P0. This signal is filtered, amplified, and added to the control signal of the actuator to cancel the differential signal through an integrator. Consequently, a negative-feedback loop is formed to maintain a constant external-cavity length. The external-cavity laser controlled in this loop is thereby stabilized. We can know both the transient and the total displacement from the initial position by evaluating, respectively, the differential signal and the integral of the signals. The absolute displacement performance of the employed actuator is readily calibrated and maintained over a long period of time. This promises an accurate displacement measurement, independent of the oscillation performance of the employed laser diode and its driving condition.

FIGURE 3.23 Schematic diagram for explaining a coupled-cavity laser displacement sensor (CCL sensor).

Ω=ωm+η −R R τΩ R

c

(1 ) nL sin ( )

2 2

3

2 0

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3.3.2.3 Displacement Sensor Performance

A small displacement sensor fabricated for use in practical implementations consists of a monolithic laser diode-photodiode device, a TaF3 ball lens 600 µm in diameter, and a piezoelectric-transducer actuator, all of which are assembled on a substrate (see Figure 3.24). The InP-based device emits a laser beam at 1.3 µm. The sensitivity of such a displacement sensor is readily estimated numerically by a slight vibration of the external mirror. The vibration generates a modulated signal whose amplitude is linearly related to the vibration amplitude. The corresponding frequency spectrum exhibits a sharp peak at the frequency of the vibration readily discriminated from broadband noises (see Figure 3.25). Maximum sensitivity for detecting displacement is estimated when the signal is equal to the noise floor. This minimal vibration is usually too small to detect and so is estimated at larger vibration regions based on the linear relationship. A typical maximum sensitivity is 0.02 nm/ at 200 Hz for an existing sensor.

FIGURE 3.24 Schematic illustration of a CCL sensor. A monolithic LD-PD with a ball lens is employed.

FIGURE 3.25 Spatial resolution measurement for a CCL sensor. The vibration amplitude is calibrated at 200 Hz.

tube-shaped piezoelectric xyz scanner

sensor module micrometer head

cantilever tip

external mirror LD-PD

ball lens

0 2 4 6

vibration amplitude (nm) 0

0.2 0.4 0.6

S (mV)

0 100 200 300 400 500

−20

−40

−60

−80

−100

power density (dBm/Hz)

frequency (Hz) (a)

(b) Signal

Hz

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An effective sensitivity level is also defined at the point where the signal equals the total noise power given by an appropriate integral range. A typical value is given as 0.8 nm in the range 0 to 500 Hz, which covers most micro-opto-mechatronic applications. This level of sensitivity substantially shows the spatial resolution of the displacement sensors.

3.3.2.4 Examples of Practical Implementation

3.3.2.4.1 Application to Scanning Probe Microscope [Katagiri and Hara, 1998]

Scanning probe microscopes (SPMs), including atomic-force microscopes using a cantilever with a sharp tip, are a powerful tool for imaging surface topography. Because the surfaces of interest have nanometer-scale structures, measurement of the resulting extremely small distortions of the cantilever is required.

The displacement sensor, which uses the coupled-cavity lasers, has shown potential for taking such measurements. Figure 3.26 shows a schematic diagram of an SPM system equipped with a detecting head with a displacement sensor. As the head approaches a sample on a mechanical stage, the tip makes contacts with the sample surface. Further displacement of the head in the same direction generates a distortion of the cantilever. Because the cantilever works together with an external mirror of the displacement sensor, the distortion is translated to the photodiode output variance. The absolute distortion of the cantilever, which is derived from the detected signal, corresponds to the load of the tip to the surface. The numerical evaluation of the load is based on a calculation using Hooke’s law with values of stiffness of the cantilever.

Hence, an optimum load is adjustable according to the hardness of the sample. Once an optimum load is determined, the controller of the displacement sensor works to cancel a temporal distortion of the cantilever, while the external-cavity length remains constant by working a tube actuator along the z-axis.

While the head is raster-scanned in the x–y plane by the tube actuator, we obtain images of the sample surface using the control signal in the z-axis direction.

A typical example is shown in Figure 3.27 for an optical-disk surface. The measured image faithfully reflects the surface with shallow tracking grooves 0.1 µm in depth and 1.6 µm in spacing.

3.3.2.4.2 Application to Dynamic Detection of Small Forces [Katagiri and Itaoh, 1998]

Dynamic measurement of small forces is of great importance for mechanical systems. In the field of high-precision information instruments, including hard-disk and optical-disk systems that must be miniaturized, the forces of interest become smaller as the system’s dimensions are reduced. For optimum design of the systems such forces must be temporally measured under operating conditions. Such small forces can be measured from the distortion of a cantilever as described above. Stiff cantilevers with a high resonance frequency are needed for measuring such forces dynamically over a wide frequency range. The distortion is obviously small, so a highly sensitive distortion measurement is essential. The displacement sensor in a coupled-cavity laser configuration is the most suitable because it exerts negligible mechanical influence on the cantilever and enables highly sensitive detection.

FIGURE 3.26 Configuration of scanning probe microscope with CCL displacement sensor.

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A typical example is presented using a simple system of measuring the dynamic friction force to verify the force-measurement scheme. The system has a cantilever with a stiffness of 500 mN µm⁻¹ (see Figure 3.28). This cantilever provides a resonant frequency of around 3 kHz and a minimum detectable force of 0.4 mN. The force-detection sensor is readily calibrated under the constant-load conditions. Figure 3.29 shows a temporal trace of the friction force that occurs when a small object is dragged over a rough surface. The trace conclusively reveals the transition process through the maximum friction-force state to the kinetic-force state. The measured friction values agree with those measured by conventional techniques with an inclined plane.

3.3.3 Mode-Locked Lasers and Their Tuning Schemes by Micro-Mechanism