Fifth Order versus Third Order Intermodulation Distortion Lisa Yong1, Mohamad Kadim Suaidi2, Anderson Kho Ngap Chai1, Awangku Abdul Rahman3 1 School of Engineering, Swinburne University of Technology (Sarawak Campus), 93576 Kuching, Sarawak, Malaysia. 2 3 Faculty of Electronics and Computer Engineering, Kolej Universiti Teknikal Kebangsaan Malaysia, Locked Bag 1200, Ayer Keroh, 75450 Melaka, Malaysia. Research and Innovation Management Centre (RIMC), Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia. Abstract – The third order intermodulation distortion (IMD3) rather than the fifth order intermodulation distortion (IMD5) is often considered when examining the performance of an analogue fibre optic system. This paper aims at looking into the possibility of the IMD5 being comparable to the IMD3. The IMD5 generated when the laser is intensity-modulated by the multiple carriers signal is simulated and compared to the IMD3. Individual IMD5 level is found to be lower than the IMD3 when the RF powers of the individual carriers are low or the modulation index per channel is low. However the composite IMD5 level at channel position could exceed the composite IMD3 especially when modulation index increases. This is contributed by the number of IMD5 being greater than that of the IMD3 based on counting. Hence indicating the IMD5 should not be neglected in examining the performance of the analogue fibre optic system driven by multiple carriers. Keywords: IMD3 ; IMD5 1. Introduction The third order intermodulation distortion (IMD3) is often considered as the limitation of the analogue optical fibre systems such as the subcarrier multiplexed systems and hybrid radio fibre systems as seen in Figure 1 [1], [2]. This distortion is produced when two or three carriers of different frequencies are supplied to a nonlinear system. The electrical-tooptical block in Figure 1 is a nonlinear device. The input to the electrical-to-optical converter (E/O) can consist of a single carrier or several equally spaced carriers. The IMD3 tends to fall in-band and might reduce the dynamic range [3]. Therefore it is 1-4244-1435-0/07/$25.00©2007 IEEE often taken into consideration when examining the performance of the analogue optical fibre systems [1]. Figure 1. Basic configuration of the hybrid radio fibre systems. The fifth order intermodulation distortion is the next odd order distortion after IMD3 which also has the tendency to fall in-band. The fifth order intermodulation distortion is often considered to be negligible as their individual levels might be a great deal lower than the second and third orders distortions. Individual IMD5s might be insignificant as compared to the IMD3s but their levels might escalate if their quantities increase and the input magnitude intensifies. Therefore the paper aims at comparing the IMD5 and IMD3 generated when the laser diode is intensity modulated by multiple carriers that are equally spaced to determine the possibility of the individual or composite IMD5 produced is comparable to the IMD3 This paper is organized as follows. The first part introduces the third and fifth order intermodulation distortions; and the analogue fibre optic system. The second part provides the methods used to model the laser diode in order to quantify the levels of the IMD5 and IMD3 produced. The individual and composite IMD5 and IMD3 produced by intensity modulating the laser diode are shown, compared and discussed in the next part. The conclusion is given the last part. im (t ) ⎡ ds(t ) d 2 s (t ) ⎤ ⎡ ds (t ) 2 = ⎢ D1 s(t ) + E1 + F1 + ⎥ − Ls(t ) + Ms(t ) qV dt dt 2 ⎦ ⎢⎣ dt ⎣ 2. Laser Model The laser diode is modeled using Volterra Series, meaning that this device is considered as a weakly non-linear system with memory. The model formulation begins by rearranging and combining the single mode laser rate equation pair as given in equations (1) and (2) [4] to give an output to input relation as given in equation (3): dN (t ) I N (t ) = − − g 0 (N (t ) − N 0 )(1 − εS (t ))S (t ) τs dt qV dS (t ) N (t ) S (t ) = Γg 0 ( N (t ) − N 0 )(1 − εS (t ) )S (t ) + Γβ − dt τs τp (1) (2) where qV is the product of electron charge and active region volume (Am3s), I is the injected current (A), τp is the photon lifetime (s), β is the spontaneous coupling coefficient, Γ is the optical confinement factor, ε is power gain compression parameter (m3), τs is the carrier lifetime (s), N0 is the transparent carrier density (m-3), g0 is the optical power gain (m3s-1), N(t) is the carrier density (m-3) and S(t) is the photon density (m-3). I 0 + ΔI N 0 ⎧ d (Sb + s (t ) ) (Sb + s(t ) ) = +⎨ + − qV dt τs ⎩ τp N ⎫⎛ 1 1 Γβ 0 ⎬⎜ + τ s ⎭⎝ Γ τ s ⎡ d (Sb + s(t ) ) (Sb + s(t ) ) N ⎤ ⎪⎫ + − Γβ 0 ⎥ ⎬ ⎢ τp τ s ⎦⎥ ⎪⎭ dt ⎣⎢ b When the laser diode is represented by Volterra Series, its output s(t) is expressed as [4], [5], [6]: s (t ) = ∑ ∫ L ∫ hn (u1 Kun ) ⋅ ∏ im (t − ul )du1 K dun ∞ ∞ n =1 ∞ −∞ n −∞ l =1 (5) where hn(u1, . . .,un) is the n-th order Volterra Series kernel. The exponential growth or probing method is used to solve equation (5) with ΔI in equations (4) and (5) is substituted with equation (6) [6], [7]: ΔI = ∑ e jωi t = e jω1t + e jω 2t + L + e jω n t n i =1 1 D1 + j ω 1 E1 − ω12 F1 H 2 (ω 1 , ω 2 ) = − ⎤ ⎥ ⎥× ⎥ ⎥⎦ (6) (7) 1 H 1 (ω1 ) ⋅ H 1 (ω 2 ) ⋅ H 1 (ω 1 + ω 2 ) ⋅ G 2 (ω1 , ω 2 ) 2 Where 2 G2 (ω1 , ω2 ) = −2 L − jM ⋅ (ω1 + ω2 ) + N TF ⋅ (ω1 + ω 2 ) (8) (9) where I0 and ΔI are the laser bias and modulating current due to the intercepted mobile signal. Sb and s(t) are the laser steady-state photon density and photon density due to the modulating current. Then, terms with Γg (1 − ε (S + s (t ) ))(S + s (t ) ) + Γ β as the b (4) H 1 (ω 1 ) = (3) 0 2 ⎡ d 2 s (t ) ⎤ ⎛ ds(t ) ⎞ 4 ds (t ) + ⎢Us(t ) 5 + Vs(t ) 3 ⎜ + Xs(t ) 4 ⎟ + Ws(t ) ⎥ +L dt dt 2 ⎦ ⎝ dt ⎠ ⎣⎢ where ωn is the n-th angular frequency of the input. The Fourier Transform of the first to fifth orders Volterra Series kernels obtained are: ⎡ ⎤ ⎢ ⎥ 1− β ⎢ ⎥ β ⎢ Γg (1 − ε (S + s (t ) ))(S + s(t ) ) + Γ ⎥ b b 0 ⎢⎣ τ s ⎥⎦ ⎧⎡ d ⎪⎪⎢ 1 + ⎨⎢ dt ⎪⎢ Γg (1 − ε (S + s(t ) ))(S + s(t ) ) + Γ β b b ⎢ 0 τs ⎩⎪⎣ 2 2 d 2 s (t ) ⎛ ds(t ) ⎞ ⎛ ds(t ) ⎞ ⎤ ⎡ 3 2 ds (t ) + 2Gs(t )⎜ + N TF ⎜ ⎟ ⎟ ⎥ + ⎢ Rs(t ) + σs (t ) 2 dt dt ⎝ dt ⎠ ⎝ dt ⎠ ⎥⎦ ⎢⎣ 2 2 2 d s(t ) ⎤ ⎡ ds(t ) ⎤ 4 2 ⎛ ds (t ) ⎞ 3 d s (t ) + Gs(t ) 2 + P2 s(t ) 3 ⎟ + P1s(t ) ⎥ + ⎢Os(t ) + Qs(t ) ⎜ ⎥ dt 2 ⎦ ⎢⎣ dt 2 dt ⎦ ⎝ dt ⎠ N TF s (t ) τs denominator in equation (3) are expanded using Taylor Series. Collecting the s(t) terms leads to equation (4) that relates the modulating input with its output [4],[5]: ⎧ ⎪⎪ 3 1 H 3 (ω 1 , ω 2 , ω 3 ) = − H 1 (ω 1 + ω 2 + ω 3 ) ⋅ ⎨ ∑ 2H 1 (ω k ) ⋅ 6 ⎪ kk ≠, mm, n≠=n1 ⎪⎩ m < n H 2 (ω m , ω n ) ⋅ G 2 (ω k , ω m + ω n ) + H 1 (ω 1 )H 1 (ω 2 )H 1 (ω 3 ) ⋅ G 3 (ω 1 , ω 2 , ω 3 ) ⎫ ⎪ ⎬ ⎪ ⎭ G3 (ω1 , ω2 , ω3 ) = 6 R + 2 jσ ⋅ (ω1 + ω2 + ω3 ) (10) where − 2G ⋅ (ω1 + ω2 + ω3 ) 2 (11) H 4 (ω 1 , ω 2 , ω 3 , ω 4 ) = −1 H 1 (ω 1 + ω 2 + ω 3 + ω 4 ) ⋅ 24 ⎧ ⎪⎪ 4 ⎨ ∑ 6 H 1 (ω k )H 3 (ω l , ω m , ω n )G 2 (ω k , ω l + ω m + ω n ) ⎪ kk ≠,l l, m≠ m, n≠=n1 ⎩⎪ l < m < n + ∑ [4 H (ω 4 k , l , m , n =1 k ≠l ≠m ≠ n k < l ,< m < n 2 k , ω l )H 2 (ω m , ω n )G 2 (ω k + ω l , ω m + ω n ) + 2 H 1 (ω k )H 1 (ω l )H 2 (ω m , ω n )G 3 (ω k , ω l , ω m + ω n )] + H 1 (ω 1 )H 1 (ω 2 )H 1 (ω 3 )H 1 (ω 4 )G 4 (ω 1 , ω 2 , ω 3 , ω 4 ) ⎫ ⎪ ⎬ ⎪ ⎭ where G 4 (ω1 , ω 2 , ω 3 , ω 4 ) = 24O + j 6 P2 ⋅ (ω1 + ω 2 + ω 3 + ω 4 ) H 5 (ω1 , ω 2 , ω3 , ω 4 , ω5 ) = 2 (13) −1 H 1 (ω1 + ω 2 + ω3 + ω 4 + ω5 ) ⋅ 120 ⎡ 5 ⎢ ⎢ ∑ [24 H 1 (ω k ) ⋅ H 4 (ωl , ω m , ω n , ω p )G2 (ω k , ωl + ω m + ω n + ω p ) ⎢ k ,l , m , n , p =1 ⎢⎣lk<≠ml ≠<mn <≠ np ≠ p + ∑ [12H (ω 5 k ,l , m , n , p =1 k ≠l ≠ m≠ n ≠ p k <l ,m< n< p 2 k ] ] ∑ [4 H (ω )⋅ H (ω , ω )⋅ H (ω 5 k , l , m , n , p =1 k ≠l ≠ m ≠ n ≠ p l < m, n< p + 1 k 2 l m 2 n ] , ω p )⋅ G3 (ω k , ωl + ω m , ω n + ω p ) ∑ [2 H1 (ωk )⋅ H1 (ωl )⋅ H1 (ωm )⋅ H 2 (ωn , ω p )⋅ G4 (ωk , ωl , ωm , ωn + ω p )] 5 k ,l , m , n , p =1 k ≠l ≠ m≠ n ≠ p k <l < m,n< p + H 1 (ω1 ) ⋅ H 1 (ω 2 ) ⋅ H 1 (ω3 ) ⋅ H 1 (ω 4 ) ⋅ H 1 (ω5 ) ⋅ G5 (ω1 , ω 2 , ω3 , ω 4 , ω5 ) ⎤ ⎥ ⎥ ⎥ ⎦ (14) where G5 (ω1 ,ω2 ,ω3 ,ω4 ,ω5 ) = 120U + j 24W ⋅ (ω1 + ω2 + ω3 + ω4 + ω5 ) − 24 X ⋅ (ω1 + ω2 + ω3 + ω4 + ω5 ) 2 (15) The RF input power is related to the modulating input current (ΔI) according to equation (16) [8]: ΔI = 2 Pi / Ri (16) where Pi is the input power and Ri is the input resistance of 50Ω. ΔI is related to the optical modulation per carrier by [9]: mopt = ΔI / (I 0 − I th ) ∑m N n =1 2 (18) opt The mtot should be less than 1 to avoid overmodulation, which can lead to higher levels of distortions [1]. The IMD5 and IMD3 generated based on two-tone IM test are looked into first, where the intersection between the IMD23 and carrier or third order input intercept point (IIP3) as well as the intersection between the IMD25 and carrier or fifth order input intercept point (IIP5) are found. Then the composite IMD5 and IMD3 produced at carrier position due to intensity modulation of the laser by six equally spaced carriers are compared. 3.1 Two-tone IM Test , ωl )H 3 (ω m , ω n , ω p )G2 (ω k + ωl , ω m + ω n + ω p ) + 6 H 1 (ω k ) ⋅ H 1 (ωl ) ⋅ H 3 (ω m , ω n , ω p )⋅ G3 (ω k , ωl , ω m + ω n + ω p ) + mtot = 3. Results and Discussion (12) − 6 P1 ⋅ (ω1 + ω 2 + ω 3 + ω 4 ) where Ith is the bias and threshold currents. When the input to the laser diode is made up of several carriers, the total modulation index (mtot) needs to be considered. The mopt is related to mtot by [1]: (17) The laser diode parameters used in this paper refers to those in [6]. The carrier frequencies, bias current and optical modulation index per carrier used for the twotone IM tests are indicated in Table 1. The lower optical modulation index translates to -37.5 dBm in Test 1. The mopt of 0.5 translates to -2.13dBm of RF input power. Table 1. Parameters used in the first two tone IM measurements simulation. Parameters First carrier frequency, f1 (MHz) Second carrier frequency, f2 (MHz) Bias current, I0 Optical modulation index per carrier, mopt Test 1 890.2 890.4 25mA 0.008519 Test 2 890.2 890.4 25mA 0.5 The IMD3 and IMD5 generated due to the twotone tests are referred to as IMD23 and IMD25, shown in Figures 2 and 3. The IMD23s appeared at 890 and 890.6 MHz, adjacent to the carriers in both figures. 890 MHz is equivalent to 2f1-f2 while 890.6 MHz, 2f2f1. The IMD23 at 890.6MHz was found to be slightly higher than that at 890MHz for both tests. The IMD23 due to mopt of 0.008519 was 133.34dB lower than the first carrier or -133.34dBc. The IMD23 increased to 62.6dBc when mopt of 0.5 was used. The IMD25 appeared at 889.8 and 890.8 MHz, adjacent to the IMD23. 889.8 MHz is equivalent to 3f1-2f2 while 890.8 MHz, 3f2-2f1. The IMD25 in the first two tone IM measurement was -174.75 dBc, 41.41dB lower than the IMD23. However, the second test indicated that the IMD25 was -33.34 dBc and 29.34 dB higher than the IMD23. As such, the IMD5 should not be totally ignored. 1 .10 18 1 .10 17 1 .10 Photon density (1/meter cube) 16 15 1 .10 1 .10 14 1 .10 13 1 .10 12 11 1 .10 1 .10 10 1 .10 9 1 .10 8 8.898 .10 8.9 .10 8 8.902 .10 8.904 .10 Frequency (Hz) 8 8 8 8.906 .10 8 8.908 .10 8 Carriers Third order IMD Fifth order IMD Photon density (1/meter cube) Figure 2. The outcome of the two-tone IM test using mopt of 0.008519. 1 .10 20 1 .10 19 1 .10 18 1 .10 17 than the IMD3. Both IMDs intersected at approximately -16.75dBm, meaning the IMD23 and IMD25 are equal at -16.75dBm or when mopt is 0.093. The IMD23 was found to be greater than the IMD25 for RF input power less than -16.75dBm. The opposite occurs when the RF input power exceeds -16.75dBm. Therefore, the IMD5 tends to be more significant than the IMD3 as the RF input power increases. The IMD25 was at the same level as the first carrier when the carriers’ RF power was 6.25dBm. Hence, the IIP5 is 6.25dBm and translates to mopt of 1.312. The IIP3 is 29.25dBm. Beyond these RF input powers, the IMD23 and IMD25 exceeded the carriers. Hence, the IIP3 alone is not sufficient to characterize the analogue fibre optic system’s nonlinearity. 3.3 Composite IMD3 and IMD5 The six equally spaced carriers from 890.2 MHz to 910.2MHz were employed in simulating the composite IMD3 and IMD5. The IMD3 of type 2f1-f2 and f1+f2-f3, and the IMD5 of type f1+f2+f3-f4-f5, 3f1-f2f3, 3f1-2f2, 2f1+f2-2f3, f1+f2+f3-2f4, 2f1+f2-f3-f4 appearing at carrier position were counted. The number of carrier position IMD5 are greater than the IMD3 as shown in Figure 5. 80 1 .10 8.898 .10 8.9 .10 8 8.902 .10 8.904 .10 Frequency (Hz) 8 8 8 8.906 .10 8 8.908 .10 Num ber of IMD 70 16 8 Carriers Third order IMD Fifth order IMD Figure 3. The two-tone IM when the mopt is 0. 5. 60 50 40 30 20 10 0 1 3.2 IMD23 and IMD25 versus RF Power The IMD23 and IMD25 for a bias current of 25mA, and RF input power ranging between -120 to 40dBm is shown in Figure 4. 3 4 5 6 Carrier Number Fifth order intermodulation distortion Third order intermodulation distortion Figure 5. The sum of IMD3 and IMD5 at carrier positions. The composite IMD5 and composite IMD3 levels at carrier positions at 25mA bias current and low mtot are given in Figure 6. 400 7.00E+11 200 6.00E+11 Photon Densities (m -3) Log(Photon density [1/meter cube]) 2 0 5.00E+11 4.00E+11 3.00E+11 2.00E+11 200 1.00E+11 120 100 80 60 40 20 RF input power (dBm) 0 20 40 First Carrier Third order IMD Fifth Order IMD Figure 4. Carrier, IMD23 and IMD25 versus input power for 25mA bias current. Both distortions in log scale increased proportionally with the RF input power in dBm. The IMD25 trend is twice steeper than that of the IMD23, signifying the IMD5 tends to increase more rapidly 0.00E+00 1 2 3 4 5 6 Carrier Number Composite fifth order intermodulation distortion level Composite third order intermodulation distortion level Figure 6. The composite IMD3 and IMD5 at carrier positions with low mtot. The composite IMD5 levels were very much lower than the composite IMD3 levels, signifying that IMD3 60.00 50.00 40.00 dB was more dominant than the IMD5 at low mtot. The composite IMD3 maximized at the middle occupied channels as more IMD3s fell onto these channels, as compared to the end channels. The composite IMD5 level maximized at the fifth carrier rather than the fourth since the frequency of the fifth carrier was higher than the fourth. The composite IMD5s and IMD3s appearing at carrier positions for the mtot of one with 25mA bias current are given in Figure 7. Both distortions increased in magnitude with the composite IMD5 levels exceeding the composite IMD3 levels. This signifies that the composite IMD5 was more dominant than the composite IMD3 at the mtot of one with low bias current. 30.00 20.00 10.00 0.00 Carrier Number Carrier-to-composite IMD3 Carrier-to-composite IMD5 Carrier Number Carrier-to-IMD Figure 9. Carrier-to-composite intermodulation distortion ratios for low mtot. 4. Conclusion 1.80E+17 1.60E+17 -3 Photon Density (m ) 1.40E+17 1.20E+17 1.00E+17 8.00E+16 6.00E+16 4.00E+16 2.00E+16 0.00E+00 1 2 3 4 5 6 Carrier Number Composite fifth order intermodulation distortion level Composite third order intermodulation distortion level Figure 7. The composite IMD3 and IMD5 at carrier positions with low mtot. The composite IMD3 and IMD5 are both lower than the carrier magnitudes for both low and high mtot. Thus resulting in positive carrier-to-composite IMD3 and carrier-to-composite IMD5 ratios as in Figures 8 and 9. 190.00 180.00 170.00 160.00 dB 150.00 140.00 130.00 120.00 110.00 100.00 1 2 Carrier-to-composite IMD3 3 4 Carrier Number IMD5 Carrier-to-composite 5 6 Carrier-to-IMD Figure 8. Carrier-to-composite intermodulation distortion ratios for low mtot. The carrier-to-composite intermodulation distortion curve seemed superimposed with the carrierto-composite IMD3 curve for low mtot. The carrier-tocomposite IMD5 curve appeared below the carrier-tocomposite IMD3 curve as given in Figure 9 as the composite IMD5 at carrier position were more dominant than the composite IMD3 for high mtot. The IMD5 has been found to be more dominant than the IMD3 when the mopt is high or RF input power increases for low bias current when the laser diode is modulated by six equally carriers. However, the opposite occurs when the mopt is low. 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