Optimized dielectric mirror coating designs for&#x00A0;quasi-harmonic cavity resonance

Li-Wei Wei; Li-Wei Wei; Harold Hollis; Benno Willke; Aaron D. Spector; Guido Mueller; Guido Mueller

doi:10.1364/AO.519987

1. INTRODUCTION

High-reflectivity mirror coatings are a common element in optical setups to reflect and guide lasers and other light sources. In comparison to metallic coatings, dielectric coatings of alternating thin-film layers of high- and low-refractive index materials have lower loss and a higher damage threshold, which are key to the operation of high-finesse and/or high-power optical cavities for a wide range of applications such as cavity ring-down spectroscopy [1], laser interferometric gravitational wave (GW) detectors [2–5], optical atomic clocks [6,7], tests of fundamental physics [8,9], and searches for axions and axion-like particles [10,11] which are leading dark matter candidates [12,13].

In most cases these optical cavities are constructed to be used with monochromatic light (albeit with some optical frequency tuning range) and are set to resonate with a single-frequency laser. The Pound–Drever–Hall (PDH) sensing technique is commonly applied and utilized in feedback control systems to achieve the resonance condition [14,15]. The single-frequency laser is phase-modulated and injected to the optical cavity, and the power in the reflection of the optical cavity is directed to a photodiode to generate the electric signal that later goes through a demodulation process. The error signal obtained via such modulation–demodulation technique then carries the information on the resonance condition between the single-frequency laser and the optical cavity and can be readily used in a feedback control system that maintains the resonance condition.

Some applications, however, require the construction of a bichromatic optical cavity. One example is the filter cavity that allows for frequency-dependent squeezing in GW detectors [16]. Since there is no coherent amplitude of the squeezed field, to achieve resonance (with controlled detuning) between the squeezed field and the filter cavity, a surrogate field is required whose optical frequency relation to the squeezed field is governed either with radio-frequency heterodyne techniques or second harmonic generation. In the latter case, a bichromatic optical cavity is effectively constructed [17–21], as illustrated in the upper part of Fig. 1.

Fig. 1. Illustration on two use cases of bichromatic cavities. Top: Bichromatic cavity for the generation of frequency-dependent squeezed vacuum state. Bottom: Bichromatic cavity for boosting signal power in a light-shining-through-a-wall experiment. BS: beam splitter; SHG: second-harmonic generation; AOM: acousto-optic modulator; M: mirror; DBS: dichroic beam splitter. The AOMs are used to shift the optical frequency of the second harmonic field. In a typical light-shining-through-a-wall experiment, the magnetic fields B provides the virtual photon field $\gamma ^*$ that boosts the interaction between the photon $\gamma$ and the axion/axion-like particle (ALP) $a$ fields, as depicted in the corresponding Feynman diagram. For the purpose of illustration, the electric feedback signals following the photodiode are used for laser frequency control in this figure. We note that the signal can also be applied to the length control of the bichromatic cavity.

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Another use of a bichromatic cavity with quasi-harmonic fields is the dual-cavity-enhanced light-shining-through-a-wall experiment [22–24] that seeks experimental evidence of hypothetical particles that may constitute dark matter, as illustrated in the lower part of Fig. 1. In a light-shining-through-a-wall experiment, a strong laser source is directed to a light-tight barrier, and the detection of a photon $\gamma$ after the wall indicates the existence of an axion or axion-like particle $a$ that allows for the process $\gamma \to a \to \gamma$ to take place in which the axion/axion-like particle $a$, traverses the wall [25] with almost no interaction. The interaction between $\gamma$ and $a$ is on the same ground, which is also very feeble. To improve the search sensitivity, which is recast from the probability of such a process, optical cavities can be used to both boost the number of photons before the wall as well as the efficiency with which photons are regenerated after the wall [26–28]. These two optical cavities are denoted as the production cavity and the regeneration cavity and are conceptually similar to the power-recycling cavity and the signal-recycling cavity, respectively, in laser interferometric gravitational detectors [29].

The Pound–Drever–Hall technique can be applied to bring the laser and the two optical cavities to mutual resonance. One issue with a direct application of the technique lies in the shot noise associated with the laser field that interrogates the regeneration cavity. The shot noise of a sensible laser beam used for cavity interrogation will overwhelm the weak regenerated photon signal detectable by a single-photon counter. For example, a transition edge sensor can exhibit an intrinsic background rate as low as $6.9 \times {10^{- 6}}\;{\rm Hz}$ [30]. Furthermore, the dc component of the interrogation laser beam will also saturate the single-photon counter. A mitigation approach is to interrogate the regeneration cavity with the second-harmonic field of the source laser and use dichroic filters to prevent the second-harmonic field from reaching the single-photon counter. Due to sufficient separation in photon energy, the residual second-harmonic field that reaches the single-photon counter can also be distinguished from the fundamental signal field. The transition edge sensor, e.g., has been demonstrated to have an energy resolution of around 11% [31]. A bichromatic cavity is effectively constructed in this approach.

Bichromatic cavities also find applications in nonlinear optics, for example, to improve the efficiency in frequency doubling [32–34] and optical parametric oscillators [35–37]. In this paper, we will focus on high-finesse bichromatic cavity applications. In particular, we will discuss the construction of a bichromatic regeneration cavity in a cavity-enhanced light-shining-through-a-wall experiment. We will present high-reflectivity coating designs optimized for the operation of a bichromatic regeneration cavity with material parameters and manufacturing errors taken into account.

In this study we use the Any Light Particle Search II (ALPS II) regeneration cavity [11] to define our coating design goals. The requirement on temperature dependence in differential reflection phase is derived, in the next section, to be ${\lesssim} 13\;{\unicode{x00B5}}{\rm rad/K}$. Based on the results published in [21], the temperature dependence in the differential reflection phase for the mirror coatings that constitute the filter cavity of the Advanced Virgo gravitational wave detector is calculated to be $\approx 681\;{\unicode{x00B5}}{\rm rad/K}$ [38]. In comparison, our proposed coating design shows a nominal temperature dependence in differential reflection phase of 4.6 µrad/K. Monte Carlo analysis also shows that the proposed coating design can maintain its nominal characteristics to fulfill the ALPS II requirements when subjected to realistic coating manufacturing tolerances.

2. HIGH-FINESSE BICHROMATIC OPTICAL CAVITIES

Figure 1 illustrates the two exemplary use cases of high-finesse bichromatic optical cavities discussed in the Introduction. To generate a frequency-dependent squeezed vacuum state with a rotation in the squeeze angle at the frequency of interest for advanced laser interferometric GW detectors, the full-width at half-maximum (FWHM) linewidth of the bichromatic cavity needs to be on the order of 50 Hz [18,19].

The regeneration cavity in light-shining-through-a-wall experiments provides a signal power gain of $\approx {\cal F}/\pi$ [27,39], where ${\cal F}$ is the cavity finesse. Light-shining-through-a-wall experiments such as ALPS II use a very long high-finesse cavity to resonantly enhance the signal field. The long cavity is necessary to encompass the super-conducting dipole magnet string that, in the case of ALPS II, has a physical length of 120 m and provides a magnetic field length product of $560\;{\rm T} \cdot {\rm m}$ [40] that greatly boosts the search sensitivity. A surrogate field is used to keep this cavity precisely on resonance with the extremely weak signal field.

As a consequence of the long baseline and the high finesse, the regeneration cavity of the ALPS II experiment has reached a cavity storage time [41] of around 7 ms [42], and aims to eventually reach a regeneration cavity linewidth of ${\approx} 10\;{\rm Hz}$ with upgraded cavity mirrors [11].

A. Bichromatic Cavity Control

High-finesse optical cavities inherently require high-precision control techniques to ensure a strict compliance with the resonance condition governed by the frequency of the laser and the length of the cavity. As discussed in the Introduction, surrogate fields are required to achieve resonance between the main laser field and the filter/regeneration cavity. Second harmonic generation is one means to govern the frequency relation between the surrogate field and the main laser field, as illustrated in Fig. 1.

In the case of frequency-dependent squeezing, a controlled detuning from exact laser–cavity resonance is required to obtain the desired rotation in the squeeze angle [43]. This can be done by frequency shifting the second harmonic field, with an acousto-optic modulator for example. In the case of the regeneration cavity, both the main laser field and its second harmonic should maintain resonance with the cavity. As we will show below, a departure from the exact harmonic relation between the surrogate and the main laser fields is in general present for their simultaneous resonance with the regeneration cavity. We therefore note the conceptual similarity in the optical setup for the control of the bichromatic cavity in these two cases illustrated in Fig. 1.

B. High-Reflectivity Dielectric Mirror Coatings

The quarter-wave stack (QWS), also known as the distributed Bragg reflector, that consists of alternate layers of high-index and low-index materials with optical thickness of one quarter-wavelength, is arguably one of the most typical high-reflectivity dielectric coating designs. At a given angular frequency $\omega$, the reflected field ${E_{{\rm refl}}}(\omega)$ and the incident field ${E_{{\rm inc}}}(\omega)$ can be related by

(1)$${E_{{\rm refl}}}(\omega) = \tilde \Gamma (\omega) \cdot {E_{{\rm inc}}}(\omega),\;\;\;\tilde \Gamma (\omega) = r(\omega) \cdot {e^{i\phi (\omega)}},$$

where $\tilde \Gamma (\omega)$ is the complex reflection coefficient that can be decomposed into a reflection coefficient $r(\omega)$ and a reflection phase $\phi (\omega)$ with respect to the incident field at the front face of the first coating layer. It is known that at the design center wavelength ${\omega _0}$ of an ideal coating, $\phi (\omega = {\omega _0})$ is either 0 or $\pi$. As the number of coating bi-layers is increased, the stop-band $\Delta \omega$ (i.e., the spectral range with high reflectivity) begins to fill out with higher and flatter maximum reflection coefficients; outside the range defined by $|\omega - {\omega _0}| \le \Delta \omega$, $r(\omega) - r({\omega _0})$ becomes significant and continues to exacerbate sharply [44–46].

To provide reflectivity also at the second-harmonic frequency $2\omega$, a straightforward option is to add a quarter-wave stack designed for high reflectivity at $2\omega$ to the coating. In our study we further consider the addition of a multiband reflector (MBR) consisting of alternate half-wave and quarter-wave layers (2:1 periodic stack) at $1.5\omega$, the mean optical frequency of the harmonics. Hereinafter we will refer to these coating bi-layers as QWS($\omega$), QWS($2\omega$), and MBR($1.5\omega$).

Figure 2 shows the reflectance and the reflection phase of these coating bi-layers assuming no material dispersion. The plot shows a difference in the reflection phase between the two harmonic fields. As we will see later, such a difference as well as other factors generally prohibits the simultaneous resonance of exact harmonic fields in a given cavity, a condition hereinafter referred to as bichromatic cavity resonance.

Fig. 2. Reflectance and the reflection phase of exemplary quarter-wave stacks QWS($\omega$), QWS($2\omega$), and the multiband reflector MBR($1.5\omega$) with respect to optical frequencies normalized to $\omega$. The stack configuration is vacuum–${[{\rm L(L)H}]^{12}}$–substrate, where L is the quarter-wave stack of the low refractive index material (${n_L} = 1.4588$) and H is that of the high refractive index material (${n_H} = 2.0564$). We use LLH bi-layers for the MBR. ${n_{{\rm air}}}$ is set to 1 and ${n_{{\rm substrate}}}$ is set to ${n_L}$. Material dispersion is omitted. The reflection phase curves are unwrapped to result in a minimum absolute value around the center wavelength of each of the types of bi-layers.

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To carefully analyze the difference in the reflection phase, the characteristic of the coating materials, in particular their chromatic dispersion, needs to be considered. Ion-beam sputtered dielectric coatings with alternate layers of silica (${{\rm SiO}_2}$) and tantala (${{\rm Ta}_2}{{\rm O}_5}$) have been shown to exhibit loss (transmissivity plus absorption and scatter) on the parts per million level [47] and are generally regarded as the gold standard of various high-finesse optical cavity applications in the visible and near-infrared [48], among which laser interferometric GW detectors serve as a prominent example [49]. For this study we have chosen silica and tantala as the low-index and the high-index coating material. Silica is also chosen as the mirror substrate for this study.

We note that emergent coating technologies such as AlGaAs-based crystalline coatings that are developed for their superior low-thermal-noise performance at room temperature [50] have also been demonstrated to exhibit loss at the parts per million level [48]. These crystalline coatings can therefore also be considered for high-finesse cavity applications whose temperature dependence characteristics can also be investigated in a fashion similar to this study.

C. Quasi-Harmonic Cavity Resonance Frequencies

We next examine the condition for simultaneous resonance of exact harmonic fields with a given optical cavity.

The on-axis electric field $E(x = 0,y = 0,z,t)$ of a single-frequency fundamental-mode Gaussian beam traveling along the z axis can be formulated as a superposition of forward-propagating (${E_ +}$) and backward-propagating (${E_ -}$) components, written as

(2)$$E(z) {e^{i\omega t}} = \left\{{{E_ +} {e^{- i[kz - \psi (z)]}} + {E_ -} {e^{i[kz - \psi (z)]}}} \right\} {e^{i\omega t}},$$

where $k = 2\pi /\lambda$ is the wave number at wavelength $\lambda$, $\psi (z) = {\rm atan}({z/{z_R}})$ is the Gouy phase shift, and ${z_R}$ is the Rayleigh range.

We construct a linear plano-concave cavity of length $L$ with a planar mirror at $z = 0$ and a spherical mirror at $z = L$. We assume identical dielectric coatings, and hence identical $\phi (\omega)$, for the two mirrors of the linear cavity. Following the sign conventions in Eqs. (1) and (2), after dropping the time dependence, the round-trip phase accumulation of the Gaussian beam in the plano-concave cavity becomes

(3)$${e^{i{\phi _{{\rm rt}}}}} = \left.{e^{- i[kz - \psi (z)]}}\right|_{z = 0}^{z = L} \cdot {e^{i\phi (\omega)}} \cdot \left.{e^{i[kz - \psi (z)]}}\right|_{z = L}^{z = 0} \cdot {e^{i\phi (\omega)}},$$

whose value is unity when the Gaussian beam resonates with the cavity such that

(4)$${\phi _{{\rm rt}}} = - 2kL + 2\psi (L) + 2\phi (\omega) = - N \cdot 2\pi ,$$

where $N$ is a positive integer. The resonance frequency $\nu = c/\lambda = \omega /(2\pi) = k \cdot c/(2\pi)$, where $c$ is the speed of light, is consequently

(5)$$\nu = \frac{c}{{2L}}\left[{N + \frac{{2\psi (L)}}{{2\pi}} + \frac{{2\phi (\omega)}}{{2\pi}}} \right].$$

Consider two optical frequencies ${\nu _1}$ and ${\nu _2}$ that satisfy the harmonic relation ${\nu _2} = 2{\nu _1}$. The cavity resonance conditions for ${\nu _1}$ and ${\nu _2}$ are

(6)$$\begin{split}{\nu _1} &= \frac{c}{{2L}}\left[{N({\nu _1}) + \frac{{2\psi (L)}}{{2\pi}} + \frac{{2\phi (\omega)}}{{2\pi}}} \right] \quad {\rm and} \\ {\nu _2} &= \frac{c}{{2L}}\left[{N({\nu _2}) + \frac{{2\psi (L)}}{{2\pi}} + \frac{{2\phi (2\omega)}}{{2\pi}}} \right],\end{split}$$

respectively. When both ${\nu _1}$ and ${\nu _2}$ are resonant with the cavity, their frequency relation is accordingly

(7)$$\begin{split}f = \frac{{{\nu _2}}}{2} - {\nu _1}& = \frac{1}{2} \cdot \frac{c}{{2L}}\left[{N({\nu _2}) + \frac{{2\psi (L)}}{{2\pi}} + \frac{{2\phi (2\omega)}}{{2\pi}}} \right] \\ &\quad - \frac{c}{{2L}}\left[{N({\nu _1}) + \frac{{2\psi (L)}}{{2\pi}} + \frac{{2\phi (\omega)}}{{2\pi}}} \right].\end{split}$$

In the highly idealized case of $\psi (L) = 0$ (plane-parallel cavity) and $\phi (\omega) = 0$, $N({\nu _2}) = 2N({\nu _1})$, $f = 0$, and the harmonic relation between ${\nu _1}$ and ${\nu _2}$ holds true. Realistic cavities have a nonzero Gouy phase shift $\psi (L)$ and $\phi (2\omega)/2 \ne \phi (\omega)$. This leads to a nonzero $f$ in Eq. (7), and the two optical frequencies (fundamental and second harmonic) are not simultaneously resonant with a given optical cavity.

In other words, simultaneous resonance with a given optical cavity can in general only be achieved with quasi-harmonic fields but not exact harmonic fields. Hereinafter we refer to the simultaneous resonance as quasi-harmonic bichromatic cavity resonance, and the nonzero $f$ as the frequency offset required for quasi-harmonic bichromatic cavity resonance.

For the control purposes of the filter cavity for frequency-dependent squeezing in GW detectors and the regeneration cavity in a dual-cavity-enhanced light-shining-through-a-wall experiment, a nonzero $f$ and hence a departure from exact harmonic relation is not an issue as long as the departure remains stable over time.

The frequency offset $f$ for quasi-harmonic bichromatic cavity resonance can depend on various parameters [38]. From the perspective of the coating design, the parameters of interest are first the difference in the reflection phase,

(8)$${\phi _{{\rm diff}}} \equiv \frac{{\phi (2\omega)}}{2} - \phi (\omega),$$

with which we can rewrite the $\phi$-related terms in Eq. (7) as

(9)$${f_{{\phi _{{\rm diff}}}}} = \frac{c}{{2L}} \cdot 2 \cdot \frac{{{\phi _{{\rm diff}}}}}{{2\pi}}.$$

In the following, we will focus on the impact of ${\phi _{{\rm diff}}}$ on $f$. A discussion on the impact of other terms in Eq. (7) can be found, for example, in [38].

In an optical cavity the thermal load on the mirrors can vary depending on whether a resonant laser field is injected. This can manifest as a change in the temperature of the coating, the reflection phase of the coating, and eventually the frequency offset $f$ for quasi-harmonic bichromatic cavity resonance. An ideal coating for quasi-harmonic bichromatic cavity control should have a temperature dependence of ${\rm d}{\phi _{{\rm diff}}}/{\rm d}T = 0$. Physically, this means that, in the presence of temperature perturbation, the coating reflection phase for the second-harmonic field should ideally change twice as much as that for the fundamental field.

D. Coating Requirements for Quasi-Harmonic Bichromatic Cavity Control

In this study we will examine a bichromatic coating designed to have minimized temperature dependence. In the following we will define the coating design goal based on the regeneration cavity proposed in ALPS II, which will operate at 1064 nm wavelength, require a finesse value of around 120000, and anticipate mirror transmissivities of 2 ppm and 25 ppm at a 1064 nm wavelength ($\omega$) [11]. Accordingly, the second-harmonic wavelength for bichromatic cavity control is 532 nm ($2\omega$), and the $1.5\omega$ wavelength is around 709 nm.

A change of $\pi$ in the reflection phase of the mirror coating corresponds to one free spectral range of a linear cavity. To stay within one-tenth of the linewidth of the ALPS II regeneration cavity, the requirement on phase stability is $\pi /120000/10 \approx 2.6\;{\unicode{x00B5}}{\rm rad}$. Assuming a fluctuation of the coating temperature of ${\pm}0.1\;{\rm K}$, the design goal for the bichromatic coating is to have a temperature dependence in differential reflection phase within

(10)$$\left| {\frac{{{\rm d}{\phi _{{\rm diff}}}}}{{{\rm d}T}}} \right| \lesssim 13\;{\unicode{x00B5}}{\rm rad/K}.$$

The transmissivity values quoted for the ALPS II regeneration cavity mirrors are [11]

(11)$$\begin{split}{{\cal T}_{1064\;{\rm nm}}} = 2\;{\rm ppm}\quad {\rm and}\quad {{\cal T}_{532\;{\rm nm}}} = 1\% ,\\{{\cal T}_{1064\;{\rm nm}}} = 25\;{\rm ppm}\quad {\rm and}\quad {{\cal T}_{532\;{\rm nm}}} = 5\% .\end{split}$$

For brevity, we will hereinafter refer to these two mirrors as the 2 ppm mirror and the 25 ppm mirror since 1064 nm is the main wavelength of interest. We also note that the exactitude of the transmissivity at 532 nm is secondary to the other two parameters. The choice of higher transmissivity at 532 nm facilitates achieving the resonance condition between the laser field and the cavity (lock acquisition) and provides better robustness in maintaining the resonance condition.

3. BICHROMATIC COATING DESIGNS

Here we describe the coating structures under study, the optical and thermal constants of the used coating materials, and the manufacturing errors of dielectric coatings. One of the goals of this study is to determine the preferred coating design structure for bichromatic cavity resonance, and the manufacturing errors that the design structure is most sensitive to.

Fig. 3. Illustration of the coating types in this study. The numbers of bi-layers in the figure are only indicative and are varied in the study.

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A. Coating Structures

On the basis of these bi-layers we examine the following types of coatings, as illustrated in Fig. 3:

Type 1a. QWS($\omega$) on top of QWS($2\omega$);
Type 1b. QWS($2\omega$) on top of QWS($\omega$);
Type 2. MBR($1.5\omega$) only;
Type 3a. QWS($\omega$) on top of MBR($1.5\omega$); and
Type 3b. MBR($1.5\omega$) on top of QWS($\omega$).

For coating Type 1 the reflectivity is constituted separately at each wavelength band of interest. For coating Type 2 the reflectivity is constituted equally at both wavelength bands when the material dispersion is omitted. For coating Type 3, the rationale is to provide additional reflectivity at 1064 nm wavelength to the multiband reflector.

We will use these straightforward coating designs for an initial understanding of the problem regarding the temperature dependence in the differential reflection phase. In case it is necessary, more sophisticated coating designs can also be considered, given the development of various advanced coating design–production techniques [51–53].

B. Coating Material Parameters

The success of a coating design also depends on the correct determination of the optical constants used. Unlike in bulk, the refractive index of a coating material in the form of thin film varies depending on the type and specific parameters of the deposition processes [54–56]. In the modern coating design-production chain, the nominal values can, however, be precisely calibrated thanks to the development of computational manufacturing and reverse engineering [57,58].

Various researchers have studied the refractive indices of silica and tantalum pentoxide in thin films [55,58–60]. We use those reported in [60] in this study for a generic investigation into optimal coating designs for a bichromatic cavity. The resultant coating designs will then be subjected to pre-production optimization using the specifically calibrated optical constants for the production process.

In a dielectric coating, the most important parameter for each layer is the optical thickness, which is the product of the refractive index $n$ and the physical thickness $d$. The influence of temperature $T$ on the optical properties of the coating material is generally categorized into thermo-elastic $(\partial d/d)/\partial T \equiv \alpha$ and thermo-refractive $\partial n/\partial T \equiv \beta$ changes in the coating layer.

Inasmuch as the refractive index of the coating material depends on specific production processes, the thermo-elastic and thermo-refractive coefficients can also vary. Physically, an increased thermal load on the coating layer can manifest as either a change in its dimension and/or density, resulting in thermo-elastic and/or thermo-refractive effects, respectively.

Temperature effects on the coating performance have been studied for the noise characterization of GW detectors [61,62], as well as the center-wavelength shift of dense wavelength-division multiplexing optical filters [63–65]. In this study, we use the thermo-elastic and thermo-refractive coefficient reported in [62] to determine the thermal dependence in the differential reflection phase of the bichromatic coating designs.

The coating material parameters used in this study are summarized in Table 1.

Table 1. Coating Material Parameters Used in This Study^a

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C. Coating Manufacturing Errors

The performance of any given coating design will depend on how precisely the design is realized in the actual production. Various techniques, optical or non-optical, direct or indirect, have been developed for the precise monitoring and control of the thickness of each coating layer [66,67]. Complementarily, the influence of manufacturing error on the coating can also be examined to ensure the delivery of its designed performance. In modern coating production, the influence of committed manufacturing errors of deposited layers can potentially be mitigated via online re-optimization of the remaining layers in the coating design [54,68–70].

The manufacturing error in coating thickness is generally divided into three categories: systematic, noncorrelated, and correlated. Systematic errors are often linked to indirect monitoring techniques in which the monitoring sample and the batch sample may experience different material deposition conditions. A so-called tooling factor is introduced to account for the difference. An error in the calibration of the tooling factor will result in a systematic error. Correlated errors are often linked to direct optical monitoring, where the committed errors propagate through subsequent deposited coating layers. Noncorrelated errors, or random errors, of the coating layers are generally modeled to follow the normal distribution and are analyzed using Monte Carlo methods in the pre-production phase.

The accuracy and sensitivity in coating layer thickness monitoring depends on the technique used, and the choice of the most accurate technique depends on the production environment and the type of optical coating to be produced [67]. An accurate error analysis of a coating design is often not possible until the coating design proceeds to its pre-production phase.

In this study we will investigate the effect of systematic and random layer thickness errors during the production of the proposed bichromatic coating design. We will assume a range of ${\pm}1\%$ for the systematic error and study the impact of normally distributed random errors with a 1%, 3%, and 5% standard deviation.

4. SIMULATION RESULTS

This section is divided into two parts. First, we verify whether any of these bichromatic coating designs can fulfill the design goals outlined in Eqs. (10) and (11). The qualifying designs are then checked for their sensitivity to manufacturing errors with Monte Carlo simulations. We will first focus on the 25 ppm mirror. The main characteristics of interest of the coatings under study are the transmissivity values at 1064 nm and 532 nm wavelengths, ${{\cal T}_{1064\;{\rm nm}}}$ and ${{\cal T}_{532\;{\rm nm}}}$, as well as the temperature dependence in the differential reflection phase ${\rm d}{\phi _{{\rm diff}}}/{\rm d}T$.

A. Nominal Characteristics of the Coatings under Study

The transmissivity values and the temperature dependence in differential reflection phase at 1064 nm and 532 nm wavelengths of the coatings under study can be obtained through the recursive propagation of the reflection response at the interface of two adjacent layers. Our calculations are based on the software implementation in [46], with adaptations to account for the dispersion and temperature dependence coefficients of coating materials. For each coating type we vary the numbers of elementary coating bi-layers to identify the combination that best matches our design goal.

Table 2 shows the transmissivity values at 1064 nm and 532 nm wavelengths as well as the temperature dependence in the differential reflection phase as a function of the number of coating bi-layers for a Type 1a coating stack. The design with 17 QWS($\omega$) bi-layers on top of six QWS($2\omega$) bi-layers best matches the transmissivity goals of Eq. (11). The temperature dependence value in the differential reflection phase is ${-}1.1\;{\rm mrad/K}$, which thereby largely exceeds the requirement in Eq. (10). In comparison, the results for coatings of Type 1b shown in Table 3 show a reduced temperature dependence in the differential reflection phase, but nevertheless fail to meet the design goal of ${\lesssim} 13\;{\unicode{x00B5}}{\rm rad/K}$ [Eq. (10)]. The design with four QWS($2\omega$) and 17 QWS($\omega$) bi-layers best matches the transmissivity goals and has a temperature dependence value in the differential reflection phase of 0.1 mrad/K.

Table 2. Characteristics of Coating Type 1a^a

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Table 3. Characteristics of Coating Type 1b^a

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The results for coatings of Type 2 are shown in Table 4. They show a temperature dependence value in the differential reflection phase of 5.4 µrad/K that meets the design goal. The transmissivity at the 532 nm wavelength, however, is far from the design goal and even lower than that at the main 1064 nm wavelength.

The results for coatings of Type 3a are shown in Tables 5 and 6. The design that best matches the transmissivity goals are 10 QWS($\omega$) bi-layers on top of eight MBR($1.5\omega$) bi-layers. The temperature dependence value in the differential reflection phase is excessive, at ${-}1.2\;{\rm mrad/K}$.

The results for coatings of Type 3b summarized in Tables 7 and 8, on the other hand, show contained temperature dependence values in the differential reflection phase. The design with seven MBR($1.5\omega$) bi-layers on top of 11 QWS($\omega$) bi-layers best matches the transmissivity design goal and has a temperature dependence value in the differential reflection phase of 4.6 µrad/K that also meets the design goal. Type 3b is thus the only coating design that can fulfill the design goals outlined in Eqs. (10) and (11).

In view of the contained and consistent temperature dependence in the differential reflection phase of coatings of Type 2, we look into the possibility to exert some control on the transmissivity values by changing the center wavelength of the multiband reflector. Conceptually, the MBR($1.5\omega + \delta \omega$) has two reflective bands centered at $\omega + 1/3 \cdot \delta \omega$ and $2\omega + 2/3 \cdot \delta \omega$ with equal width in the stop-band $\Delta \omega$. A detuned center wavelength can therefore introduce a differential change in the reflectivity at $\omega$ and $2\omega$.

Table 4. Characteristics of Coating Type 2^a

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Table 5. Characteristics of Coating Type 3a^a

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Table 6. Characteristics of Coating Type 3a, Continued^a

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Table 7. Characteristics of Coating Type 3b^a

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Table 8. Characteristics of Coating Type 3b, Continued^a

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The characteristics of the Type 2 coatings with shifted center wavelengths are shown in Fig. 4. We see that it is indeed possible to differentially adjust the reflectivity. The temperature dependence in the differential reflection phase, however, also changes and sets constraints on the detuning of the coating center wavelength. We also note the two notches in the temperature dependence curve. This indicates that center wavelength detuning can also be used as a means to further minimize the temperature dependence in the differential reflection phase.

Fig. 4. Characteristics of coating Type 2 with offsets in the center wavelength. N: number of MBR bi-layers.

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The coating of Type 3b with seven MBR($1.5\omega$) bi-layers on top of 11 QWS($\omega$) bi-layers remains the only design that fulfills the requirement goals and is referred to as the 25 ppm test coating design in the following. The spectrum of the main characteristics of the 25 ppm test coating design is shown in Fig. 5. The plotted wavelength range is far greater than the typical tuning range of a few tens of GHz of commercially available Nd:YAG NPRO lasers. The nominal characteristics of the 25 ppm test coating design within the plotted wavelength range nevertheless fulfills our design goals.

Fig. 5. Transmission coefficients and temperature dependence in differential reflection phase versus wavelength of the 25 ppm test coating design (7 MBR($1.5\omega$) bi-layers on top of 11 QWS($\omega$) bi-layer).

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A temperature dependence of around 114 Hz/K of ${f_{{\phi _{{\rm diff}}}}}$ [Eq. (9)] was reported for the 285 m long Advanced Virgo filter cavity [21]. If we assume an identical contribution from the two cavity mirrors, this corresponds to a temperature dependence of around 681 µrad/K of ${\phi _{{\rm diff}}}$ [Eq. (8)] per mirror, which is on the same order of magnitude of the ${\phi _{{\rm diff}}}$ temperature dependence of the bichromatic coatings in this study other than Type 2 and Type 3b.

Figure 6 shows the electric field intensity (EFI) distribution [71,72] of the 25 ppm test coating design. In the case of the ALPS II regeneration cavity, despite the high finesse, the circulating power at the 1064 nm and the 532 nm wavelengths will both be very low. The former is due to the extremely low probability of the light-shining-through-a-wall process, and the latter is intended such that the background (stray light) level can be kept low. The laser-induced damage threshold (LIDT) is therefore not a major concern for our coating design study. The EFI distribution is shown here for the potential use of the test coating design in high-power optical cavity applications.

Fig. 6. Electric field intensity (EFI) versus position into the dielectric coating of the 25 ppm test coating design.

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B. Sensitivity to Manufacturing Errors

In the following we examine the sensitivity in performance of the 25 ppm test coating design to manufacturing errors. Figure 7 shows the characteristics of the 25 ppm test coating design when systematic errors in the range of ${\pm}1\%$ in the coating layer thickness are introduced. As one would expect, the transmissivity minima shift toward longer/shorter wavelengths when positive/negative errors are introduced. The temperature dependence in differential reflection phase in the vicinity of the (532 nm, 1064 nm) harmonic wavelength pair remains within our design goal, with a range of around ${-}1\;{\unicode{x00B5}}{\rm rad/K}$ to ${+}10\;{\unicode{x00B5}}{\rm rad/K}$ for ${\pm}1\%$ errors.

Fig. 7. Characteristics of the 25 ppm test coating design in the presence of systematic errors in layer thickness.

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Based on these simulation results for optimally constrained temperature dependence in differential reflection phase, one can also consider introducing a systematic increase in the coating layer thickness to the 25 ppm test coating design to result in a nominal ${\rm d}{\phi _{{\rm diff}}}/{\rm d}T$ of zero. The tolerance in the manufacturing error projected from the design goal of Eq. (10) in this case is ${\pm}2.6\%$ in systematic errors.

Next, we consider the introduction of normally distributed random errors in the layer thickness to the 25 ppm test coating design. The results of 1000 Monte Carlo trials for the nominal harmonic wavelength pair is shown in Fig. 8. The percentages of trials that fulfill the design goal of Eq. (10) are 100%, 97.90%, and 83.10% with random errors of 1%, 3%, and 5% standard deviation, respectively.

Fig. 8. Characteristics of the 25 ppm test coating design in the presence of systematic errors in layer thickness with standard deviation of 1%, 3%, and 5% presented as histograms of the results of the Monte Carlo simulations. The bin widths for ${{\cal T}_{1064\;{\rm nm}}}$, ${{\cal T}_{532\;{\rm nm}}}$, and ${\rm d}{\phi _{{\rm diff}}}/{\rm d}T$ are 0.5 ppm, 0.2%, and 2 µrad/K, respectively. We note that there are few tail points in the simulated data with 3% and 5% errors that fall into bins outside the limits of the plots.

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Finally, we examine the case with the presence of both systematic and random errors, as shown in Fig. 9. Systematic errors in the range of ${\pm}1\%$ in steps of 0.1% are assumed, and 1000 Monte Carlo trials are made at each systematic error value. We see that the result with systematic and random errors combined can basically be treated as the superposition of the impact of each of the errors. Concerning the temperature dependence in the differential reflection phase, the range of the centroids of the histograms is around 10 µrad/K for ${\pm}1\%$ errors, as in the systematic-error-only case shown in Fig. 7. The range encircled by the 0.1 probability contour also matches that of the random error only case shown in Fig. 8.

Fig. 9. Characteristics of the 25 ppm test coating design in the presence of both systematic and random errors in the layer thickness. On the left panel the x axis is the systematic error, the y axis is the parameter of interest, and the contour denotes the probability value of the normalized histograms of the Monte Carlo results at each systematic error value; normally distributed random errors with 1% standard deviation are used in the left panel. The data with different systematic errors are then combined to produce the histograms in the right panel; the systematic error is assumed to be uniformly distributed. Data using random errors with 3% and 5% standard deviation are additionally plotted in the right panel. The bin widths of the histograms are the same as in Fig. 8.

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The percentages of trials with combined systematic and random errors that fulfill the design goal of Eq. (10) are 99.91%, 93.59%, and 80.17% with random errors of 1%, 3%, and 5% standard deviation, respectively.

To complete our study, we examine the combination with nine MBR($1.5\omega$) bi-layers on top of 13 QWS($\omega$) bi-layers, which is effectively the 2 ppm test coating design, for its sensitivity to manufacturing errors in a similar fashion to our 25 ppm test coating design. The results are presented in Fig. 10. The percentages of trials with combined systematic and random errors that fulfill the design goal of Eq. (10) are 100%, 95.57%, and 82.64% with random errors of 1%, 3%, and 5% standard deviation, respectively.

Fig. 10. Characteristics of the 2 ppm test coating design in the presence of both systematic and random errors in the layer thickness. See the description in Fig. 9; the bin widths for ${{\cal T}_{1064\;{\rm nm}}}$, ${{\cal T}_{532\;{\rm nm}}}$, and ${\rm d}{\phi _{{\rm diff}}}/{\rm d}T$ are 0.05 ppm, 0.05%, and 2 µrad/K, respectively.

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5. DISCUSSION

A. Choice of Coating Structure for Minimal Temperature Dependence in the Differential Reflection Phase

We have presented a bichromatic coating design, namely the 25 ppm test coating design, with an optimized temperature dependence in the differential reflection phase. The performance of the 25 ppm test coating design is also examined when subjected to manufacturing errors in the layer thickness, and the results are promising. The constrained temperature dependence in the differential reflection phase can be attributed to the use of multiband reflectors as follows.

The principle of dielectric coatings lies in the interference condition characterized by the wavelength $\lambda$ of the light and the optical thickness $n \cdot d$ of the layered material. In this study we set our focus on the material aspect, but it should be evident that a change in $n \cdot d$ can also be effectively represented as a change in $\lambda$ to account for the interference condition. In other words, the influence of the thermo-elastic and thermo-refractive effects on the temperature dependence in the differential reflection phase can also be estimated, at least qualitatively, from the group delay $\partial \phi /\partial \omega$ of the coating. The group delays at the harmonic wavelengths are almost identical for the multiband reflector in Fig. 2, while those for the quarter-wave stacks differ substantially.

When material dispersion is considered, the response of the multiband reflector is no longer perfectly common-mode for the harmonic wavelengths. Additionally, the addition of quarter-wave stacks for the fundamental wavelength is shown to further reduce the temperature dependence in the differential reflection phase. This can be explained by the opposite and hence compensating group delay of the two types of coating bi-layers.

Finally, our simulated data show that bichromatic coatings of Type 3a fail to meet our design goals in terms of the temperature dependence in the differential reflection phase, while those of Type 3b succeed. To account for such an observation, it is probably best to consider the concept of penetration depth [73]. When the multiband reflector is placed on top, most of the power of the harmonic fields are reflected off the multiband reflector. When the quarter-wave stack QWS($\omega$) is placed on top, a considerable amount of power at $\omega$ is reflected off the quarter-wave stack, while the power at $2\omega$ is only reflected deeper into the coating off the multiband reflector. Conceptually, this can be seen as adding amplitude weighting to the group delay.

B. Manufacturability and Further Pre-Production Optimization

The 25 ppm test coating design is subjected to errors of ${\pm}1\%$ in our simulations. As discussed earlier, the exact extent of plausible manufacturing errors can only be accurately estimated when a coating design proceeds to its pre-production phase. With ${\pm}1\%$ systematic errors and normally distributed random errors of 1% standard deviation, the probability of our 25 ppm test coating design fulfilling the design goal is greater than 99.9%.

Limiting the layer thickness errors to 1% may be optimistic, but nevertheless realistic [52]. Our choice of relatively simplistic coating bi-layers and structure should facilitate the optimization of the monitoring strategy during production. The relatively slow deposition rate of the ion-beam sputtering coating production technique, which is regarded as the gold standard for high-power and/or high-finesse cavity mirror coatings in the visible and the near-infrared, also facilitates the precise manufacture of a coating design [70].

The 25 ppm test coating design presented in this study was manufactured and tested for its temperature dependence in the differential reflection phase in an experimental cavity setup [38]. The experimental characterization of such a Type 3b coating shows good agreement with the numerical studies presented in this study.

So far, we have focused on the 25 ppm mirror. Based on the results presented in Table 8, the coating designs of Type 3b can also be used to produce the 2 ppm mirror. There are quite some combinations that produce the desired performance in terms of reflectivity coefficients and the temperature dependence in the differential reflection phase. In principle, for a realistic lossy cavity, one can consider all the designs that have a ${{\cal T}_{1064\;{\rm nm}}} \le 2\;{\rm ppm}$ and an acceptable temperature dependence in the differential reflection phase. At the ppm level the total cavity round-trip loss will be dominated by other sources of losses such as mirror surface scattering. The additional losses of the mirrors and the cavity will also most likely dominate the cavity impedance condition along with the 25 ppm mirror.

6. CONCLUSION

We have studied dielectric coating designs that are optimized for the bichromatic cavity resonance of quasi-harmonic fields $\omega$ and $2\omega$. Bichromatic cavity resonance for exact harmonic fields is generally inhibited due to the difference in the reflection phase and the Gouy phase shift in a realistic optical cavity.

With simulations that take into account coating material parameters and manufacturing errors, we have examined the performance of coating structures that consist of only quarter-wave stack QWS($\omega$) and multiband reflector MBR($1.5\omega$) bi-layers.

The coating design with the best performance consists of MBR($1.5\omega$) on top of QWS($\omega$). We have presented the specific designs for the ${{\cal T}_{1064\;{\rm nm}}} \approx 25\;{\rm ppm}$ and 2 ppm mirrors anticipated for the ALPS II experiment. The proposed designs show satisfactory results in our simulations.

The reflectance of the proposed coating design for the main wavelength of interest can be tailored by the number of QWS($\omega$) without affecting its temperature dependence in the differential reflection phase.

The constrained temperature dependence is largely due to the use of MBR($1.5\omega$), which provides a similar reflection phase to the optical frequency relation $\phi (\omega)$ at $\omega$ and $2\omega$.

The temperature dependence of MBR($1.5\omega$) can further be reduced by choosing another type of bi-layer that has the proper compensating $\phi (\omega)$ relation, which in our proposed design is the QWS($\omega$). Another possible means to control the temperature dependence is the offset of the center wavelength of the multiband reflector.

The coating design with the best simulated performance proposed in this study is therefore very versatile and can also be used to construct high-finesse bichromatic cavities of other finesse values and impedance conditions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	Symbol	$ω$	$1.5 ω$	$2 ω$	Unit
	Wavelength	1064	709.33	532	nm
${S i O}_{2}$	$n$	1.4588	1.4621	1.4668
	$α$	0.51			ppm/K
	$β$	8			ppm/K
${T a}_{2} O_{5}$	$n$	2.0564	2.0822	2.1220
	$α$	3.6			ppm/K
	$β$	14			ppm/K

		Number of QWS ( $2 ω$ ) bi-layers
		6	7	8	9
Number of QWS ( $ω$ )	15	92.2 ppm	62.7 ppm	92.2 ppm	62.7 ppm
bi-layers		10.86%	5.34%	2.59%	1.25%
		$- 1.4 m r a d / K$	$- 1.4 m r a d / K$	$- 1.4 m r a d / K$	$- 1.4 m r a d / K$
	16	46.4 ppm	31.5 ppm	46.4 ppm	31.5 ppm
		9.39%	4.60%	2.23%	1.07%
		$- 1.3 m r a d / K$	$- 1.3 m r a d / K$	$- 1.3 m r a d / K$	$- 1.3 m r a d / K$
	17	23.4 ppm	15.9 ppm	23.4 ppm	15.9 ppm
		7.94%	3.88%	1.87%	0.90%
		$- 1.1 m r a d / K$	$- 1.1 m r a d / K$	$- 1.1 m r a d / K$	$- 1.1 m r a d / K$
	18	11.8 ppm	8.0 ppm	11.8 ppm	8.0 ppm
		6.70%	3.26%	1.57%	0.75%
		$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$
	19	5.9 ppm	4.0 ppm	5.9 ppm	4.0 ppm
		5.70%	2.77%	1.33%	0.64%
		$- 0.9 m r a d / K$	$- 0.9 m r a d / K$	$- 0.9 m r a d / K$	$- 0.9 m r a d / K$

	Number of QWS ( $2 ω$ ) bi-layers
		4	5	6	7
Number of QWS ( $ω$ )	15	92.2 ppm	245.9 ppm	92.2 ppm	245.8 ppm
bi-layers		7.44%	3.63%	1.75%	0.84%
		0.1 mrad/K	0.3 mrad/K	0.1 mrad/K	0.4 mrad/K
	16	46.4 ppm	123.8 ppm	46.4 ppm	123.7 ppm
		7.65%	3.73%	1.80%	0.86%
		0.1 mrad/K	0.3 mrad/K	0.1 mrad/K	0.4 mrad/K
	17	23.4 ppm	62.3 ppm	23.4 ppm	62.2 ppm
		7.95%	3.88%	1.87%	0.90%
		0.1 mrad/K	0.3 mrad/K	0.1 mrad/K	0.4 mrad/K
	18	11.8 ppm	31.3 ppm	11.8 ppm	31.3 ppm
		8.36%	4.09%	1.97%	0.95%
		0.1 mrad/K	0.3 mrad/K	0.1 mrad/K	0.4 mrad/K
	19	5.9 ppm	15.8 ppm	5.9 ppm	15.8 ppm
		8.87%	4.34%	2.10%	1.01%
		0.1 mrad/K	0.3 mrad/K	0.1 mrad/K	0.4 mrad/K

Number of MBR ( $1.5 ω$ ) bi-layers
19	20	21	22	23
36.2 ppm	20.0 ppm	11.1 ppm	6.1 ppm	3.4 ppm
19.3 ppm	10.3 ppm	5.5 ppm	2.9 ppm	1.6 ppm
5.4 µrad/K	5.4 µrad/K	5.4 µrad/K	5.4 µrad/K	5.4 µrad/K

		Number of MBR ( $1.5 ω$ ) bi-layers
		4	5	6	7
Number of QWS ( $ω$ ) bi-layers	8	1118.5 ppm	618.3 ppm	341.7 ppm	188.9 ppm
		51.57%	32.02%	18.64%	10.44%
		$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$
	9	563.0 ppm	311.2 ppm	172.0 ppm	95.0 ppm
		55.61%	34.97%	20.50%	11.54%
		$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$
	10	283.4 ppm	156.6 ppm	86.5 ppm	47.8 ppm
		56.95%	35.93%	21.10%	11.89%
		$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$
	11	142.6 ppm	78.8 ppm	43.6 ppm	24.1 ppm
		55.10%	34.51%	20.17%	11.33%
		$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$
	12	71.8 ppm	39.7 ppm	21.9 ppm	12.1 ppm
		50.71%	31.28%	18.11%	10.11%
		$- 1.1 m r a d / K$	$- 1.1 m r a d / K$	$- 1.1 m r a d / K$	$- 1.1 m r a d / K$
	13	36.1 ppm	20.0 ppm	11.0 ppm	6.1 ppm
		45.09%	27.29%	15.62%	8.66%
		$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$
	14	18.2 ppm	10.0 ppm	5.6 ppm	3.1 ppm
		39.39%	23.41%	13.25%	7.30%
		$- 0.9 m r a d / K$	$- 0.9 m r a d / K$	$- 0.9 m r a d / K$	$- 0.9 m r a d / K$
	15	9.1 ppm	5.1 ppm	2.8 ppm	1.5 ppm
		34.28%	20.05%	11.25%	6.17%
		$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$

Optimized dielectric mirror coating designs for quasi-harmonic cavity resonance

Abstract

1. INTRODUCTION

2. HIGH-FINESSE BICHROMATIC OPTICAL CAVITIES

A. Bichromatic Cavity Control

B. High-Reflectivity Dielectric Mirror Coatings

C. Quasi-Harmonic Cavity Resonance Frequencies

D. Coating Requirements for Quasi-Harmonic Bichromatic Cavity Control

3. BICHROMATIC COATING DESIGNS

A. Coating Structures

B. Coating Material Parameters

C. Coating Manufacturing Errors

4. SIMULATION RESULTS

A. Nominal Characteristics of the Coatings under Study

B. Sensitivity to Manufacturing Errors

5. DISCUSSION

A. Choice of Coating Structure for Minimal Temperature Dependence in the Differential Reflection Phase

B. Manufacturability and Further Pre-Production Optimization

6. CONCLUSION

Disclosures

Data availability

REFERENCES

Data availability

Cited By

Figures (10)

Tables (8)

Equations (11)

Applied Optics

		Number of MBR ( $1.5 ω$ ) bi-layers
		8	9	10	11
Number of QWS ( $ω$ ) bi-layers	8	104.4 ppm	57.7 ppm	31.9 ppm	17.6 ppm
		5.73%	3.10%	1.67%	0.90%
		$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$
	9	52.5 ppm	29.0 ppm	16.0 ppm	8.9 ppm
		6.34%	3.44%	1.86%	1.00%
		$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$
	10	26.4 ppm	14.6 ppm	8.1 ppm	4.5 ppm
		6.54%	3.55%	1.91%	1.03%
		$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$
	11	13.3 ppm	7.4 ppm	4.1 ppm	2.2 ppm
		6.22%	3.37%	1.82%	0.98%
		$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$	$- 1.2 m r a d / K$
	12	6.7 ppm	3.7 ppm	2.0 ppm	1.1 ppm
		5.53%	3.00%	1.61%	0.86%
		$- 1.1 m r a d / K$	$- 1.1 m r a d / K$	$- 1.1 m r a d / K$	$- 1.1 m r a d / K$
	13	3.4 ppm	1.9 ppm	1.0 ppm	0.6 ppm
		4.72%	2.55%	1.37%	0.74%
		$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$	$- 1.0 m r a d / K$
	14	1.7 ppm	0.9 ppm	0.5 ppm	0.3 ppm
		3.97%	2.14%	1.15%	0.62%
		$- 0.9 m r a d / K$	$- 0.9 m r a d / K$	$- 0.9 m r a d / K$	$- 0.9 m r a d / K$
	15	0.9 ppm	0.5 ppm	0.3 ppm	0.1 ppm
		3.34%	1.80%	0.97%	0.52%
		$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$	$- 0.8 m r a d / K$

		Number of MBR ( $1.5 ω$ ) bi-layers
		4	5	6	7
Number of QWS ( $ω$ ) bi-layers	8	1161.1 ppm	644.1 ppm	356.7 ppm	197.4 ppm
		15.16%	8.43%	4.61%	2.49%
		6.9 µrad/K	6.3 µrad/K	5.9 µrad/K	5.7 µrad/K
	9	584.5 ppm	324.2 ppm	179.5 ppm	99.3 ppm
		14.23%	7.89%	4.31%	2.33%
		4.4 µrad/K	5.0 µrad/K	5.2 µrad/K	5.3 µrad/K
	10	294.2 ppm	163.2 ppm	90.4 ppm	50.0 ppm
		13.45%	7.44%	4.06%	2.19%
		2.1 µrad/K	3.7 µrad/K	4.5 µrad/K	4.9 µrad/K
	11	148.0 ppm	82.1 ppm	45.5 ppm	25.2 ppm
		12.83%	7.09%	3.86%	2.08%
		0.0 µrad/K	2.6 µrad/K	3.9 µrad/K	4.6 µrad/K
	12	74.5 ppm	41.3 ppm	22.9 ppm	12.7 ppm
		12.36%	6.82%	3.71%	2.00%
		$- 1.7 µ r a d / K$	1.6 µrad/K	3.4 µrad/K	4.3 µrad/K
	13	37.5 ppm	20.8 ppm	11.5 ppm	6.4 ppm
		12.05%	6.65%	3.61%	1.95%
		$- 3.3 µ r a d / K$	0.8 µrad/K	2.9 µrad/K	4.1 µrad/K
	14	18.9 ppm	10.5 ppm	5.8 ppm	3.2 ppm
		11.89%	6.55%	3.56%	1.92%
		$- 4.6 µ r a d / K$	0.1 µrad/K	2.5 µrad/K	3.8 µrad/K
	15	9.5 ppm	5.3 ppm	2.9 ppm	1.6 ppm
		11.88%	6.54%	3.55%	1.92%
		$- 5.6 µ r a d / K$	$- 0.5 µ r a d / K$	2.2 µrad/K	3.7 µrad/K

		Number of MBR ( $1.5 ω$ ) bi-layers
		8	9	10	11
Number of QWS ( $ω$ ) bi-layers	8	109.2 ppm	60.4 ppm	33.4 ppm	18.4 ppm
		1.34%	0.72%	0.38%	0.21%
		5.5 µrad/K	5.5 µrad/K	5.4 µrad/K	5.4 µrad/K
	9	54.9 ppm	30.4 ppm	16.8 ppm	9.3 ppm
		1.25%	0.67%	0.36%	0.19%
		5.3 µrad/K	5.3 µrad/K	5.3 µrad/K	5.3 µrad/K
	10	27.6 ppm	15.3 ppm	8.4 ppm	4.7 ppm
		1.18%	0.63%	0.34%	0.18%
		5.1 µrad/K	5.2 µrad/K	5.3 µrad/K	5.3 µrad/K
	11	13.9 ppm	7.7 ppm	4.3 ppm	2.4 ppm
		1.12%	0.60%	0.32%	0.17%
		5.0 µrad/K	5.1 µrad/K	5.2 µrad/K	5.3 µrad/K
	12	7.0 ppm	3.9 ppm	2.1 ppm	1.2 ppm
		1.08%	0.58%	0.31%	0.17%
		4.8 µrad/K	5.1 µrad/K	5.2 µrad/K	5.3 µrad/K
	13	3.5 ppm	1.9 ppm	1.1 ppm	0.6 ppm
		1.05%	0.56%	0.30%	0.16%
		4.7 µrad/K	5.0 µrad/K	5.2 µrad/K	5.2 µrad/K
	14	1.8 ppm	1.0 ppm	0.5 ppm	0.3 ppm
		1.03%	0.55%	0.30%	0.16%
		4.5 µrad/K	4.9 µrad/K	5.1 µrad/K	5.2 µrad/K
	15	0.9 ppm	0.5 ppm	0.3 ppm	0.2 ppm
		1.03%	0.55%	0.30%	0.16%
		4.4 µrad/K	4.9 µrad/K	5.1 µrad/K	5.2 µrad/K