Four-Wave Mixing in Optical Fibers and Its Applications
by Osamu Aso *, Masateru Tadakuma * and Shu Namiki *Four-wave mixing (FWM) is a phenomenon that must be avoided in DWDM
transmission, but depending on the application it is the basis of important second-generation optical devices and optical device measurement technology. This paper discusses the theory of FWM, and then introduces one of its applications --a broadband all-optical simultaneous wavelength converter developed using a high nonlinearity dispersion fiber (HNL-DSF)
that efficiently produces FWM. The conversion bandwidth extends to 23.3 nm HWHM (half widthat half maximum), the widest yet reported for wavelength conversion using non-polarizationmaintaining fiber. As a further application, a novel technique is introduced for measuring the nonlinear coefficient of optical fibers by evaluating FWM generating efficiency. With this technique it
is now possible to effect simultaneous measurement of the chromatic dispersion and nonlinear
coefficient of fiber.

ABSTRACT1.

INTRODUCTION

When a high-power optical signal is launched into a fiber,
the linearity of the optical response is lost. One such nonlinear effect, which is due to the third-order electric susceptibility is called the optical Kerr effect. 1), 2) Four-wave
mixing (FWM) is a type of optical Kerr effect, and occurs
when light of two or more different wavelengths islaunched into a fiber. Generally speaking FWM occurs
when light of three different wavelengths is lauched into afiber, giving rise to a new wave (know as an idler), thewavelength of which does not coincide with any of the others. FWM is a kind of optical parametric oscillation.
In the transmission of dense wavelength-division multiplexed (DWDM) signals, FWM is to be avoided, but for certain applications, it provides an effective technological basis
for fiber-optic devices. FWM also provides the basic technology for measuring the nonlinearity and chromatic dispersion of optical fibers. This paper discusses those aspects of
R & D into FWM applications that the authors have carriedout recently in connection with broadband all-optical simultaneous wavelength conversion and a technique for measuring the nonlinear coefficient of optical fibers.

f idler = f p1 + f p2 - f probe(1)

where: fp1 and fp2 are the pumping light frequencies, andfprobe is the frequency of the probe light. 1), 2)
This condition is called the frequency phase-matchingcondition. When the frequencies of the two pumping
waves are identical, the more specific term "degenerated
four-wave mixing" (DFWM) is used, and the equation forthis case may be written

f idler = 2f p - f probe(2)

where: fp is the frequency of the degenerated pumping
wave.
Continuous-wave DFWM may be expressed by the following nonlinear coupled-mode equations 1)
Pumping light

Probe light
Idler light

Frequencya) 2-channel pump wave

2.

THEORY OF FWM

Figure 1 is a schematic diagram that shows four-wave
mixing in the frequency domain. As can be seen, the lightthat was there from before launching, sandwiching the two
pumping waves in the frequency domain, is called theprobe light (or signal light). The idler frequency fidler may
then be determined by

Pumping light

Probe light
Idler light

Frequencyb) 1-channel pump wave (degenerated FWM)

Figure 1*

WP Team, Opto-technology Lab., R & D Div.

Schematic of four-wave mixing in the frequency
domain63

dE p
1+ α Ep = i γ
dz2( Ep 2+2 Eprobe 2+2 Eidler 2) Ep+2i γ E*pEprobeEidlerexp(i∆β z)

1
dE probe+
α E probe = i γ
2dz
dE idler1+
α E idler = i γ
dz2( Eprobe 2+2 Eidler 2+2 Ep 2 )Eprobe+2i γ E*idlerEp2exp(-i ∆β z)( Eidler 2+2 Ep 2+2 Eprobe 2)Eidler+2i γ E*probeEp2exp(-i ∆β z)where: z is the longitudinal coordinate of the fiber, α isthe attenuation coefficient of the fiber, and Ep,
Eprobe and Eidler are the electric field of the pumping, probe and idler waves.

γ is the nonlinear coefficient, and is obtained by 1)γ ≡2 π fp . n2
c
A eff

(4)

where: n2 is the nonlinear refractive index, Aeff is the effective area of the fiber and c is the speed of light ina vacuum.
The term ∆β in Equation (3) represents the phase mismatch of the propagation constant, and may be defined as∆β = β probe + βidler - 2 β pump = -8 π f p2
D( fp)( fprobe - fp)c

To generate FWM efficiently, it is required that pumpwavelength conincides with the fiber zero-dispersion
wavelength. 3) The first term on the right side of Equation(3) represents the effects of self-phase modulation (SPM)
and cross-phase modulation (XPM) resulting from theoptical Kerr effect.

WAVELENGTH CONVERSION BY FIBER
FOUR-WAVE MIXING

3.1 Significance of Wavelength Converters
Wavelength converter is simply a device for converting theinjected signal light from one wavelength to another. 8)~13) It
therefore is seen to have great promise in configuring thephotonic networks of the future using optical cross connects. A number of methods of wavelength conversion
have been proposed, of which parametric conversion
using optical fiber FWM offers two major advantages: highconversion speed and the ability to effect simultaneousconversion of signals within a wavelength bandwidth.
3.2 Wavelength Conversion in the Fiber
The most important characteristics desired of wavelength
converters using parametric conversion are high conversion efficiency and broad bandwidth.
To achieve this kind of wavelength conversion, the following conditions must be met:(a) pump wavelength must coincide with zero-dispersion
wavelength;(b) chromatic dispersion variation in the longitudinal64

direction of the fiber should be minimized; and(c) states of polarization of the pump and signals mustcoincide.
As has already been argued in the literature, 6), 7) in orderto broaden the conversion bandwidth, consideration must
additionally be given to coherence length. The arguments
concerning efficient DFWM generation may be summarized as follows: Letting ∆f be the frequency spacing
between the pumping light and the signal (or idler) light,
fiber length L must, to produce effective DFWM across thefrequency band, satisfy the condition

L ≤ L coh ≡2πc. 1 ∝ 1=∆β∆f 2
4 π 2 f p D( f p ) ∆ f 2(6)

where: Lcoh is coherence length, a parameter having alength dimension.

(5)

where: D is the chromatic dispersion coefficient.

3.

(3)

As Equation (6) shows, fiber length must be reduced toeffect broadband simultaneous wavelength conversion atlarge values of ∆f. Reducing fiber length is also significant
in terms of condition (b), since it results in a homogeneouschromatic dispersion distribution along the fiber. Reducing
fiber length is also effective in satisfying condition (c).
Unless polarization-maintaining fiber (PMF) is used, thestate of polarization at launching is not maintained until
output. This is due to variations in polarization in thelength direction caused by birefringence within the fiber.
Even if the state of polarization is aligned at the time oflaunching into the fiber, the relative phase difference
between the pumping light and the signal light can be
expressed, if birefringence ∆n is present, as

∆φ =

2π
∆ n. ∆ f.L
c(7)

One way of achieving a broader conversion band ∆f isto reduce ∆n . It has been reported 12) that broadband
simultaneous wavelength conversion, with a ∆n of effectively zero at 36.0 nm HWHM has been successfully
achieved taking advantage of DFWM in the eigenstate ofpolarization using PMF. If, however, fiber length L is
reduced, even the limited ∆n can to some extent control
the problem of mismatching of polarization.
If the fiber is shortened, however, its length will be insufficient to produce nonlinear interactions. To compensate
for this, it was decided to use HNL-DSF.

Furukawa Review, No. 19. 2000

Conversion efficiency (dB)

Relative refractive index difference ∆ (%)

0.0

3.00

2.00

∆ + =2.75%
1.00

0.00

SiO2 level∆ -=-0.55%

-1.00
-15.0

-10.0

-5.0

0.0

5.0

10.0

-10.0
-20.0
-30.0
-40.0
-50.0
-60.0
0.0

15.0

Table 1

10.0

20.0

30.0

40.0

λidler - λp (nm)Fiber radius (µm)

Figure 2

L = 24.5 km
L = 1.2 km
L = 200 m

Refractive index profile of HNL-DSF

Figure 4

Measured values of conversion efficiency for fibersof selected lengths

Transmission Characteristics of HNL-DSF
Characteristic

Measured value

Attenuation coefficient

0.61 dB/km

Zero-dispersion wavelength1565.5 nm

Dispersion slope (at zero-dispersion wavelength)0.029 ps/nm2/km
13.8 W-1km-1Nonlinear coefficient

Polarization controller
HNL-DSF

Pumping lightsourceEDFA

10-dB coupler

During measurement, the pumping wavelength was madeto agree with the zero-dispersion wavelength of the fiber.
The injected pumping power was set at 100 mW (20dBm), and signal power was 1 mW (0 dBm).
From Figure 4 it can be seen that as the length of the
HNL-DSF is reduced, the bandwidth broadens, reaching
23.3 nm HWHM at a length of 200 m--the greatest bandwidth heretofore achieved using non-polarization maintaining fiber. 13)

Optical spectrum
analyzer

5.

Signal lightsourceEDFAMEASUREMENT OF NONLINEAR COEFFICIENT AND CHROMATIC DISPERSION

Polarization controller

Figure 3

4.

Setup for wavelength conversion experiment

EXPERIMENTS IN BROADBAND SIMULTANEOUS ALL-OPTICAL WAVELENGTHCONVERSION USING HNL-DSF

Figure 2 shows the refractive index profile of the HNLDSF used in these experiments, and Table 1 shows transmission characteristics. The fiber was made by vaporphase axial deposition, and had a nonlinear coefficient γ
of 13.8 W-1km-1, approximately five times the value forordinary DSF. Figure 3 shows the experimental setup.
Both the pumping and probe (signal) were continuous
waves. The lightwaves amplified by the erbium-doped
fiber amplifiers (EDFAs) were coupled using a 10-dB coupler. There are polarizers at the output terminal of the coupler, and the states of polarization of the pumping and signal at input into the HNL-DSF are in alignment. The output
was measured by an optical spectrum analyzer to findidler optical power. In this way it was possible to find conversion efficiency Gc, which may be stated as

G c=

P idler (z=L)
P probe (z=0)(8)

Figure 4 shows the measured values of conversion efficiency obtained for fibers 24.5, 1.2 and 0.2 km in length.

5.1 Nonlinear Coefficients
The explosive growth in long-haul telecommunications
achieved in recent years has been largely attributable to
DWDM technology and the role played by EDFAs, 14) but
the nonlinear effects of signals amplified by EDFAs haveresulted in the degradation of system performance.
Attention has recently been focused on dispersion managed systems as a means of suppressing FWM. 1 5 )Reverse-dispersion fiber (RDF) is used in combination
with conventional single-mode fiber (SMF). 16) At 1550 nm,
RDF has a chromatic dispersion of the same magnitude
as SMF but of opposite sign (normal dispersion), and thedispersion slope is reversed. Thus it can compensate forboth dispersion and dispersion slope simultaneously. The
results of high-capacity WDM experiments using dispersion-managed systems consisting of SMF and RDF havebeen reported. 17), 18)
A number of methods have been developed for measuring the nonlinear coefficient γ, including the use of selfphase modulation 19), cross-phase modulation 20) and fourwave mixing. 21), 22) In the present paper a technique wasconsidered that was applicable to a comparatively wide
normal dispersion domain, and yet measurements could
be carried out by all-optical means. 23), 24) This was because
it was realized that as dispersion-managed systems
become more widely used and the demand for RDF andother fiber having normal dispersion increases, so will theneed to evaluate it.

655.2 Principles of Measurement
Let us discuss measurement in terms of the pump undepleted approximation proposed by Stolen and Bjorkholm, 1), 7)
which in the case of DFWM is accomplished by Equation(9).
This is an approximation in which the attenuation coefficient α of Equation (3) is zero and the pumping power istaken to be so large as to be dominant. For this reason
the pumping light is not subject to DFWM-induced reaction. The signal light and idler light are of about the same
magnitude, and interact together through DFWM.

Bandpass filter
EDFA

Signal lightsourcePolarization controller

Figure 5

Solving Equation (9) analytically, conversion efficiency
Gc in the normal dispersion domain of the fiber may be
represented 1) as

G c= γ 2P p2L 2sin(gL)gL

1 ∆β ∆β γ(+4 P p )
4

-5.0

1.5

-10.0

1.0

-15.0

0.5

-20.0
0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.0
0.28

Figure 6

Measured values of nonlinear coefficient n2/Aeff (•)
and chromatic dispersion D (o) for 10-km fiber with a
pumping wavelength of 1553.0 nm(11)

MEASUREMENT OF NONLINEAR COEFFICIENT OF RDF

6.1 System Setup
Figure 5 shows the experimental setup. It is substantiallythe same as that shown in Figure 3, except that a bandpass filter is used to reduce the amplified spontaneousemission of the EDFA amplifying the pump. Also a 15-dB
coupler is used to couple the probe light and pumping
light. Since the polarization controller (PC) is positioned
after the EDFA, an attenuator is used. This results in a
reduction in the power of the pumping light after coupling,

66

2.0

∆λ (nm)(10)

The following is an explanation of the principles of measurement using the above terms.
If in Equations (10) and (11) Pp is a variable and fiber
length L is known, γ and ∆β are the unknowns. It is possible to find by measurement the two conversion efficiencies Gc corresponding to two different values of pumping
power P p . Mathematically, this may be regarded as
obtaining two simultaneous equations with respect to thetwo unknowns γ and ∆β. Solving this simultaneous equation yields the nonlinear coefficient and the chromatic dispersion. Actually, to minimize unavoidable measurement
errors as much as possible, conversion efficiency Gc wasfound for successive values of pumping power without
solving the equation, and γ and ∆β were obtained by theLevenberg-Marquardt least square method. 25)

6.

Setup for simultaneous measurement of nonlinear
coefficient and chromatic dispersion

0.0

2

wherein g is termed parametric gain, and can beobtained byg≡

D (ps/nm/km)

dEidler= 2i γ Ep 2Eidler+2i γ E *idlerEp2exp(-i ∆β z)
dz

Optical spectrum
analyzer

n2/Aelf (10-9) (W-1)(9)

Fiber under measurement (RDF)PC
3-dB couplerdEp= i γ E p 2E p
dz
dEprobe= 2i γ Ep 2Eprobe+2i γ E *idlerEp2exp(-i ∆β z)dz

Attenuator

Pumping lightsourceso it is input into the fiber under measurement with thestates of polarization of the probe light and pumping lightcarefully aligned and without the use of a polarizer (seeFigure 3). Measurements of input probe light were takenwith an optical power meter and of output power with an
optical spectrum analyzer (OSA) having 0.01-nm resolution, to find the conversion efficiency.
6.2 Optimizing Measurement Conditions
To make an accurate evaluation of γ and ∆β using
Equations (10) and (11), it was found necessary to givesome consideration to the measurement conditions
because: a) pumping power had to be operative below thestimulated Brillouin scattering (SBS) threshold value
determined by the fiber under measurement and the line
width of the pumping light source; and b) the optical powerof the probe was set 25 dB lower than pumping power.
This was to ensure the assumption that in a DFWM system using Equation (9), in which pumping light is assumed
to be dominant.
Measurements and evaluations were then made at thetwo conditions described above. Measurements were carried out on an RDF having a total length of 10 km. The
pump wavelength was set at 1553 nm. Specifically, n2/Aeff
was evaluated from γ using Equation (4) and chromatic
dispersion coefficient D was evaluated from ∆β using
Equation (5). Figure 6 shows the results obtained. The
horizontal axis shows the wavelength spacing ∆λ between
the pump and probe, and the vertical axis shows the corresponding measured value.

Furukawa Review, No. 19. 2000

-30.0

-14.5

2.0

-15.0

1.5

-15.5

1.0

-16.0

0.5

D (ps/nm/km)

Minimum value
-40.0

-45.0

-50.0

-55.0
0.00

0.05

0.10

0.15

0.20

0.25

0.30

-16.5
1540

0.35

1545

∆λp - ∆λprobe (nm)1550

n2/Aelf(10-9) (W-1)

Idler power (dBm)

-35.0

0.0
1560

1555

Wavelength (nm)

Relationship of idler power to probe wavelength for a
10-km fiber with a pumping wavelength of 1553.0 nm

Figure 8

Measured values of nonlinear coefficient n2/Aeff (•)
and chromatic dispersion D (o) for 830-m RDF

As can be seen from Figure 6, the values of n2/Aeff generally depend on λ, and the value changes greatly at ∆λ ofabout 0.22 nm. When the chromatic dispersion coefficientfor identical fiber was measured independently by thephase-shift method, it was found to be 15.45 ps/nm/km at
a wavelength of 1553 nm, demonstrating that at values of∆λ greater than 0.22 nm, accurate evaluation was not
obtained. This means that approximation by means ofEquation (10) cannot be applied.
We measured the idler power by changing the probe
wavelength while fixing the pump wavelength. The resultis shown in Figure 7. It is known that the conversion efficiency decreases as ∆λ increases. The conversion efficiency has a minimal value when ∆λ=0.22 nm, after which
it increases as ∆λ decreases. Equation (6), which wasintroduced in the discussion of coherent length, can also
be considered, in determining fiber length, as limiting the
bandwidth, This limited bandwidth corresponds to the minimum value in Figure 7. 1) In other words this technique isuseless unless ∆λ is less than 0.22 nm.
Based on Figure 6, the value of ∆λ evaluated as optimum was 0.21 nm. These results demonstrated thatunless the optimal value was selected for wavelength
spacing ∆λ, the evaluation would include errors, so thatmeasurements corresponding to those in Figure 7 werecarried out for all fibers. This showed the need to take themeasurements and carry out evaluations at the approximate value of ∆λ that yielded the first minimum value of
conversion efficiency as ∆λ was increased, and this conclusion was confirmed by measurements made using a
number of different RDFs.

Table 2Comparison of simultaneously measured values ofnonlinear coefficient n2/Aeff with those by crossphase modulation, and of chromatic dispersion Dwith those by the phase-shift method in RDFs

Figure 7

6.3 Results of RDF Measurements
Measurements were made using RDFs of four different
lengths: 0.83, 5, 10 and 20 km, at pumping wavelength
increments of 3 nm. Figure 8 shows the results, fromwhich the dispersion slope was obtained.
Table 2 compares the results of evaluations of n2/Aeff forthe four RDFs measured with results obtained independently by cross-phase modulation (XPM). Similarly the
results of evaluations of chromatic dispersion and dispersion slope are shown against those made by the phaseshift method.

Fiber length

L (km)

Nonlinear

Chromatic dispersion

Dispersion slope

coefficient n2/Aeff@ 1550 nm@ 1550 nm(W -1)(ps/nm/km)(ps/nm2/km)

by 4WM by XPM by 4WM by PSM by 4WM by PSM

0.83 km

1.200

1.20

-15.36-15.30-0.053-0.049

5 km

0.764

1.19

-15.05-15.30-0.042-0.049

10 km

0.520

1.17

-15.35-15.30-0.048-0.049

20 km

0.366

1.39

-15.11-15.10-0.039-0.024The values for chromatic dispersion and dispersion
slope for different wavelengths were in substantial agreement with the values measured by the phase-shiftmethod. For n2/Aeff, on the other hand, it was found thaterror gradually increased with fiber length. This is attributed to the failure to account for the effects of the attenuation coefficient α, which cannot be ignored at longer fiber
lengths, in the approximation using Equation (9).
6.4 Discussion Relating to Long-Length Fibers
In applying the method described above to long-length
fibers, the attenuation coefficient has to be taken into
account. For this reason an approximation, in which
pumping light and probe light are attenuated independently of homogeneously with DFWM has been developed 26)
and may be expressed by

G c = γ 2 P p 2 L 2 exp(-3 α L)sin(gL)
gL

2

(12)

and

g≡

1
∆β ( ∆β +4 γ P p e - α L )4(13)

When the evaluation was repeated using these equations, it was confirmed that the value of n2/Aeff agreed withthe value obtained by XPM, irrespective of fiber length.

67

7.

CONCLUSION AND OUTLOOK FOR THE
FUTUREIn this paper the authors have examined techniques forachieving broadband all-optical simultaneous wavelength
conversion by taking advantage of four-wave mixing(FWM) occurring in the fiber, together with techniques forthe simultaneous measurement of the nonlinear coefficient and chromatic dispersion.
It has been demonstrated that the use of short-length
HNL-DSF simultaneously solves the problems of chromatic dispersion variance along the longitudinal direction andpolarization mismatch of probe and pump. It has been
experimentally demonstrated that simultaneous wavelength conversion is possible over a bandwidth of 23.3
nm, the widest for non-polarization-maintaining fibers.
The authors have developed a technique for measuring
the nonlinear coefficient without electrical signal processing by combining DFWM technology with the least square
method for nonlinear functions. Measurement conditions
have been optimized for the application of this technique,and it has been demonstrated that simultaneous measurement of nonlinear coefficient and chromatic dispersion are
possible under these optimized conditions. The values
obtained are in good agreement with those obtained using
the conventional XPM and phase-shift methods. The present technique should also, in theory, be applicable to theanomalous dispersion domain and in the vicinity of zerodispersion.
ACKNOWLEDGMENTS
The authors wish to thank Y. Suzuki, T. Yagi, R. Sugizaki,
S. Arai and K. Mukasa of Fiber Development Center forproviding HNL-DSF and RDF. We also thank H. Ogoshi
for fruitful discussions. Last but not least, we thank H.
Miyazawa for his continuous encouragement.

68

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Furukawa Review, No. 19. 2000