Integrated optical isolator based on double-arm phase modulator and optical isolation method
By using a combination of a double-arm phase modulator and a micro-ring resonator in the optical isolator, the problems of large footprint and high power consumption in the prior art are solved, and low half-wave voltage and high isolation are achieved Effect.
Patent Information
- Application Number
- CN202510009398.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-03
- Publication Date
- 2025-02-18
- Estimated Expiration
- 2045-01-03
AI Technical Summary
When existing electro-optical modulation-based optical isolators need to increase the length of the modulation area electrode or cascade phase modulator when achieving low half-wave voltage and high isolation, they need to increase the device footprint and make it difficult to achieve large-scale integration.
A two-arm phase modulator is used to combine a micro-ring resonator and a cross waveguide. By adjusting the frequency and parameters of the radio frequency signal, the complete transmission of the optical signal and the effective suppression of the reflected optical signal are achieved, thereby reducing the half-wave voltage of the phase modulator.
While ensuring that the chip length remains unchanged, the half-wave voltage of the phase modulator is reduced by twice, significantly reducing the additional power consumption of the integrated optical isolator, and it is simple to operate and stable performance.
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Figure CN119472089A_ABST
Abstract
Description
Technical Field
[0001] The present disclosure relates to the technical field of optical isolators, and in particular to an integrated optical isolator based on a dual-arm phase modulator and an optical isolation method. Background Art
[0002] Optical isolators are key components widely used in optical systems such as lasers and amplifiers to block reflected light from entering the resonant cavity and suppress signal noise. With the continuous development of integrated microwave photonic technology, on-chip integration of optical isolators has become a key technology for realizing efficient and complex functional photonic integrated circuits (PICs). There are currently three main methods to achieve on-chip integration of optical isolators, including magnetic bias, optical nonlinearity, and spatiotemporal modulation. However, methods based on magnetic bias usually face challenges in material integration and have high optical losses. Methods based on nonlinear optics either require additional pump lasers and filters or rely on the input optical power of the light source. In contrast, methods based on electro-optical modulation do not require the integration of heterogeneous materials and are not affected by optical power, making them an attractive option for realizing on-chip isolators.
[0003] The method based on electro-optical modulation uses the different modulation effects produced when forward and reverse light passes through the phase modulator to break the optical reciprocity and obtain isolation. In order to achieve complete suppression of reflected light, the phase modulator needs to reach a certain modulation depth, so the half-wave voltage of the phase modulator is required to be low enough to reduce the additional power consumption generated in this process. In order to reduce the half-wave voltage of the phase modulator, it is usually necessary to increase the length of the modulation area electrode or accumulate the phase change by cascading two phase modulators, but both of the above methods will sacrifice the device's footprint and are not conducive to large-scale integration. Summary of the invention
[0004] 1. Technical issues to be resolved
[0005] The present invention provides an integrated optical isolator based on a dual-arm phase modulator and an optical isolation method, which are used to at least partially solve one of the above technical problems.
[0006] (II) Technical solution
[0007] According to a first aspect of the present disclosure, an integrated optical isolator based on a dual-arm phase modulator is provided, comprising: a microring resonator for screening input optical signals to retain optical signals in a specific frequency domain; a dual-arm phase modulator for phase modulating optical signals output by the microring resonator or reflected optical signals; an optical delay waveguide for connecting optical waveguides in the dual-arm phase modulator; and a cross waveguide for realizing cross transmission of optical signals.
[0008] According to an embodiment of the present disclosure, the microring resonator includes: a first straight waveguide, used to input the received optical signal into the ring waveguide through the first coupling region; the ring waveguide, used to form a resonant cavity of the microring resonator, and perform frequency domain filtering on the input optical signal; a second straight waveguide, used to output the optical signal after resonance processing to the dual-arm phase modulator through the second coupling region; a metal thermoelectrode, arranged above the ring waveguide, used to adjust the resonant wavelength of the microring resonator; and a semicircular waveguide, used to avoid end face reflection.
[0009] According to an embodiment of the present disclosure, a dual-arm phase modulator includes: an optical waveguide group for transmitting an optical signal; and a traveling wave electrode group for modulating the optical signal transmitted in the optical waveguide group.
[0010] According to an embodiment of the present disclosure, the traveling wave electrode group includes: a signal electrode for transmitting a radio frequency signal; and a ground electrode group for terminating the electric field lines emitted from the signal electrode.
[0011] According to an embodiment of the present disclosure, the optical waveguide group includes a first optical waveguide and a second optical waveguide, the ground electrode group includes a first ground electrode and a second ground electrode, the first optical waveguide is located between the signal electrode and the first ground electrode, and the second optical waveguide is located between the signal electrode and the second ground electrode.
[0012] According to an embodiment of the present disclosure, an optical delay waveguide includes: a bent optical waveguide for transmitting an optical signal, wherein two ends of the bent optical waveguide are respectively connected to a first optical waveguide and a second optical waveguide.
[0013] According to an embodiment of the present disclosure, the cross waveguide includes: a multimode interference coupler for preventing the light field from lateral divergence at the cross point through the self-image principle; and a wedge-shaped waveguide for connecting the waveguide and the multimode interference coupler.
[0014] According to a second aspect of the present disclosure, there is provided an optical isolation method based on the above-mentioned integrated optical isolator, comprising: inputting an optical signal based on a first straight waveguide and inputting an RF signal based on a signal electrode; in the case where the optical signal and the RF signal are transmitted in reverse, adjusting the parameters of a dual-arm phase modulator and the frequency of the RF signal to enable complete transmission of the optical signal; in the case where the optical signal and the RF signal are transmitted in the same direction, transferring the energy of the optical signal to a sideband through phase modulation and frequency conversion, and filtering the sideband through a microring resonator.
[0015] According to an embodiment of the present disclosure, when the optical signal and the radio frequency signal are transmitted in the reverse direction, the phase accumulation of the optical signal after passing through the dual-arm phase modulator is a periodic function of the propagation distance in the dual-arm phase modulator. When the frequency of the radio frequency signal meets a specific value, the phase accumulation is 0, thereby achieving complete transmission of the optical signal in the transmission direction.
[0016] According to an embodiment of the present disclosure, when the optical signal and the radio frequency signal are transmitted in the same direction, the phase accumulation of the optical signal after passing through the dual-arm phase modulator is a linear function of the propagation distance in the dual-arm phase modulator. When the power of the radio frequency signal meets a specific value, the modulation depth reaches 0.765π, achieving complete suppression of the reflected optical signal at the center frequency.
[0017] (III) Beneficial effects
[0018] The integrated optical isolator based on the dual-arm phase modulator and the optical isolation method provided by the present disclosure include at least the following beneficial effects:
[0019] The half-wave voltage of the phase modulator is halved while keeping the chip length unchanged, which significantly reduces the extra power consumption of the integrated optical isolator. The operation is simple and the performance is stable. BRIEF DESCRIPTION OF THE DRAWINGS
[0020] The above and other objects, features and advantages of the present disclosure will become more apparent through the following description of the embodiments of the present disclosure with reference to the accompanying drawings, in which:
[0021] Figure 1 The structure diagram of the integrated optical isolator based on the dual-arm phase modulator provided according to the embodiment of the present disclosure is schematically shown;
[0022] Figure 2 The structure diagram of the microring resonator provided according to the embodiment of the present disclosure is schematically shown;
[0023] Figure 3 Schematically shows a structural diagram of a cross waveguide provided according to an embodiment of the present disclosure;
[0024] Figure 4 The figure schematically shows a numerical simulation diagram of the optical isolation method provided according to an embodiment of the present disclosure using the Bessel function (up to the fourth order) as the modulation index.
[0025] [Reference Signs]
[0026] 110-microring resonator; 1101-first straight waveguide; 1102-annular waveguide; 1103-second straight waveguide; 1104-metal hot electrode; 1105-semicircular waveguide; 120-dual-arm modulator; 1201a-first optical waveguide; 1201b-second optical waveguide; 1202-signal electrode; 1203a-first ground electrode; 1203b-second ground electrode; 130-optical delay waveguide; 140-cross waveguide; 1401-multimode interference coupler; 1402-wedge waveguide; 001-first coupling region; 002-second coupling region. DETAILED DESCRIPTION
[0027] In order to make the purpose, technical scheme and advantages of the present invention more clearly understood, the present invention is further described in detail below in combination with specific embodiments and with reference to the accompanying drawings. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.
[0028] The terms used herein are only for describing specific embodiments and are not intended to limit the present invention. The terms "comprise", "include", etc. used herein indicate the existence of the features, steps, operations and / or components, but do not exclude the existence or addition of one or more other features, steps, operations or components.
[0029] In the present invention, unless otherwise clearly specified and limited, the terms "installed", "connected", "connected", "fixed" and the like should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection, an electrical connection, or can communicate with each other; it can be a direct connection, or an indirect connection through an intermediate medium, it can be the internal connection of two elements or the interaction relationship between two elements. For ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.
[0030] In the description of the present invention, it should be understood that the terms "longitudinal", "length", "circumferential", "front", "rear", "left", "right", "top", "bottom", "inside", "outside", etc., indicating the orientation or positional relationship are based on the orientation or positional relationship shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the subsystem or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention.
[0031] Throughout the drawings, the same elements are represented by the same or similar reference numerals. Conventional structures or configurations will be omitted when they may cause confusion in the understanding of the present invention. The shapes, sizes, and positional relationships of the components in the drawings do not reflect the actual size, proportion, and actual positional relationship. In addition, in the claims, any reference symbol between brackets shall not be construed as a limitation to the claims.
[0032] Similarly, in order to simplify the present invention and help understand one or more of the various disclosed aspects, in the above description of exemplary embodiments of the present invention, the various features of the present invention are sometimes grouped together into a single embodiment, figure, or description thereof. The description with reference to the terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representation of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner.
[0033] In addition, the terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, the features defined as "first" and "second" may explicitly or implicitly include one or more of the features. In the description of the present invention, the meaning of "plurality" is at least two, such as two, three, etc., unless otherwise clearly and specifically defined.
[0034] Figure 1 The structure of an integrated optical isolator based on a dual-arm phase modulator provided according to an embodiment of the present disclosure is schematically shown.
[0035] like Figure 1 As shown, the integrated optical isolator based on the dual-arm phase modulator provided by the embodiment of the present disclosure includes: a microring resonator 110, a dual-arm modulator 120, an optical delay waveguide 130, and a cross waveguide 140.
[0036] The microring resonator 110 is used to filter the input optical signal to retain the optical signal in a specific frequency domain. The microring resonator 110 can be used as a filter to suppress the sideband of the reflected light.
[0037] Figure 2 The structure of a microring resonator provided according to an embodiment of the present disclosure is schematically shown.
[0038] like Figure 2 As shown, in some embodiments, the microring resonator includes a first straight waveguide 1101 , a ring waveguide 1102 , a second straight waveguide 1103 , a metal hot electrode 1104 , and a semicircular waveguide 1105 .
[0039] The first straight waveguide 1101 is used to input the received optical signal into the ring waveguide 1102 through the first coupling region 001 .
[0040] The ring waveguide 1102 is used to form a resonant cavity of the microring resonator and perform frequency domain filtering on the input optical signal. When the optical signal is input to the ring waveguide, the light wave with a wavelength that meets certain conditions can form effective propagation and interference in the ring waveguide, thereby forming a stable standing wave mode in the ring waveguide, generating a resonance phenomenon, and realizing frequency domain filtering.
[0041] The second straight waveguide 1103 is used to output the optical signal after the resonance processing to the dual-arm phase modulator through the second coupling region 002.
[0042] The metal thermode 1104 is disposed above the annular waveguide and is used to adjust the resonant wavelength of the microring resonator. When the metal thermode is powered on and heated, the heat is transferred to the annular waveguide, causing the refractive index of the annular waveguide to change, thereby adjusting the resonant wavelength of the microring resonator. The semicircular waveguide 1105 is used to avoid end face reflection.
[0043] The dual-arm phase modulator 120 is used to phase modulate the optical signal and / or the reflected optical signal output by the microring resonator. The reflected optical signal is an optical signal formed by the optical signal output after passing through the integrated optical isolator and reflected by other devices. By phase modulating the optical signal and / or the reflected optical signal by the dual-arm phase modulator, the reflected light transmission can be effectively suppressed while ensuring the forward light transmission.
[0044] In some embodiments, the dual-arm phase modulator includes an optical waveguide group and a traveling wave electrode group. The optical waveguide group includes a first optical waveguide 1201a and a second optical waveguide 1201b, which are used to transmit optical signals. The traveling wave electrode group includes a signal electrode 1202 and a ground electrode group, which are used to modulate the optical signal transmitted in the optical waveguide group. The signal electrode 1202 is used to transmit radio frequency signals, and the ground electrode group is used to terminate the electric field lines emitted from the signal electrode. Among them, the ground electrode group includes a first ground electrode 1203a and a second ground electrode 1203b, the first optical waveguide 1201a is located between the signal electrode 1202 and the first ground electrode 1203a, and the second optical waveguide 1201b is located between the signal electrode 1202 and the second ground electrode 1203b.
[0045] The optical delay waveguide 130 is used to connect the optical waveguides in the dual-arm phase modulator.
[0046] In some embodiments, the optical delay waveguide includes a curved optical waveguide for transmitting optical signals, one end of which is connected to the first optical waveguide 1201a and the other end of which is connected to the second optical waveguide 1201b. The waveguide used in the integrated optical isolator provided in the embodiment of the present disclosure is a ridge waveguide.
[0047] The cross waveguide 140 is used to realize cross transmission of optical signals to ensure that the optical signals are cross-transmitted with minimum interference and loss.
[0048] Figure 3 The structure of a cross waveguide provided according to an embodiment of the present disclosure is schematically shown.
[0049] like Figure 3 As shown, in some embodiments, the crossing waveguide includes a multimode interference coupler 1401 and a wedge-shaped waveguide 1402. The multimode interference coupler 1401 is used to prevent the light field from lateral divergence at the crossing point through the self-image principle. The wedge-shaped waveguide 1402 is used to connect the waveguide and the multimode interference coupler.
[0050] Based on the above-mentioned integrated optical isolator based on the dual-arm phase modulator, the present disclosure also provides an optical isolation method.
[0051] The optical isolation method comprises: inputting an optical signal based on a first straight waveguide and inputting a radio frequency signal based on a signal electrode; in the case where the optical signal and the radio frequency signal are transmitted in reverse, adjusting the parameters of a dual-arm phase modulator and the frequency of the radio frequency signal to enable complete transmission of the optical signal; in the case where the optical signal and the radio frequency signal are transmitted in the same direction, transferring the energy of the optical signal to a sideband through phase modulation and frequency conversion, and filtering the sideband through a microring resonator.
[0052] The following will be combined Figure 1 The method is described in detail with the integrated optical isolator based on the dual-arm phase modulator shown.
[0053] like Figure 1 As shown, the wavelength of the light source input from the first straight waveguide is The optical signal is transmitted to the ring waveguide through the first coupling region 001. When the wavelength of the optical signal overlaps with the resonant wavelength of the microring resonator, the light of this wavelength can form a stable standing wave mode in the ring waveguide, generating a resonance phenomenon. The expression of the resonant wavelength is:
[0054]
[0055] in, is the resonant wavelength, L r is the circumference of the ring waveguide, n eff is the effective refractive index of the ring waveguide, and m is an integer.
[0056] By changing the voltage applied to the metal thermode, the effective refractive index of the ring waveguide can be changed, thereby adjusting the resonant wavelength of the microring resonator. Resonance wavelength of the microring resonator When overlapping, the optical signal can be output to the first optical waveguide of the dual-arm phase modulator through the second coupling region.
[0057] While the optical signal is input from left to right into the first optical waveguide, the radio frequency signal is input into the signal electrode and transmitted from right to left through the signal electrode, forming electric fields in opposite directions on the cross sections of the first optical waveguide and the second optical waveguide. The frequency of the radio frequency signal is f s , the peak voltage is A. At this time, the optical signal and the RF signal are transmitted in the opposite direction.
[0058] Since the transmission direction of the input optical signal is opposite to that of the electrical signal (i.e., the RF signal), the total phase accumulation of the optical signal after being transmitted through the first optical waveguide can be expressed as:
[0059]
[0060] Among them, R f π is the modulation index when the optical signal and the electrical signal are transmitted in reverse, L is the electrode length of the two-arm phase modulator, α is the microwave power attenuation coefficient, n opt is the light group refractive index, n RF is the microwave refractive index. According to the expression of total phase shift accumulation, when the optical signal and the electrical signal are transmitted in the opposite direction, the phase accumulation is a periodic function of the propagation distance in the two-arm phase modulator. Assuming that the microwave power attenuation coefficient is 0, when the RF signal frequency f s satisfy , the phase accumulation is 0. Then, the optical signal enters the second optical waveguide through the optical delay waveguide, wherein the modulation process experienced by the optical signal in the second optical waveguide is exactly the same as that in the first optical waveguide. Therefore, the phase accumulation of the optical signal in the second optical waveguide is still 0 (i.e., the optical signal is not modulated), realizing the complete transmission of the optical signal in the current transmission direction, and output after passing through the cross waveguide.
[0061] After the optical signal is output, it is reflected by other devices to form a reflected optical signal. The reflected optical signal is reflected back to the dual-arm phase modulator through the cross waveguide. At this time, the transmission direction of the reflected optical signal is the same as that of the RF signal, that is, the optical signal and the RF signal are transmitted in the same direction.
[0062] The output light field of the reflected light after passing through the second optical waveguide, the optical delay waveguide and the first optical waveguide can be expressed as:
[0063]
[0064] Where, E0 is the amplitude of the optical carrier, is the angular frequency of the optical carrier, is the time delay between the first optical waveguide and the second optical waveguide, ,in, is the length of the optical waveguide between the input ends of the first ground electrode and the second ground electrode, and its length is equal to the sum of the length of the optical delay waveguide and the length of the signal electrode, , where A is the peak voltage of the microwave signal, is the half-wave voltage of the single-arm phase modulator during forward modulation, f s is the frequency of the RF signal.
[0065] When the delay ΔT satisfies When , the modulation index of the optical signal output by the dual-arm phase modulator in the disclosed embodiment is twice that of the single-arm modulation, that is, β=2·π·R b . Where k is an integer.
[0066] In summary, when the amplitude A of the RF signal satisfies ; frequency f s satisfy , where m is an integer; the length of the delay waveguide is L s satisfy , where k is an integer, the integrated optical isolator provided in the embodiment of the present disclosure achieves an optical isolation effect with maximum isolation at minimum power consumption.
[0067] Figure 4 The figure schematically shows a numerical simulation diagram of the optical isolation method provided according to an embodiment of the present disclosure using the Bessel function (up to the fourth order) as the modulation index.
[0068] like Figure 4 As shown in the figure, the dot is the working point of the integrated optical isolator provided by the embodiment of the present disclosure. According to the Bessel function expansion, when the modulation index β is about 0.765π, that is, A is about 0.3825 When f c The reflected light at the frequency f can be completely depleted and the power can be distributed to the c +nf s At the sideband of , n is an integer.
[0069] Afterwards, the modulated reflected light enters the second straight waveguide. Since the wavelength corresponding to the sideband is consistent with the resonant wavelength λ of the microring resonator, m There is no overlap, and ultimately only a tiny amount of reflected light at the suppressed f0 is coupled into the first straight waveguide and then enters the light source.
[0070] Therefore, a higher degree of isolation can be obtained through the integrated optical isolator and optical isolation method based on a dual-arm phase modulator proposed in the present disclosure. In summary, the integrated optical isolator and optical isolation method based on a dual-arm phase modulator provided in the present disclosure reduces the half-wave voltage of the phase modulator by half while ensuring that the chip length remains unchanged, significantly reduces the additional power consumption of the integrated optical isolator, and is simple to operate and has stable performance.
[0071] The embodiments of the present invention are described above. However, these embodiments are only for the purpose of illustration, and are not intended to limit the scope of the present invention. Although each embodiment is described above, it does not mean that the measures in each embodiment cannot be used in combination advantageously. Without departing from the scope of the present invention, those skilled in the art may make various substitutions and modifications, which should all fall within the scope of the present invention.
Claims
1. An integrated optical isolator based on a dual-arm phase modulator, characterized in that: The integrated optical isolator comprises: A microring resonator is used to filter the input optical signal to retain the optical signal in a specific frequency domain; A dual-arm phase modulator, used for phase modulating an optical signal output by the microring resonator or a reflected optical signal; An optical delay waveguide, used to connect the optical waveguide in the dual-arm phase modulator; Cross waveguides are used to achieve cross transmission of optical signals.
2. The integrated optical isolator according to claim 1, characterized in that: The microring resonator comprises: A first straight waveguide, used to input the received optical signal into the ring waveguide through the first coupling region; The ring waveguide is used to form the resonant cavity of the microring resonator and perform frequency domain filtering on the input optical signal; A second straight waveguide, used for outputting the optical signal after the resonance processing to the dual-arm phase modulator through the second coupling region; A metal thermode, disposed above the ring waveguide, for adjusting the resonant wavelength of the microring resonator; Semicircular waveguide, used to avoid end face reflection.
3. The integrated optical isolator according to claim 1, characterized in that: The dual-arm phase modulator comprises: An optical waveguide group, used for transmitting optical signals; The traveling wave electrode group is used to modulate the optical signal transmitted in the optical waveguide group.
4. The integrated optical isolator according to claim 3, characterized in that: The traveling wave electrode group comprises: A signal electrode, used for transmitting a radio frequency signal; The ground electrode group is used to terminate the electric field lines emitted from the signal electrode.
5. The integrated optical isolator according to claim 4, characterized in that: The optical waveguide group includes a first optical waveguide and a second optical waveguide, the ground electrode group includes a first ground electrode and a second ground electrode, the first optical waveguide is located between the signal electrode and the first ground electrode, and the second optical waveguide is located between the signal electrode and the second ground electrode.
6. The integrated optical isolator according to claim 1, characterized in that: The optical delay waveguide comprises: The bent optical waveguide is used for transmitting optical signals, and two ends of the bent optical waveguide are respectively connected to the first optical waveguide and the second optical waveguide.
7. The integrated optical isolator according to claim 1, characterized in that: The cross waveguide comprises: Multimode interference coupler, used to avoid lateral divergence of the light field at the intersection point through the principle of self-image; Wedge-shaped waveguides are used to connect waveguides and multimode interference couplers.
8. An optical isolation method based on the integrated optical isolator according to claims 1 to 7, characterized in that: The optical isolation method comprises: Inputting an optical signal based on the first straight waveguide and inputting a radio frequency signal based on the signal electrode; In the case where the optical signal and the radio frequency signal are transmitted in reverse, adjusting the parameters of the dual-arm phase modulator and the frequency of the radio frequency signal so that the optical signal is completely transmitted; In the case where the optical signal and the radio frequency signal are transmitted in the same direction, the energy of the optical signal is transferred to the sideband through phase modulation and frequency conversion, and the sideband is filtered through the microring resonator.
9. The optical isolation method according to claim 8, characterized in that: When the optical signal and the radio frequency signal are transmitted in reverse, the phase accumulation of the optical signal after passing through the dual-arm phase modulator is a periodic function of the propagation distance in the dual-arm phase modulator. When the frequency of the radio frequency signal meets a specific value, the phase accumulation is 0, thereby achieving complete transmission of the optical signal in the transmission direction.
10. The optical isolation method according to claim 8, characterized in that: When the optical signal and the radio frequency signal are transmitted in the same direction, the phase accumulation of the optical signal after passing through the dual-arm phase modulator is a linear function of the propagation distance in the dual-arm phase modulator. When the power of the radio frequency signal meets a specific value, the modulation depth reaches 0.765π, so that the reflected optical signal is completely suppressed at the center frequency.
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