## Abstract

The reflection Airy distribution (RAD) of a Fabry-Pérot (F-P) resonator is deduced with the consideration of the mode coupling between the cavity resonance field and the initial back-reflected field at the input F-P resonator facet, as well as the influence of the loss/gain factor. A waveguide loss/gain measurement method is proposed based on the measurement of the finesse of the RAD, which is intrinsically free from the influence of the coupling loss and the substrate scattering noise. The waveguide loss can be measured with a simple single-facet coupling setup, considerably reducing the coupling difficulty and the complexity of the measurement system while achieving the same or better measurement accuracy as that of the transmission F-P method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A Fabry-Pérot (F-P) resonator is a typical multi-beam interferometric device and has been widely used as the prototype device for laser oscillation, spectrum measurement [1], loss/gain characterizations [2–8], facet coating measurement [9] and etc. Although there are well-established theories which can be found in many standard textbooks [1,10], the theory still keeps evolving to accommodate new applications and situations [11].

In most of the applications based on the F-P resonator theory, the transmission characteristics of an F-P resonator are widely used, while its reflection characteristics are seldom considered. This can be attributed to several reasons. For a high-Q F-P resonator, its transmission filed distribution is a series of bright narrow lines situated among a dark background. While for the reflection distribution, it is a series of narrow dark lines situated among a bright background. It is more convenient to detect bright lines than the dark lines in practical measurements. For resonators with low Q values, such as uncoated waveguide resonators, the transmission and reflection spectrum may look alike, but the reflection spectrum should still be used with caution. The reason is that the reflection spectrum is usually “contaminated” by the partially coherent initial back-reflected field (IBRF). While for the transmission spectrum, there is no such influence and the cavity resonance information can be revealed more accurately.

In most of the textbooks, the incident wave of an F-P resonator is usually assumed to be a plane wave. With such an assumption, the measurement based on the reflection spectrum shall yield the same accuracy as that based on the transmission spectrum, on condition that the IBRF and the resonance field of the cavity are in total coherence with each other. The transmission and the reflection spectra are complementary under plane wave assumption. However, in real situations, especially when a waveguide is considered, the IBRF can be rarely in total coherence with the resonance field of the cavity due to mode or polarization mismatch between the two fields. The simple addition of the IBRF term to the reflection spectrum will not be generally valid for waveguide resonators. This is also the reason why measurements based on the reflection spectrum can hardly obtain accurate results [12,13].

In this paper, we deduce the reflection Airy distribution (RAD) of an F-P resonator with the consideration of the mode coupling between the cavity resonance field and the IBRF. The analysis shows that the finesse of the RAD is not influenced by the above-mentioned mode coupling, and can be used to characterize the loss/gain factor of the F-P resonator. Based on this theory, we propose and demonstrate an on-chip waveguide loss measurement method by analyzing the finesse of the F-P reflection spectrum through a single-facet coupling setup. The waveguide loss can be either calculated using an explicit analytical expression or extracted using a simple curve fitting method, which is more intuitive and straightforward. The measurement results using the proposed method show comparable accuracy as that obtained from the F-P transmission contrast method [2–5]. In the case of misalignment, the proposed method shows better measurement accuracy than the transmission method, due to its immunity to the influence of the background noise level. This scheme only requires a single-facet coupling, capable of simplifying the waveguide measurement system setup and alignment procedure, lowering the system cost and increasing the measurement efficiency.

## 2. Theoretical Analysis

#### 2.1. Reflection Airy distribution and finesse

Figure 1 shows the schematic electric field inside and outside of a waveguide Fabry-Perot resonator illuminated by incident light (electric field *E _{0}* and intensity

*I*). At the input facet (the left facet in Fig. 1), part of the incidence field will couple into the waveguide and excite the eigen modes of the waveguide resonator with mode coupling coefficients determined by the overlap integrals of the mode fields of the incidence

_{0}*E*and the waveguide eigen modes. For a single mode waveguide resonator, the mode coupling coefficient between the incidence field and the cavity mode is designated as

_{0}*η*. Inside the cavity, the excited mode will circulate around and form a steady cavity intensity distribution.

_{inc-cav}According to the circulating-field approach [14], the electric field inside the cavity *E*_{1} can be written as:

*r*and

_{1}*r*are the amplitude reflection coefficient of the input facet 1 and the output facet 2, respectively,

_{2}*t*

_{1}is the amplitude transmission coefficient at facet 1,

*G = e*is the intensity gain (or loss) per pass,

^{–αL}*α*is the absorption coefficient in the cavity,

*φ*is the round-trip phase shift. The cavity field will transmit out of the cavity as the forward transmission field

*E*and backward transmission field

_{FT}*E*:

_{BT}*E*, which reveals the internal resonance characteristics of the cavity. However, the reflection intensity distribution arises from the contribution of

_{FT}*E*and the initial back-reflected field

_{BT}*E*. The IBRF has to be treated with care for the case of waveguide F-P resonator, since

_{IBRF}*E*is only in partial coherence with

_{IBRF}*E*due to mode and polarization mismatch. We can decompose

_{BT}*E*into the coherent term

_{IBRF}*E*and the incoherent term

_{IBRF,C}*E*with respect to

_{IBRF,IC}*E*. The coherent part

_{BT}*E*will contribute to the formation of an ideal reflection intensity distribution. The incoherent part

_{IBRF,C}*E*, however, will leads to the change of the DC level of the reflection intensity distribution, resulting in a deviation of the contrast ratio from that of the transmission spectrum. This is why the waveguide loss measurement based on the contrast ratio method can only rely on the evaluation of the transmission spectrum, but not the reflection spectrum.

_{IBRF,IC}For the coherent IBRF part, *E _{IBRF,C}* can be written as:

*η*is the coherent coefficient between the IBRF and

_{IBRF}*E*, which includes the contribution of the partially reflection from the left of facet 1 and the mode coupling between the IBRF and

_{BT}*E*, the minus sign represents a phase-shift of π after reflection.

_{BT}The combined reflection field is the supposition of the IBRF and *E _{BT}*:

*η*characterizes the coupling coefficient between the resonance cavity field

_{m}= η_{inc-cav}/η_{IBRF}*E*and IBRF. In an ideal case with a plane wave assumption where

_{BT}*η*,

_{IBRF}= η_{inc-cav}*η*1,

_{m}=*E*reduces to the common reflection spectrum as discussed in textbooks(e.g., equation 9.49 in Ref [10]). In general,

_{R,C}*η*and

_{IBRF}*η*are unknown and vary with different coupling conditions. The presence of

_{m}*η*

_{m}in the coherent reflection spectrum will influence the contrast ratio of the reflection intensity distribution even in the absence of the incoherent term

*E*.

_{IBRF,IC}For the incoherent part *E _{IBRF,IC}*, its time average will produce a DC component in the final reflection intensity distribution. The generic form of the reflection intensity is:

*I*is the intensity of the input light. If we assume that

_{0}= E_{0}^{2}*r*, and

_{1}= r_{2}= r*t*, which is a common case for most waveguide resonators, and define the reflection Airy distribution (RAD) to be

_{1}= t_{2}= t*A*, then:

_{R}≡I_{R,C}/I_{0}*R = r*,

^{2}*T = t*,

^{2}*U = R + Tη*.

_{m}*R + T =*1 is assumed in the above deduction.

*A*represents the reflected flux-density distribution without the influence of the incoherent IBRF term. Equation (7) can be alternatively expressed as:

_{R}*A*is the generic Airy distribution [11] with an inclusion of the gain/loss factor. It represents the contribution of the intracavity enhancement from the waveguide cavity.

_{circ}If IBRF is in total coherence with the back-transmission field *E _{BT}*,

*η*,

_{IBRF}= η_{inc-cav}*η*= 1 and

_{m}*U = R + Tη*= 1, Eqs. (7) and (8) reduce to:

_{m}The maximum and minimum value of *A _{R}* occurs at

*φ =*(2m + 1)π and

*φ*= 2mπ,

*m =*any integer, respectively.

*η*between the back-transmitted resonance cavity filed and the IBRF. Since the incoherent DC component and the coupling coefficient are unknown and vary with coupling condition, the contrast ratio

_{m}*C*cannot generally be used to determine the waveguide gain/loss.

_{R}Now, we will proceed to calculate the finesse ℱ and show that ℱ is not influenced by the IBRF. For the reflection spectrum, it is more convenient to use the full width at half minimum (FWHM) of the valley to characterize ℱ, as shown in Fig. 2. We can get the half width at half minimum *φ _{1/2}* by solving the equation:

*φ*of FWHM can be found to be:The factor

_{1/2}*U*that influences the reflection intensity distribution

*I*and contrast ratio

_{R}*C*disappears in Eq. (15). Figure 2 shows three reflection spectra for different mode coupling factor

_{R}*η*of 100%, 70%, and 40%, which is plotted using Eq. (7). It shows that the FWHM of the three curves keep exactly the same regardless of

_{m}*η*, which means that the width of the reflection valley (or the peak) is independent of the partial mode coupling between the back-reflected field and the IBRF. It only depends on the reflectivity

_{m}*R*and the gain/loss term

*G*. Then the finesse ℱ can be calculated to be:

*G*and

*α*can be determined from ℱ using Eq. (16) as:

*η*), between the back-transmitted cavity field and the coherent IBRF (

_{inc-cav}*η*), and the incoherent IBRF

_{IBRF}*E*can alter the contrast ratio of the reflection intensity distribution, but have no influence on the finesse.

_{IBRF,IC}#### 2.2. Discussion

Using Eq. (18), the loss of a waveguide resonator with a known facet reflectivity can be measured by calculating the finesse ℱ of its reflection intensity distribution. Since the finesse expressions are the same for both the transmission and the reflection intensity distributions, the discussion on the measurement based on the transmission ℱ in Ref [7] also holds true for the reflection ℱ. Figure 3 shows the dependence of ℱ on the loss *G = e ^{–αL}* for several typical uncoated waveguide resonators based on different material systems. It can be found that a higher facet reflectivity will result in a better resolution in the loss measurement due to the higher sensitivity of ℱ to

*G*. Hence, it will be easier to obtain a relatively accurate loss value for waveguides with high reflectivity (higher than 0.2), such as InP (refractive index

*n =*3.17@1.5μm, material reflectivity

*R*≈0.27), GaAs (

*n*≈3.39@1.5μm,

*R*≈0.30), or Si (

*n*= 3.51@1.3μm,

*R*≈0.31) waveguides. For waveguides with medium reflectivity, such as LiNbO

_{3}(

*n*≈2.21@1.5μm, R≈0.142) and Si

_{3}N

_{4}(n≈2.00@1.5μm,

*R*= 0.11), careful measurement of ℱ will be required. However, for waveguides with very low reflectivity such as glass (

*n*≈1.50@1.5μm,

*R*= 0.04) or anti-reflection coated waveguides, it will be very difficult to obtain reliable results.

Another fact is that the measurement resolution of the loss value decreases with the increase of the waveguide loss *G*, regardless of the reflectivity. When measuring a sample, it would be advisable to choose a short sample to reduce the waveguide loss, so as to increase the measurement accuracy. For a waveguide with high facet reflectivity (*R*>0.2), *αL*<1 might be reasonable for a loss measurement resolution *|dℱ/d(αL)|>*0.1.

Next, we will compare the measurement errors using the reflection finesse and the transmission contrast ratio. For both cases, the measurement errors mainly come from the following three reasons: (a) the estimated end face reflectivity deviation from the actual value; (b) measurement error due to the limitation of the measurement accuracy of the finesse value or the contrast ratio; (c) DC component error due to substrate radiation light, stray light, etc.

Let *G _{F}* and

*G*be the loss measured using the reflection finesse (RF) method and the transmission contrast (TC) method, respectively, as shown in Eqs. (19) and (20):

_{K}*K = I*is the contrast ratio of the peak to the valley of the transmission intensity distribution. The term

_{max}/I_{min}*R*has the same contribution to both expressions, leading no difference in the measurement errors between the two methods.

The contributions of the measurement uncertainty of finesse and contrast ratio are plotted in Fig. 4. For a measurement error fluctuation of ± 5%, the deviation of the measured loss value from the actual loss value is calculated using different end face reflectivity. It can be clearly observed that as the waveguide reflectivity increases, the loss deviation caused by the finesse measurement error becomes less obvious; for a fixed reflectivity around 0.30, the standard error of the measurement will be below 10% for *αL*<3dB; but for the reflectivity of 0.142, the standard error will be above 20% for *αL*>3dB. The reflection finesse measurement method is more suitable for the low-loss high-reflectivity waveguide. As for the transmission contrast measurement method, we observed the same trend as the reflection finesse method. However, for the same reflectivity *R* and the same loss *αL*, the standard deviation of the transmission contrast method is smaller than the standard deviation of the reflectivity finesse method, but still on the same order of magnitude.

In the contrast ratio method, there will inevitably be some DC component coupled into the measurement setup, causing a raising of the DC floor in the transmission curve, thereby reducing *K* and further resulting in a larger measurement loss than the actual value. The finesse method is free from the influence of the DC component, capable of producing more reliable result than the contrast ratio method in the case of misalignment or noisy coupling.

## 3. Experiment application

When measuring low-loss waveguide structures, usually two non-destructive measurement methods can be used. One is the ring resonator resonance method [15] and the other is the F-P resonance method [2–5,8,16]. The ring resonance method is suitable for devices where the fabrication precision can be well controlled. However, the additional bending loss associated with ring structure should be taken into consideration in the measurement, and the test sample with a ring resonator structure need to be customized. The F-P resonance method is suitable for a single-mode waveguide with well-defined cleavage or coated facets, where III/V semiconductor material waveguides fall into this category. The waveguide loss has been traditionally calculated using the intensity transmission contrast ratio [2–5]. There is no bending loss error associated with the F-P resonance method, and the test structure is also a simple straight waveguide, making this method widely used in on-chip waveguide loss measurement. However, traditional F-P resonance transmission method requires fiber coupling accessible to both facets of the waveguide, resulting in a time-consuming coupling process and complex coupling system. In addition, in situations where only a single waveguide facet can be accessible, such as active or passive waveguide chip soldered on submounts, the double-facet-coupling cannot be easily applied.

By using the theory proposed in the previous section, it is possible to obtain the waveguide loss through a single-facet coupling. Figure 5 shows the experimental setup for measuring the waveguide loss using the reflection distribution. The setup consists of a tunable laser for wavelength sweeping, an optical circulator to launch the sweeping laser and receive the reflection light, and a photodiode (PD) to convert the reflected light into an electric signal. A lensed fiber mounting on a translation stage is used for optical coupling. The reflected interferometric pattern of the F-P cavity is gathered by the same lensed fiber. A computer is used to control the tunable laser and records the corresponding photocurrent.

In this experiment, shallow-etch ridge InGaAsP/InP waveguides were used as test waveguides, which consisted of a 400-nm InGaAsP waveguide layer (undoped, *E*_{g} = 1.2μm) sandwiched by undoped InP cladding layers. The waveguide were 3 and 4 μm in width and cleaved to about 4 mm in length. The reflectivity of the two cleavage facets were estimated to be *R = R*_{1} *= R*_{2}≈0.27 using the Fresnel formula.

During the test, the transmission spectrum was also simultaneously recorded at the other side of the test sample (not shown in Fig. 5). Figure 6 shows the measured reflection (upper data points) and the transmission (lower data points) intensity distribution. The upper and lower curves are the fitting curves using Eq. (7) and Eq. (21), respectively. Both curves fit well with the experimental data, showing the equivalent effectiveness of the reflection finesse and the transmission contrast ratio methods. The loss can be readily calculated using the analytical expression Eq. (7) and Eq. (21), or by parameter extraction through the fitted curves. Nine waveguides were measured and the loss values calculated using the two measurements schemes are listed in Table 1. In each scheme, both the analytical and the curve fitting methods were applied for further comparison, where the analytical methods of transmission and reflection spectrum were used by extracting the contrast and finesse respectively. The values obtained from the curve fitting method based on the transmission scheme were used as the references. The deviation of each measurement from the reference value is listed on the right side of the loss value. It can be clearly observed that the analytical reflection finesse method shows the higher measurement accuracy than that of the transmission contrast method. It can be noticed that the sample No.7 has a large measurement deviation, as compared to other samples. It is beacause the total loss value (*αL*>4dB) is too large to ensure the measurement accuracy.

Then another experiment was carried out to verify the robustness of the reflection finesse method under poor coupling conditions. Two samples were tested by gradually moving the lens fiber away from the waveguide facets so that more light was coupled outside of the waveguides, such that more fraction of the stray light is contained in the collected spectrum. As shown in Fig. 7, there is a high degree of consistency in the loss values obtained from the two fitting methods regardless of the alignment conditions, indicating that the fitting value obtained either from the reflection or the transmission spectrum can be used as a reliable source for loss measurement. The loss value based on the analytical parameter extraction from the transmission contrast method showed obviously growing deviation from the original well-coupled state, whereas the analytical parameter extraction from the reflection finesse method shows much less deviation. This shows the effectiveness and robustness of the reflection method under poor coupling conditions. In practical applications, the reflection finesse method cannot only reduce the system cost and the coupling difficulty but also effectively eliminate the influence of the DC noise caused by the substrate radiation.

## 4. Conclusion

In this paper, the reflection Airy distribution of a waveguide F-P resonator is deduced. We show that the initial back reflected field from the incident cavity facet can alter the contrast ratio of the reflected field but has no influence on the finesse of the reflection spectrum. A waveguide loss measurement method based on the reflected interferometric pattern of an F-P cavity is proposed and demonstrated. The loss value obtained from the reflection finesse method has the same measurement accuracy as that obtained from the F-P transmission contrast method. The proposed method is free from the influence of background noise, relaxing the coupling requirement as compared to the transmission contrast method. It only requires a single-end coupling system, considerably reducing the coupling difficulty and simplifying the measurement system. The accurate, low-cost, easy-to-operate and reliable characteristics of the reflection finesse method make it can a promising method for waveguide gain/loss measurement.

## Funding

National Key Research & Development (R&D) Plan (2016YFB0402301).

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