九州大学大学院総合理工学報告第 46 巻 第 1 号 1-6 頁 令和 6 年 9 月

Engineering Sciences Reports, Kyushu University
Vol. 46, No. 1, pp. 1-6, SEP, 2024

Photon-photon resonance phenomenon in active
multimode interferometer laser as Fabry-Perót laser
He XIAO*1, Yudai KAWANO*1, Haisong JIANG*2†,
and Kiichi HAMAMOTO*2
E-mail of corresponding author: jiang.haisong.447@m.kyushu-u.ac.jp†(Received January 10, 2024, accepted March 11, 2024)
Photon-photon resonance (PPR) has attracted attention for its capability to extend modulation
speed of directly modulated laser (DML). PPR is phenomenon resulted from the interference of twolasing modes. For the general Fabry-Perót (F-P) lasers, though multiple wavelengths exist, it has been
considered as a structure that could not explore PPR phenomenon. This is because the lasing modes in
F-P laser are all orthogonal. Active-MMI (multimode interferometer) laser as a F-P laser, however, hasbeen observed the PPRs experimentally. In this paper, the existence of PPR in active-MMI LD has
been confirmed theoretically. This calculation method provides a way to evaluate the location of PPR
in the design.
Key words: PPR, DML, modulation bandwidth, Fabry-Perót lasers

1.

Introduction

Directly modulated laser (DML) is expected
as one of the suitable data-transmitters
especially in computing system 1). The directly
modulated laser’s modulation bandwidth ingeneral, is restricted by so-called relaxation
oscillation frequency (or carrier photon
resonance (CPR) frequency) 2).
Lots of efforts
have been paid to extend DML modulation
bandwidth 3-10). One possible way to overcome
the restriction of relaxation oscillation is toutilize PPR (photon-photon resonance) which
realizes higher frequency resonance comparedto CPR 11-25). From the previous works, all theconfigurations that have been utilized to
exploit PPR phenomenon share the common
grating. All the reported DMLs with PPR are
those distributed Bragg reflector (DBR) lasers
or distributed feedback (DFB) lasers. PPR is
known as a resonance caused by the interaction
between two lasing modes. PPR phenomenon is
not able to be observed in Fabry-Perót (F-P)
laser because of the mode orthogonality 26-27)
among the longitudinal modes. In 2017,
however, the existence of the PPR phenomenonin F-P laser has been proved 28-32). By applying*1 Department of Applied Science for Electronics andMaterials. Graduate Student*2 Department of Advanced Energy Science andEngineering.

active-multimode interferometer (MMI) laser,
the plural PPRs phenomenon was observed
which contradicts the principle.
In this paper, the mode orthogonality in F-P
laser will be clarified. Then, non-orthogonality
in active-MMI for multiple PPRs is explained.
Since active-MMI laser naturally accesses
multiple waveguides, different oscillators exist
that non-orthogonal mode do exist in activeMMI laser which provides the possibility to
exploit PPR.

2. Non-orthogonality in active-MMI
laser diodeTo explain the mode orthogonality in FabryPerót laser, longitudinal lasing modes in F-P
laser is clarified first 33). Figure 1 is a schematic
diagram of F-P laser. The F-P cavity whose
length is 𝐿 , is filled with semiconductormaterial with a refractive index 𝑛𝑟 . At the bothends, there are two reflection mirrors withreflective
index
𝑅1
and
𝑅2 .
The
electromagnetic wave that exists in this cavity
is expressed in2

Photon-photon resonance phenomenon in active multimode interferometer laser as Fabry-Perót laser𝑑ф(𝑡) 𝛼= [𝐺2 (𝑁(𝑡) − 𝑁𝑡ℎ )]𝑑𝑡
2
𝑆1 (𝑡)−𝜅𝑠𝑖𝑛 ф(𝑡) − 𝛥𝜔𝑆2 (𝑡)

L

z

ⅆ𝑁(𝑡)
ⅆ𝑡

𝑅2
𝑅1
Fig. 1 Schematic diagram of Fabry-Perót laser.
𝑖2𝜋𝑛𝑟 𝑧𝜆0

(1)

] 𝑒𝑥𝑝[(𝑔 − 𝛼𝑖 )𝑧]In the expression, 𝜆0 is the wavelength in freespace. 𝛼𝑖 is the internal loss coefficient. 𝑔 is
gain coefficient. The condition for the existenceof an electromagnetic wave is that, when thiselectromagnetic wave is reflected to its origin,its amplitude has to be equal to the starting
value. For the lasing modes, the electronic
fields are𝑧)𝜋

𝐸𝑚 = 𝐴𝑚 𝑒 𝑗𝑚(𝐿(2)
𝐴𝑚 is the amplitude. 𝐿 is the cavity length.
𝑚 is the order of the longitude mode. The PPR
phenomenon
enhancethemodulation
bandwidth as the interaction between twomodes increase. The strong lasing mode iscalled main mode. The weak lasing mode iscalled sub mode. The coupling efficiency 𝜂𝑧
between main and sub modes is important to
PPR phenomenon. The coupling efficiency
between different modes is explained as thespatial overlap between different modes asshown in Eq. (3).
2𝐿𝜂𝑧 = ∫0 𝐸𝑚𝑎𝑖𝑛 𝐸𝑠𝑢𝑏 𝑑𝑧(3)
As the lasing modes in Fabry-Perót laser
originate from Fabry-Perot resonance in thesame cavity, all the longitudinal lasing modes
are orthogonal to each other. In this case, thecoupling efficiency 𝜂𝑧 , which is expressed in Eq.
(3) is calculated as below:
2𝐿

𝑧

𝑧

𝜂𝑧 = ∫0 𝐴𝑚𝑎𝑖𝑛 𝑒 𝑗𝑚𝑚𝑎𝑖𝑛 (𝐿)𝜋 𝐴𝑠𝑢𝑏 𝑒 𝑗𝑚𝑠𝑢𝑏 (𝐿)𝜋 𝑑𝑧=

𝐴𝑚𝑎𝑖𝑛 𝐴𝑠𝑢𝑏

𝑁(𝑡)
𝜏𝑟

− 𝐺1 [𝑁(𝑡) − 𝑁𝑡𝑟 ] × [𝑆1 (𝑡)] −𝐺2 [𝑁(𝑡) − 𝑁𝑡𝑟 ] × [𝑆2 (𝑡)]𝑛𝑟

𝐸 = 𝑒𝑥𝑝 [=𝐽−𝑚𝑚
𝑗𝜋( 𝑚𝑎𝑖𝑛 + 𝑠𝑢𝑏 )𝐿𝐿𝑧𝑒 𝑗(𝑚𝑚𝑎𝑖𝑛 +𝑚𝑠𝑢𝑏) 𝐿𝜋 |2𝐿
0 = 0 (4)And the coupling efficiency 𝜂𝑧 is to be zero.
The result means that all the lasing modes in
the Fabry-Perót cavity are orthogonal and thus
any PPR phenomenon is not caused any more.
According to the PPR rate equation 31),
𝑑𝑆1 (𝑡)= 𝐺1 [𝑁(𝑡) − 𝑁𝑡ℎ ] × [𝑆1 (𝑡)]𝑑𝑡
𝑑𝑆2 (𝑡)= 𝐺2 [𝑁(𝑡) − 𝑁𝑡ℎ ] × [𝑆2 (𝑡)]𝑑𝑡
+ 2𝜅√[(𝑆1 (𝑡)] × [𝑆2 (𝑡)]𝑐𝑜𝑠ф(𝑡)(5)𝑆1 , 𝑆2 , are the photon number of two lasing
modes in laser diode. 𝑁(𝑡) and 𝑁𝑡ℎ , 𝑁𝑡𝑟 , 𝛼
are carrier number and the threshold carrier
number,transparency
carriernumber,linewidth-enhancement factor, respectively. 𝜏𝑟is carrier lifetime. 𝐺1 , 𝐺2 is the gain coefficientof the main lasing mode and sub-lasing mode.
𝛥𝜔 is the frequency difference between twolasing modes. 𝐽 is the current injection. 𝜅 isthe coupling parameter. In the F-P lasers,
because of the orthogonality of all lasing modes,
there is no interaction between different modes.
The interaction terms related to coupling
parameter 𝜅 in Eq. (5) are zero that the rateequations with PPR degenerate to the general
rate equations without PPR. Thus, the PPR
phenomenon is rarely observed in F-P lasers.
The configuration of active-MMI laser [32]reported to observe multiple PPRs is shown in
Fig. 2.
Upper arm
Path1
Center arm Path 2
Lower arm Path 3

MMI area
15 μm
Output part350 μm50 μm490 μmFig. 2 Schematic diagram of active-MMI laser.

The situation of Fabry-Perót resonance in
active-MMI LD is slightly different from theabove-discussed general case of Fabry-Perót
LD. Even though in a single device, there areseveral different light-path inside of the device.
There are three parts in this device. In the
middle is the MMI areas. The LHS of MMI area
are three arm parts, called upper arm, middle
arm and lower arm respectively.
In the RHSof the MMI area, it is another waveguideworking as modulation port. There are threedifferent optical paths named path 1, path 2
and path 3 in that cavity.
The coupling
efficiency 𝜂𝑧 in active-MMI laser diode is
2𝐿𝑚𝑎𝑖𝑛𝜂𝑧 = ∫
0

𝐴𝑚𝑎𝑖𝑛 𝑒𝑗𝑚𝑚𝑎𝑖𝑛𝑧𝜋
𝐿𝑚𝑎𝑖𝑛 𝐴

𝑠𝑢𝑏 𝑒

𝑗𝑚𝑠𝑢𝑏

𝑧𝜋
𝐿𝑠𝑢𝑏 𝑑𝑧令和 6 年度=

九州大学大学院総合理工学報告

𝐴𝑚𝑎𝑖𝑛 𝐴𝑠𝑢𝑏
𝜋𝑚1𝑚𝑎𝑖𝑛 𝜋𝑚𝑠𝑢𝑏
+
)
𝐿
𝐿

𝑗(

𝑒

𝑚
𝑚
𝑗( 𝑚𝑎𝑖𝑛 + 𝑠𝑢𝑏 )𝑧𝜋 2𝐿𝑚𝑎𝑖𝑛
𝐿𝑚𝑎𝑖𝑛 𝐿𝑠𝑢𝑏

|0

(6)

As the optical cavity in active-MMI laser diodefor different path is different, the coupling
efficiency 𝜂𝑧 is𝐴𝑚𝑎𝑖𝑛 𝐴𝑠𝑢𝑏𝜂𝑧 =𝑚𝑚𝑎𝑖𝑛 𝑚𝑠𝑢𝑏 ×
𝑗𝜋(
+
)
𝐿𝑚𝑎𝑖𝑛
𝐿𝑠𝑢𝑏
(𝑒

𝐿
𝑗(2𝑚𝑚𝑎𝑖𝑛 +2𝑚𝑠𝑢𝑏 𝑚𝑎𝑖𝑛 )𝜋𝐿𝑠𝑢𝑏− 1) ≠ 0

Therefore, non-orthogonal lasing modes exist
in active-MMI laser diode. To confirm this path
difference in the active-MMI LD shown in Fig.
2, we do simulations by using the FDTD (finitedivision time-domain) method 20).
The initial light is injected from three arms
respectively. We monitored the phase of the
propagated light at the single output
waveguide end. If the path length is the sameamong three different injection conditions, theachieved phase as a function of wavelengthmust be completely matched to each other. The
simulation results are shown in Fig. 3 (a1). The
phase conditions for each path are different
from each other. This confirms our thought that
there are non-orthogonal modes in active-MMI
LD.
360[degree ]Path 1
Path 2

Phase

Path 301549

1549.5 1550 1550.5
Wavelength [nm]
(a1)

1551

1549.51551

Intensity [a.u.]

1
0.8
0.6
0.4
0.2
0
1549

1550

1550.5

Wavelength [nm](a2)Fig. 3 (a1) Phase in each path in active-MMI laser;(a2) Transmittance of each arm in active-MMI laser.

第 46 巻

第 1 号

3From Fig. 3 (a2), the wavelength whosetransmittance is equal to 1 are modes that may
exist in the path. Hence, the effective optical
path length for each path. For each path, thosewavelengths whose transmittance is equal to 1
are called 𝜆𝑝𝑒𝑎𝑘 . In the simulation range, there
are three 𝜆𝑝𝑒𝑎𝑘 for each path. With therelationship expressed in Eq. (2), the optical
path for each path is achieved and shown in
Table 1.
Table 1 Optical path for three paths in active-MMI LD.
𝜆𝑝𝑒𝑎𝑘1𝜆𝑝𝑒𝑎𝑘

ΔλOptical path

[nm]

[nm]

[nm]

[µm]

Path 1

1549.35

1550.10.75

500.33

Path 2

1549.25

1549.975

0.725

517.52Path 31549.2

1549.925

0.725

517.49Based on the research on the PPR
phenomenon, the spectrum of laser is very
important information to confirm the PPR
phenomenon. From all the works concentrated
on the PPR phenomenon, experimentally,
except for the main lasing wavelength, there isanother weak lasing wavelength. The locationof PPR is derived from𝑓𝑝𝑝𝑟 = 𝑐(

1

𝜆𝑚𝑎𝑖𝑛

−

1
𝜆𝑠𝑢𝑏

)

(7)

where 𝜆𝑚𝑎𝑖𝑛 and 𝜆𝑠𝑢𝑏 are wavelengths for thestrong lasing mode and weak lasing mode. 𝑐 isthe light velocity in vacuum. It indicates thatthe location of PPR is related to the spectrum.
Thus, the predictable spectrum will help us toutilize PPR to extend the modulation
bandwidth.

3.

Spectrum of active-MMI LD

3.1

Simulation results

To get the spectrum, the intensity ratio of
each arm has to be clarified. Because of the selfimage phenomenon 34), the ratio of opticalpower from each path is not the same weight in
active-MMI LD. As shown in Fig. 4, when thelight is injected from the single-end side, thepercentage of the power in each path isdifferent. The majority of power is propagatedthrough path 1 which is accessed with theupper arm.
The second most power is
propagated through path 2 connected to the
middle arm. Path 3 related to the lower arm
contributes the least to light propagation.

4

Photon-photon resonance phenomenon in active multimode interferometer laser as Fabry-Perót laser
Table 2 The calculated PPR location.

1.0

Δλ
[nm]
PPR location[GHz]
0

Intensity [a.u.]

0.6

63

75

-30

1550.7nm

-40
-50
-60
-70
1549

Path 1

1549.51550

1550.51551

Wavelength [nm]Path 2
Fig. 6 Calculated spectrum of active-MMI laser.

Path 3

0.4

3.2

0.2
0
1549

24

Power [dB]

Taking this intensity ratio into account, thetransmittance of each arm will be shown in Fig.
5.

0.8

0.6

-20

Fig. 4 Intensity ratio of each path in active-MMI laser.

1

0.5

1550.1nm
1549.9nm
1550.6nm

-10

0

0.2

1549.51550

1550.51551

The small signal modulation response of the
proposed 1×3 active-MMI LD 31) is shown isshown in Fig. 7.
30

It is necessary to clarify that, when thisconfiguration is working as an active device,
the lasing mode generated by path 1 may not
be the main lasing mode, and the lasing mode
related to another path may not be the sublasing mode either. This is because all thesimulations we carried out did not consider thegain. Thus, it is unclear which one is the main
lasing mode until the laser is lasing. However,
the wavelength difference does not change.
From the wavelength difference, the location of
PPR is obtained. The spectrum from passive
device simulation is calculated. The results arecalculated with the condition that lasers havebeen through 6 times round-trip times. Figure
6 displays four possible lasing wavelengths1549.9 nm, 1550.1 nm,1550.6 nm and 1550.7
nm. Though we don’t know which one is themain lasing laser, we assume that the laser
with wavelength of 1550.1 is the main lasing
mode. The details are listed in Table 2. Basedon that assumption, there are three possible
locations for PPRs. It is 24 GHz, 63 GHz and 75
GHz.

Output Power [dB]

Wavelength [nm]Fig. 5 Transmittance of each arm in active-MMI laser.

Experimental results

20
10
0
-10
-20
-30
-40

0

10

20 30

40 50

60 70

Frequency [GHz]Fig. 7 Calculated spectrum of active-MMI laser.

Two PPRs are observed. One is located at 20
GHz and another is located at 60 GHz. The
experiment confirms the prediction of thespectrum. The location of PPR in theexperiment is almost in accordance with thetheoretical calculation. The reason why there isa difference between theoretical calculationand experiment lies in the lasing condition. The
current injection of laser changes the refractive
index of the device, which leads to the change
of PPR location. From Table 2, there might a
PPR located around 75 GHz which is over therange of the experiment equipment.

令和 6 年度

4.

九州大学大学院総合理工学報告第 46 巻Summary and Conclusion

In this paper, non-orthogonal modes in
active-MMI LD are explained. According to thecalculation of non-orthogonal modes in activeMMI LD, the theoretical calculation of PPR
location in active-MMI LD has been presented.
The simulation result is in accordance with theexperiment result. This method provides thepossibility to calculate the PPR location that
benefits
themodulation
bandwidth
enhancement by utilizing PPR.
Acknowledgments
This work was partly supported by Kakenhi#19H02201 and #22H01556.
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