1. Introduction

Electro-optic modulators (EOM) are essential components in ultra-precision optical systems that require high purity phase modulation of the incident light field [1]. Some prominent applications that employ EOMs to phase modulate optical fields are; laser frequency stabilization [1–3], trace gas detection using frequency modulation (FM) spectroscopy [4], gravity wave detection [5, 6] and signal processing with fiber optic gyroscopes [7]. In these applications and others, any modulation of the light field’s amplitude at the modulation frequency is an unwanted noise source that can significantly limit the overall performance of the system. Unfortunately, it is well-known that all EOMs used for optical beam phase modulation generate unwanted residual amplitude modulation (RAM) that ultimately limits the sensitivity and the resolution possible [8–13].

RAM has been observed and investigated for more than 20 years with the aim of understanding it sufficiently, to reduce or remove it. Through the body of work that has been performed, RAM has been found to depend on many intertwined factors dependent upon the EOM medium and its environment, the optical arrangement and the properties of the optical and modulating fields. Some of the known mechanisms through which RAM is produced are Fabry-Perot etalon effects due to reflections from the facets of the phase modulator crystal and other reflective surfaces [2, 10, 11], beam polarization effects [12], thermal effects in the EOM medium [10–12], scattering by impurities and other defects [10, 14], spatial inhomogeneities of the electric field inside the medium [10, 11] as well as through piezoelectrically induced vibrations inside the crystal [11]. Recent studies [12, 13] have shown that RAM can be suppressed to the 10⁻⁵ level through adequate control of beam and medium properties. Although a practical and useful outcome, our understanding of RAM is not complete and observations such as its irreproducible temporal behavior remain unexplained and prevent suppression of this noise to even lower levels.

Recently, it was shown that photo-induced refractive index changes in magnesium-oxide-doped lithium niobate (MgO:LiNbO₃) EOMs produced intensity dependent RAM even at low incident field light irradiances (10-100 mW/mm²) [14]. Here, we present results of an experimental investigation into the variation of the phase and magnitude of the observed RAM and its dependence on incident intensity. The results show that as the intensity increases so does the variance in RAM magnitude while the RAM phase becomes less well defined, with the combination of effects producing temporal variation in RAM. To qualitatively explain this we present a phenomenological model that is based on the gradual formation and erasure of intensity dependent photorefractive scattering centers randomly distributed throughout the MgO:LiNbO₃ electro-optic phase modulator.

In this paper we present for the first time a simple optical technique for reducing the photorefractive RAM produced in EOMs. We have found that the light induced refractive index changes in the phase modulator crystal can be erased by uniformly illuminating the EOM crystal with a second intense optical beam, which decreases the overall RAM. This approach is of much interest to those working with EOMs in low noise applications, seeking easily applicable means to reduce the level of RAM produced.

2. Photorefractive effect in Lithium Niobate

Lithium Niobate (LiNbO₃) is a popular choice for EOM phase modulators due to its large electro-optic coefficients [15, 16]. However, pure LiNbO₃ has proven to have a high photorefractive sensitivity, which makes it impractical for phase modulation of laser beams with intensities in the 50 mW/mm² range (e.g. 10 mW laser beam in a 0.25 mm beam radius at 514.7 nm) [17]. Photorefraction, also known as ‘optical damage’, was first reported by Ashkin et al. [17] and is generally regarded as visually observable distortion of the spatial distribution of the beam transmitted through the modulator due to light induced refractive index changes. It is caused by the photoexcitation of electrons and holes [18] from multiple trap states in the medium’s band-gap that originate from numerous impurities, dopants and defects [19]. Once excited, these charges migrate throughout the medium with the resulting charge redistribution producing a space charge field, which creates an inhomogeneous refractive index change through the Pockels effect.

Since the photorefractive effect is a serious limitation for lithium niobate phase modulators [17], manufacturers have investigated the use of various dopants (Zn, Mg, etc) in order to control the nonlinear optical properties of the medium, as it has been shown that dopants alter the photoconductivity thus reducing the sensitivity to photorefractive changes [20]. Consequently, it has been found that 5% MgO:LiNbO₃ crystals exhibit a high ‘optical damage threshold’ when compared to other derivatives of lithium niobate, due to a 100 fold increase in photoconductivity upon illumination in the visible range [20]. This has led to the misconception that 5% MgO:LiNbO₃ does not exhibit any photorefractive effect, as no visual evidence of distortion of the spatial distribution of the incident field is usually observed. Recently, however, it has been shown that the weak photorefraction that is displayed by 5% MgO:LiNbO₃ when illuminated at low intensities in the visible, is observable [21] and it produces RAM when the medium is used as a phase modulator [14].

Ideally, when a time varying electric modulating field at frequency$\Omega$, is applied across an electro-optic medium to phase modulate a laser beam, sidebands of equal magnitude are produced. However, in practice, the relative transmission of these different frequency components through the electro-optic medium will not be the same [3, 9, 10] and as a result, the total transmitted field will in general contain both phase modulated (PM) and amplitude modulated (AM) contributions. As described previously, the magnitude of the AM sideband when compared to the PM sideband magnitude will determine the level of RAM observed, $R_{AM}$and is given by the difference in sideband transmission through the medium [14]:

Here,$T(\omega {}_0+\Omega )$ and $T(\omega {\text{}}_0-\Omega )$represent the intensity transmission functions of the upper sideband and lower sideband, respectively, with the optical carrier frequency given by$\omega {\text{}}_0$. As Eq. (1) shows, whenever an imbalance exists in the transmission of the sidebands of a phase modulated laser field [9, 10] RAM will be produced. In the MgO:LiNbO₃ medium used here, intensity dependent RAM will arise since the sidebands have different optical frequencies and are spatially offset [9], leading to differing levels of photorefractively induced scatter and absorption, producing an imbalance in sideband transmission.

3. Experimental observations of photorefractive RAM

To analyze the temporal variation of RAM and its dependence on intensity, both the magnitude and phase of the RAM produced were measured. The measurements were performed with a continuous wave 532 nm Nd:YAG laser that was phase modulated at 800 kHz with a MgO-doped lithium niobate EOM to a modulation depth β that did not exceed 0.4 rad. For β << 1 we can assume that the phase modulated field consists of a carrier and the two first order sidebands. The residual amplitude modulation impressed upon the optical field was measured with a phase-sensitive detector. The maximum output power of the laser was 20 mW with a beam waist of 0.16 mm positioned at the centre of the EOM crystal. The electro-optic crystal was passively thermally stabilized. The laser output was a single TEM₀₀ mode, which was linearly polarized in a vertical plane with a polarization ratio > 100:1. A Glan-Thompson polarizer was used to ensure the linearly polarized incident field had a purity of better than 10000:1. The optic axis of the MgO:LiNbO₃ crystal was kept parallel to the laser polarization direction in order to eliminate RAM due to polarization misalignment. The most dominant source of RAM in the EOM is the etalon effect, which can be very large (~10⁻²) and must be minimized with anti-reflection (AR) coatings and by adjusting the angle of incidence of the laser, to ensure it does not completely overshadow the much weaker intensity dependent photorefractive RAM (~10⁻⁴-10⁻⁵). This condition was satisfied by angling the AR coated crystal end faces to the laser beam by ~1.6⁰ that yielded a minimum RAM level ~10⁻⁵.

The magnitude of the RAM was determined from the measured ratio of the AC to DC intensity components across a low noise silicon photodetector. The AC component was measured with a lock-in-amplifier. The lock-in signals can be expressed by a magnitude R and phase$\varphi$, which are given by:

Figure 1 shows the measured RAM levels and its associated phase angle$\varphi$, at five different input intensities. At each intensity, 5400 measurements were recorded at a sampling rate of 1 measurement per second for a duration of 90 minutes. The RAM measured at the highest intensity was observed to have a magnitude and a standard deviation (indicated by the error bars in Fig. 1(b)) that was almost 10 times higher than that at the lowest intensity. As shown in Fig. 1(b) the average magnitude of the RAM increases more or less linearly with the intensity. At all intensities, the maximum level of RAM was observed when it was in phase with the reference phase angle. However, an interesting observation was that the distribution of measured RAM phases increased with intensity as shown in Fig. 1(a). At the highest intensity the full width half maximum (FWHM) of the phases distributed around the maxima was 33°, which fell to only 9° for the lowest intensity. In addition, as shown in Fig. 1(a), at the highest intensity significantly more measurements revealed phases outside of the FWHM (50%) than were observed at lowest intensity (25%). Therefore, as the intensity increases the phase of the observed RAM becomes less well defined. This spreading of the phase at higher intensities manifests itself at the detector as an increase in the fluctuations of the RAM signal about the average value.

 

Fig. 1 (a). Measured intensity dependent RAM as a function of the phase angle$\varphi$for five different input intensities over a time period of 90 minutes and 1 (b) the average RAM values for the five intensities. The FWHM of the distributions in Fig. 1(a) are 9°, 7°, 17°, 35° and 33° in the increasing order of input intensity.

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At intensities between 10 – 100 mW/mm², the formation and erasure of photorefractive changes is slow and for lithium niobate and its derivatives (e.g. MgO:LiNbO₃, Fe:LiNbO₃), these changes typically occur over time scales of hundreds of milliseconds or much longer [18]. This is because the photorefractive time constants vary inversely with irradiance, since the time it takes for the light to rearrange charges in the crystal depends on the light intensity and also on how fast the charges migrate through the crystal. Light at lower intensities takes longer to photoexcite the number of charges required to produce the same space charge field than more intense light [22]. This is consistent with our observations, where the peak values of RAM are 2 × 10⁻⁵ and 3 × 10⁻⁵ at 12 mW/mm² for time periods of 30 minutes and 90 minutes respectively, while at the higher intensity (190 mW/mm²) these values are the same (2 × 10⁻⁴) for both time periods. Hence, at the low intensity, it takes 1 to 2 hours to reach a quasi-equilibrium state, whereas at higher intensities the response time is much less.

The temporal characteristics of RAM are complex as it depends on many factors, which include intensity dependent and intensity independent contributions. Here we propose a model, based on beam fanning [23], to explain some of our observations of the intensity dependent RAM. The MgO:LiNbO₃ medium used here is anisotropic and photorefractively self-defocusing, which means that the refractive index is lower in regions where the optical field is most intense. This acts to continually spread the TEM₀₀ Gaussian mode light field away from the beam axis. In addition, it is well established that defects scatter light [22] within the illuminated volume with the scatter amplified by photorefractive two wave mixing with the incident field [24]. These scatter sites are randomly distributed throughout the 40 mm length of the optically thick medium. Since the scatter is relatively weak, it can take many hours for the self-defocusing to reach steady state [25], and prior to that the amount of scatter at any site within the interaction volume continues to vary due to continual writing and erasing of photorefractive index changes as depicted in Fig. 2. For example, the scatter at site A grows due to photorefractive amplification, which then impacts on the scatter ‘downstream’ at site B, where less scatter is produced due to pump depletion, while at site C scatter increases due to the greater levels of photorefractive scatter emanating from site A. In essence, this self-defocusing process acts like scatter centers randomly distributed throughout the illuminated volume with scattering powers that continually evolve. Evidence that this occurs in photorefractive lithium niobate is provided by numerous light transmission observations that describe spatial light distributions that continue to evolve through beam fanning over time scales of many hours at low light intensities [23, 25].

 

Fig. 2 The propagation of an optical beam through a self-defocusing photorefractive medium with numerous scatterers, where the beam brightness is proportional to the intensity.

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It has been reported [26] that the distribution of defects in the crystal can increase or decrease the photorefraction in MgO:LiNbO₃. Here a numerical simulation was performed to model the impact of photorefractive scattering centers randomly distributed throughout the medium, on the level of RAM. The photorefractive index change at each site consisted of two components; the self-defocusing resulting from the TEM₀₀ profile of the Gaussian incident field [27] and the randomly varying photorefractively amplified scatter as given by:

Here n_e is the extraordinary refractive index of the medium, w is the radius of the beam, r is the transverse distance from the beam axis, $\Delta n_G$ gives the magnitude of the intensity dependent refractive index change due to Gaussian spatial profile of the field, and $\delta \text{}{n}_s$is the level of scatter that varies randomly from 0 to a maximum level that is proportional to intensity.

When the phase modulating RF field is applied to the medium, the sidebands that are produced are spatially offset from one another [3, 9]. As a consequence, each sideband experiences a different level of scatter as it propagates through the medium creating a sideband transmission imbalance that results in RAM. Since the distribution of photorefractive scatterers in the medium can take many minutes or hours to reach equilibrium, the level of RAM will also fluctuate. Based on this model, both the average magnitude of RAM and the level of fluctuation will be dependent on the photorefractive scattering amplitude$\delta n_s$, which increases with intensity.

To investigate this, a numerical simulation was performed by modeling the volume of the photorefractive MgO:LiNbO₃ crystal illuminated by the incident field as an array with the 40 mm propagation distance split into 75 000 columns and the beam cross section (defined by its e⁻⁴ intensity points) split into 800 rows. A photorefractive scatterer with a refractive index of $\delta \text{}{n}_s$that varied randomly between 0 to a maximum level proportional to intensity, was randomly placed in one cell of each of the 75 000 columns. The transmission of each side- band through the resulting refractive index distribution was then calculated assuming a spatial offset of 0.4 μm between the two sidebands [9]. The maximum photorefractive index change at any scatter site was varied between 10⁻² and 10⁻⁹ and the corresponding magnitude of the R_AM was obtained. The simulation was run 100 times for each value of $\delta n_s$ and the average RAM magnitude and the level of fluctuation evaluated with the results shown in Fig. 3.

 

Fig. 3 Numerical simulations showing the linear relationship between photorefractive scattering amplitude and R_AM.

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As shown in Fig. 3, the simulations show that both the average level of RAM and the fluctuations display a linear dependence on the maximum photorefractive index change of the scatter site. Since the photorefractive index change at low intensities such as those used here are proportional to intensity, both the average level of RAM and its standard deviation should increase more or less linearly. This is in good agreement with the observed behavior of intensity dependent RAM as shown in Fig. 1(b). Therefore this simple phenomenological model qualitatively describes the key intensity dependent features observed with RAM.

4. Reduction of RAM through photorefractive erasure

Various techniques have been proposed to suppress RAM or its impact at the detector [2, 10, 11, 28]. One example is to suppress RAM with an active servo at the output of the EOM [10, 11], a second employs a dual-beam method where the RAM is rejected in the post optical detector electronics [2] while a third method applies two harmonically related, phase-shifted radio frequency waves to the electro-optic modulator [28]. Using these RAM reducing methods, the R_AM can be decreased to < 10⁻⁵ (50 dB) below the DC level. While successful, these methods add to the complexity of the overall system as they require sophisticated electronics.

One of the great advantages of using photorefractively induced index changes to create holograms, waveguides or to store data, is that they can be easily removed by illuminating the medium with an erasing beam that neutralizes the space charge field [23, 29]. Here we use the photorefractive nature of intensity dependent RAM to reduce it. Since RAM is produced by intensity dependent randomly distributed photorefractive scattering centers, it is possible to reduce RAM with an erasing beam. To investigate this we used the same arrangement as described in section 3 to measure RAM, with the laser intensity set to 12 mW/mm². A second 532 nm laser (Nd:YAG 2) was introduced as the erasing field with its output counter- propagating through the EOM as shown in Fig. 4.

 

Fig. 4 Experimental setup for the photorefractive RAM erasure. (GTH: Glan-Thompson Polarizer; EOM: Electro-Optic Modulator; BS: Beam Splitter; PD: Photodetector, Nd:YAG 1 and 2 are 532 nm frequency doubled Nd:YAG lasers with maximum output power of 20 mW and 50 mW, respectively).

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Initially, the RAM generated by the 12 mW/mm² beam from Nd:YAG 1 was allowed to reach steady state over a 1.5 hour period in the absence of the erasing beam from Nd:YAG 2. After this time, the erasing beam with an intensity of 59 mW/mm² was turned on for a period of 4.5 hours. The beam waist (0.16 mm) location of the erasing laser was at ~ + 30 mm with respect to the laser exit aperture, which was at a distance of 350 mm from the centre of the 40 mm long EOM crystal. As shown in Fig. 5, a dramatic reduction in both the magnitude and fluctuation in RAM was observed. The erasing beam was then turned off after 4.5 hours and the RAM measured. The RAM is observed to increase towards its original level after the erasing beam is turned off (Fig. 5), but can take several hours to reach the same average value (1.44 × 10⁻⁵) due to the much lower intensity (12 mW/mm²) of the modulated beam.

 

Fig. 5 Photorefractive erasure and recovery of residual noise as a function of irradiation time for an erasure beam intensity of 59 mW/mm²; 5 (a) shows the time dependent RAM values for a total period of write (I) erasure (II) and recovery (III), whereas 5 (b) shows the average RAM level for each 1.5 hours of irradiation, with the error bars indicate the standard deviation of the measured RAM values.

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The trial was repeated at different erasing beam intensities of 19 mW/mm², 38 mW/mm² and 59 mW/mm². As shown in Fig. 6, the reduction in RAM produced by the erasing beam is more pronounced at higher intensities.

 

Fig. 6 Erasure of photorefractive RAM for a total irradiation period of 4.5 hours with each data points corresponds to a time interval of 1.5 hours. The common point gives the average RAM values with only the writing beam turned on (Nd:YAG 1) for 1.5 hours and the other data points represent the corresponding values of RAM on erasure, with the error bars indicate the standard deviation of the measured RAM values.

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This behavior is consistent with our phenomenological model, as the introduction of an erasing beam decreases the maximum photorefractive index change$\delta n_s$ at each scatter site and thereby reduces both the magnitude and the level of fluctuation in RAM.

5. Conclusion

Our results show that the observed fluctuations in RAM level are tied to the medium’s nonlinear photorefractive response and in particular, to the continual evolution of the self-defocusing refractive index changes that occur in the photorefractive medium. These photorefractive index changes are more pronounced at higher input intensities because they increase the imbalance in the transmission of the two first order sidebands of the phase modulated light field. This will in turn lead to higher levels of RAM than are observed at lower input intensities. This, in combination with the observation that the phase of the modulator RAM becomes less well defined at higher input intensities, results in RAM fluctuations that increase as the input intensity is raised. These observations agree well with the findings of the presented phenomenological model that is based on the gradual formation and erasure of intensity dependent photorefractive scattering centers randomly distributed throughout the optically thick medium. The results of this study suggest that the RAM generated by the EOM, is far more stable when low powers are used, which is a useful consideration in low noise applications such as optical frequency references, gravity wave detectors and fiber optic sensing, where reduction of RAM is demanded. We have demonstrated a technique to further suppress RAM where a counter-propagating laser beam was used to erase the photorefractive increased scatter in the phase modulator crystal. With this method, we observed a factor of 6 reduction in both the magnitude and fluctuation level of the observed RAM allowing it to be suppressed to below the 10⁻⁵ level.