Abstract:
A variable chirp optical modulator is provided. An optical waveguide is split for part of its length into first and second waveguide arms. Electrode pairs are positioned to be proximate a first portion of corresponding waveguide arms. The lengths of each of the electrodes are different and are selected to provide a predetermined level of chirp.

Description:
This invention pertains to optical systems, in general, and to optical modulators for use in optical systems in particular. 
     BACKGROUND OF THE INVENTION 
     The term “optical system” as used herein refers to any system that utilizes light waves to convey information between one node and one or more other nodes. Much of the optical communications network in place utilizes optical fibers. One property of optical fibers that is of concern is dispersion. Dispersion in optical fiber occurs as a result of variation in the refractive index of the optical fiber with wavelength. Modulation of an optical signal results in optical harmonics of the modulation frequency about the carrier frequency. When modulated light is passed through a length of optical fiber that exhibits chromatic dispersion, the phase of the light at the distal end of the fiber varies as a function of its frequency thus producing phase modulation. Detection of optical signals causes mixing of the various frequency components, but because the various frequency components have differing phases, the mixing results in the amplitude of the detected signal changing on account of the linewidth of the transmitted signal. 
     In other words, in a dispersive medium, different wavelengths of light travel at slightly different velocities. This causes optical pulses to broaden in wavelength as they travel down optical fiber links, causing difficulty at a receiver when reconstructing an electrical pulse from a received optical pulse. With the advent of erbium doped fiber amplifiers, longer distances are traversed over optical fiber. The problems caused by dispersion are referred to as “chirping”. Chirping becomes increasingly more significant of a problem at higher modulation frequencies such as frequencies at 10 GHz and above. One limiting factor on the length of links in long haul transmission of optical signals is chromatic dispersion that occurs because a transmitter has a real optical linewidth and the refractive index of optical fiber varies, dependent upon the wavelength of the optical signals. Optical linewidth of a transmitter is determined by two factors. The factors are the inherent linewidth at DC and the broadening of the linewidth introduced by modulation. Broadening of the linewidth introduced by modulation is referred to as “static” chirp. Other optical components may introduce a shift to the center frequency of the optical linewidth; this is referred to as dynamic chirp. 
     Static and dynamic chirp introduce a pulse width change or phase modulation and an amplitude shift or intensity modulation in the optical signal. The intensity modulation changes are such that there may be a combination of link length, dispersion and frequency that completely nulls out the signal to be detected. In the case of pulse width changes, positive dynamic chirp will broaden the width of a pulse propagating down a fiber and negative dynamic chirp will narrow the pulse. Either of the two effects can render a modulated signal undetectable. 
     The chirp effects can be compensated for by deliberate introduction of an offsetting chirp in modulated signals. Various modulators providing controlled chirp have been described in the prior art. Typically such modulators are based upon designs that form the modulator on a substrate. The substrate material is frequently lithium niobate (LiNbO 3 ) although other electro-optic materials may be used. 
     The electro-optic effect in LiNbO 3  depends on the direction of the electric field relative to the orientation of the crystalline structure of the substrate along which the optical wave propagates. In an orientation referred to as an “X-cut” the optic axis is parallel to the plane of the substrate and at right angles to the direction of propagation of the optical wave, In an orientation referred to as “Z-cut” the optic axis is normal to the plane of the substrate. 
     SUMMARY OF THE INVENTION 
     An optical modulator in accordance with the invention includes an optical waveguide that splits for part of its length into first and second waveguide arms, with the waveguide arms recombining to an output waveguide portion. A first electrode proximate the first waveguide arm subjects a first optical length of said first waveguide to a first modulating electric field. A second electrode proximate the second waveguide arm subjects a first optical length of said second waveguide arm to a second modulating electric field. In accordance with the principles of the invention, the length of the second electrode is shorter that the first length of the first electrode. The lengths of the first and second electrodes are selected to provide a predetermined amount of chirp. 
     In one embodiment of the invention, the first electrode is one electrode of a first electrode pair proximate said first waveguide arm. The electrodes of the first electrode pair are arranged with respect to each other and the first waveguide arm to subject a first optical length of said first waveguide to a first modulating electric field. The second electrode is one electrode of a second electrode pair proximate the second waveguide arm. The electrodes of the second electrode pair are arranged with respect to each other and to said second waveguide arm to subject a first optical length of said second waveguide arm to a second modulating electric field, the second electrode pair is shorter than the first electrode pair. In accordance with the principles of the invention, the lengths of each of the first electrode pair and the second electrode pair are selected to provide a predetermined amount of chirp. 
     In accordance with one aspect of the invention, the first electrode pair and the second electrode pair share a common electrode. 
     In accordance with another aspect of the invention, the gap between the first electrode pair is equal to the gap between the second electrode pair in the regions in which the first and second waveguide arms are subjected to the first and second electric fields, respectively. 
     In accordance with the principles of the invention, the amount of chirp provided by the modulator is varied from a fixed chirp by varying the power split between the two waveguide arms, and/or the voltage applied to the first and second electrodes and/or a bias voltage applied to bias electrodes. 
    
    
     BRIEF DESCRIPTION OF THE DRAWING 
     The invention will be better understood from a reading of the following detailed description in conjunction with drawing, in which like reference designations in the various drawing figures are used to identify like elements, and in which: 
     FIG. 1 illustrates a portion of a prior art integrated optic Mach Zehnder Interferometer modulator; 
     FIG. 2 is a partial cross-section of the prior art device of FIG. 1; 
     FIG. 3 illustrates a second prior art integrated optic modulator; 
     FIG. 4 is a partial cross-section of the prior art device of FIG. 3; 
     FIG. 5 depicts an integrated optic modulator in accordance with the principles of the invention; 
     FIG. 6 is a partial cross-section of the modulator of FIG. 5; 
     FIG. 7 illustrates the optical waveguide paths of a Mach Zehnder Interferometer modulator in accordance with the invention; 
     FIG. 8 illustrates a second integrated optic modulator in accordance with the principles of the invention; 
     FIG. 9 is a partial cross section of the modulator of FIG. 8 taken along lines  9 — 9 ; 
     FIG. 10 illustrates a third integrated optic modulator in accordance with the principles of the invention; 
     FIG. 11 is a partial cross-section of the modulator of FIG. 10 taken along lines  11 — 11   
     FIG. 12 illustrates a fourth integrated optic modulator in accordance with the invention; 
     FIG. 13 is a partial cross-section of the modulator of FIG. 12 taken along lines  13 — 13 ; and 
     FIG. 14 is a partial cross-section of the modulator of FIG. 12 taken along lines  14 — 14 . 
    
    
     DETAILED DESCRIPTION 
     Turning now to FIG. 1, a portion of a prior art integrated optic Mach Zehnder Interferometer modulator  10  is shown. Mach Zehnder Interferometer type modulators are typically utilized in prior modulator designs. A Mach Zehnder Interferometer modulator configuration comprises an optical waveguide splitter coupled to two waveguide arms and a waveguide combiner. Electrodes are associated with each of the two waveguide arms and provide a modulation voltage to one or both electrodes. The modulation voltages change the optical index of the waveguide arms and thereby change the relative phase of the two light beams. A differential phase change may result in both beams combining in phase to give a maximum intensity signal. This is the “on” state. A λ/2 degree phase shift difference results in beam extinction or an “off” state. Chirp can be created by uneven modulation between the two waveguide arms of the Mach Zehnder Interferometer modulator. 
     The Intensity Modulation effect Vpi is the voltage differential between the Mach Zehnder Interferometer modulator electrodes. A trade off between the intensity modulation effect and the amount of chirp is required. That is, larger chirp requires a higher Vpi 
     Modulator  10  includes an optical waveguide  12  that is split by an optical splitter  14  into two waveguide arms  16 ,  18 . Optical waveguide arms  16 ,  18  are rejoined with an optical coupler  20 . FIG. 2 shows modulator  10  in cross section. The modulator is formed on a substrate  1  that is typically lithium niobate. Optical waveguide arms  16 ,  18  are formed in substrate  1 . Electrodes  22 ,  24 ,  26  are formed on substrate  1  proximate optical waveguide arms  16 ,  18 . The distance between electrode  22  and electrode  24  is different from the distance between electrode  24  and electrode  26 , providing uneven gaps  28 ,  30  between electrode pairs. This approach is utilized in the integrated optical modulator of U.S. Pat. No. 6,052,496. 
     A second prior art integrated optic Mach Zehnder Interferometer modulator  210  is shown in FIGS. 3 and 4. Modulator  310  includes an optical waveguide  312  that is split by an optical splitter  314  into two waveguide arms  316 ,  318 . Optical waveguide arms  316 ,  318  are rejoined with an optical coupler  320 . FIG. 4 shows modulator  310  in cross section. The modulator is formed on a substrate  301  that is typically lithium niobate. Optical waveguide arms  316 ,  318  are formed in substrate  301 . Electrodes  322 ,  324 ,  326  are formed on substrate  301  proximate optical waveguide arms  316 ,  318 . One waveguide arm  318  is disposed such that the adjacent electrodes  322 ,  324  are located on either side of waveguide arm  318 . Electrode  326  is disposed above optical waveguide arm  318 . The distance between the electrode pair comprising electrode  322  and electrode  324  is the same as the distance between the electrode pair comprising electrode  324  and electrode  326 , providing even gaps  328 ,  330  between electrode pairs. 
     Turning now to FIGS. 5 and 6, the approach in accordance with the present invention is illustrated. In accordance with the invention, an integrated optic modulator  510  includes an optical waveguide  512  that is split by an optical splitter  514  into two waveguide arms  516 ,  518 . Optical waveguide arms  516 ,  518  are rejoined with an optical coupler  520 . FIG. 6 shows modulator  510  in cross section. The modulator  510  is formed on a substrate  501  that is lithium niobate. As will be appreciated by those skilled in the art, substrate  501  may be of other material including, but not limited to LiTaO 3 . Optical waveguide arms  516 ,  518  are formed in substrate  501 . Electrodes  522 ,  524 ,  526  are formed on substrate  501  proximate optical waveguide arms  516 ,  518 . The electrodes  522 ,  524 ,  526  are arranged to form two electrode pairs with electrode  524  being a common electrode in the two electrode pairs. That is, electrodes  222 ,  224  form one electrode pair and electrodes  224 ,  226  form another electrode pair. The distance between electrode  522  and electrode  524  is the same as the distance between electrode  524  and electrode  526 , providing even gaps  528 ,  530  between electrode pairs. However, in contrast to the prior art approach of the device of FIGS. 1 and 2, electrode  526  is a different length than electrode  522 . 
     LiNbO 3  substrate modulators used to perform intensity modulation can in general also impress a phase modulation on an optical signal. The degree of phase modulation is expressed as the chirp parameter              α   ≡            φ          t           1   2               I          t                   (   1   )                                
     where φ is the phase shift imparted to the output signal and I is the intensity of the light output. 
     The most common intensity modulator using lithium niobate is the Mach Zehnder Interferometer modulator. Turning now to FIG. 7, only the optical path or a Mach Zehnder Interferometer is shown. The optical waveguide  712  is split by splitter or “y”  714  into two waveguide arms  716 ,  718  that rejoin at coupler  720 . The input signal at an amplitude A i  is split into two signals of respective amplitudes ρA i  and σA i  which propagate along the lengths L 1  and L 2  of waveguide arms  716 ,  718  before being recombined. The lengths L 1 , L 2  are the optical path lengths due to intrinsic characteristics of the waveguide arms and any applied electrode voltages. Coupler  720  combines the signals on the two waveguide arms  716 ,  718 . The intensity of the light in each arm is equal. The combined output amplitude, A, is the output amplitude of the optical signals in each waveguide arm  716 ,  718 . The output amplitudes for signals on the two waveguide arms  716 ,  718  are “r” and “s”, respectively, with                r   =         2     2          A   i        ρ                                (       ω                 t     -     kL   1       )             ,              and           (   2   )               s   =         2     2          A   i        σ                                  (       ω                 t     -     kL   2       )         .               (   3   )                                
     The resultant output is                A   =       r   +   s     =         2     2          A   i          {       ρ                          -                        kL   1           +     σ                          -                        kL   2             }                                ω                 t             ,           (   4   )                                
     which may be rewritten as                    A   =                    2     2          A   i                                  ω                 t       [                             -                        k        (       L   1     +     L   2       )         2                            -                        tan     -   1            {         ρ   ·   σ       ρ   +   σ                     tan                     k        (       L   1     ·     L   2       )       2       }         ·                                               ρ   2     +     σ   2     +     2      ρ                 σ                   cos              [     k        (       L   1     -     L   2       )       ]                           (   5   )                                
     Now                φ   =       -       k        (       L   1     -     L   2       )       2       -       tan     -   1            {         ρ   -   σ       ρ   +   σ                     tan                     k        (       L   1     -     L   2       )       2       }           ,              and           (   6   )                        φ     /        t       =     -       (     k   /   2     )          [              (       L   1     -     L   2       )            t       +       (       ρ   2     -     σ   2       )         ρ   2     +     σ   2     +     2      ρ                 σ                   cos              [     k        (       L   1     -     L   2       )       ]             ]           ,           (   7   )               I   =       1   2            I   i     (       ρ   2     +     σ   2     +     2      ρ                 σ                   cos              [     k        (       L   1     -     L   2       )       ]                     (   8   )                      I          t       =       1   2            I   i     (       -   2        ρ                 σ                     sin              [     k        (       L   1     -     L   2       )       ]     ·   k                              (       L   1     -     L   2       )            t       .                   (   9   )                                
     The chirp factor, as noted above, is defined as:              α   ≡              φ          t           (     1     2      I       )               I          t           .             (   10   )                                
     Recognizing that ρ 2 =1−σ 2 , allows the expression for the chirp factor to be rewritten as:                    α   =                    1   -     2                   σ   2           2                 σ          1   -     σ   2                         sin              [     k        (       L   1     -     L   2       )       ]         +                                  1   +     2                 σ          1   -     σ   2                         cos              [     k        (       L   1     -     L   2       )       ]           2                 σ          1   -     σ   2                         sin              [     k        (       L   1     -     L   2       )       ]         ·                (       L   1     +     L   2       )            t                (       L   1     -     L   2       )            t         .                     (   11   )                                
     Alternatively, the chirp factor may be expressed in terms of the power, P 1 ,P 2 , in each waveguide arm  716 ,  718 , recognizing that ρ 2 =P 1 , and σ 2 =P 2 , with P 1 , being the power in waveguide arm  716  and P 2  being the power in waveguide arm  718 , as:                    α   =                    (     1   -       P   2       P   1         )       2                       P   2       P   1                         sin              [     k        (       L   1     -     L   2       )       ]         +                                  1   +       P   2       P   1       +     2                       P   2       P   1                         cos              [     k        (       L   1     -     L   2       )       ]           2                       P   2       P   1                         sin              [     k        (       L   1     -     L   2       )       ]         ·              (       L   1     +     L   2       )            t                (       L   1     -     L   2       )            t                         (   12   )                                
     Turning now to FIG. 8, a modulator  810  in accordance with the principles of the invention is shown. Integrated optic modulator  810  includes an optical waveguide  812  that is split by an optical splitter  814  into two waveguide arms  816 ,  818 . Optical waveguide arms  816 ,  818  are rejoined with an optical coupler  820 . FIG. 9 shows modulator  810  in cross section taken along lines  9 — 9 . The modulator  810  is formed on a substrate  801  that is lithium niobate. As will be appreciated by those skilled in the art, substrate  801  may be of other material including, but not limited to GaAs. Optical waveguide arms  816 ,  818  are formed in substrate  801 . Electrodes  822 ,  824 ,  826  are formed on substrate  801  proximate optical waveguide arms  816 ,  818 . The distance between electrode  822  and electrode  824  is the same as the distance between electrode  824  and electrode  826 , providing even gaps  828 ,  830  between electrode pairs. Electrodes  822  and  824  are of substantially equal effective length, l 2 , proximate waveguide arm  818 . Electrode  826  is configured so as to provide a different effective electrode length, l 1 , proximate waveguide arm  816 . The effective optical path lengths of the waveguide arms  816 ,  818  are L 1  and L 2 , respectively. The path lengths are affected by the modulation voltage induced path length and a bias induced path length, L bias . The optical path lengths of the wave guide arms in the absence of modulation and bias are L 10  and L 20 . From these identities, 
     
       
           L   1   =L   10   +γl   1   V ( t )+L bias , and  (13) 
       
     
     
       
           L   2   =L   20   −γl   2   V ( t )−L bias .  (14) 
       
     
     From the above, various portions of the other equations may be determined as follows: 
     
       
           L   1   +L   2   =L   10   +L   20   +γV ( t )(l 1 −l 2 ),  (15) 
       
     
     
       
         
           
             
               
                 
                   
                     
                       
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     For small signals, φ 0  is much smaller than 1. If the power ratio is defined as          X   ≡       P   2       P   1         ,                          
     then the chirp equation becomes:              α   =         (     1   -   X     )       2        X          sin              [     φ   bias     ]         +         (       l   1     -     l   2       )       (       l   1     +     l   2       )       ·           (     1   +   X   +     2        X                   cos                   φ   bias           ]       2      X                   sin              [     φ   bias     ]         .                 (   21   )                                
     To understand the effect of varying the length of the electrodes, the following may be considered, If the electrodes are of equal length L 0 , the phase shift is φ 0 =2KL 0 , where K contains all the various coefficients in the relevant equations including, but not limited to, the voltage and overlap coefficients. If one electrode is shortened by an amount “z” and the other is lengthened by an amount “w”, then 
     
       
           L   e1   ≡L   0   −z=l   1 ,  (22) 
       
     
     and 
     
       
           Le   2   ≡L   0   +w=l   2 ,  (23) 
       
     
     and 
     
       
         φ= K ( L   0   −z )+ K ( L   0   +w )=2 KL   0   +K ( w−z ).  (24) 
       
     
     For a 10% increase in voltage (directly changing K) to produce the same phase shift, the equations may be solved to show that z=1+0.182L 0 , and the modified electrode lengths are L e1 =0.818L 0 −w, and L e2 =L 0 +w. If electrode length alone is used to achieve a chirp parameter of α=−0.7, then            -   0.7     =       -     (       2      l     +     0.182        L   0         )         1.818        L   0           ,                          
     yielding l=0.545L 0 , so that L e1 =0.273L 0 , and L e2 =1.545L 0 . 
     The use of power adjusting between the two waveguide arms  816 ,  818  may also be used. Using the above values for L e1  and L e2  in the chirp parameter equation provides:              α   =         1   -   X       2        X        sin                   (     φ   b     )         -     0.7                       (     1   +   x   +     2        X        cos                   (     φ   b     )         )       2        X        sin                   (     φ   b     )         .                 (   25   )                                
     If it is desired to adjust the power to achieve ±0.2 chirp, and assuming φ b =π/2, the first term in the equation dominates and        α   =       ±   0.2     =         1   -   X       2        X         .                              
     Solving for X, yields X=1.488, 0.672. With equal power levels in both waveguide arms, i.e., X=1.0 or P 1 =P 2 , a chirp factor of α=−0.7 is obtained. With X =1.488, a chirp factor of α=−0.9 is obtained. With X =0.672, a chirp factor of α=−0.5 is obtained. At either power ratio, an on/off ratio of less than −20 dB is obtained. 
     From the above analysis, chirp, α, and extinction or on-off ratio data has been calculated for different electrode length ratios, L e2 /L e1 , bias deviation from π/2 and, the power ratio, X between the waveguide arms. The data is set forth in TABLE 1. 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 electrode 
                 bias 
                   
                   
                   
               
               
                 length 
                 deviation 
                 power 
                   
                 on/off 
               
               
                 ratio 
                 from π/2 
                 ratio 
                 chirp 
                 ratio 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 2.67 
                 −0.05 π 
                 1.60 
                 −0.78569 
                 −14.2231 
               
               
                 2.67 
                 −0.05 π 
                 1.65 
                 −0.80347 
                 −14.0227 
               
               
                 2.67 
                 −0.05 π 
                 1.70 
                 −0.82088 
                 −13.8258 
               
               
                 2.67 
                 −0.05 π 
                 1.75 
                 −0.83794 
                 −13.6331 
               
               
                 2.67 
                 −0.05 π 
                 1.80 
                 −0.85468 
                 −13.445 
               
               
                 2.67 
                 −0.05 π 
                 1.85 
                 −0.87111 
                 −13.2617 
               
               
                 2.67 
                 −0.05 π 
                 1.90 
                 −0.88725 
                 −13.0834 
               
               
                 2.67 
                 −0.05 π 
                 1.95 
                 −0.9031 
                 −12.9101 
               
               
                 2.67 
                 −0.05 π 
                 2.00 
                 −0.91869 
                 −12.7418 
               
               
                 2.67 
                 −0.025 π 
                 1.60 
                 −0.74283 
                 −17.0438 
               
               
                 2.67 
                 −0.025 π 
                 1.65 
                 −0.76044 
                 −16.6611 
               
               
                 2.67 
                 −0.025 π 
                 1.70 
                 −0.77769 
                 −16.2999 
               
               
                 2.67 
                 −0.025 π 
                 1.75 
                 −0.79459 
                 −15.9589 
               
               
                 2.67 
                 −0.025 π 
                 1.80 
                 −0.81118 
                 −15.6366 
               
               
                 2.67 
                 −0.025 π 
                 1.85 
                 −0.82746 
                 −15.3316 
               
               
                 2.67 
                 −0.025 π 
                 1.90 
                 −0.84344 
                 −15.0427 
               
               
                 2.67 
                 −0.025 π 
                 1.95 
                 −0.85915 
                 −14.7687 
               
               
                 2.67 
                 −0.025 π 
                 2.00 
                 −0.87459 
                 −14.5085 
               
               
                 2.67 
                 0 
                 1.60 
                 −0.70483 
                 −18.639 
               
               
                 2.67 
                 0 
                 1.65 
                 −0.72239 
                 −18.0935 
               
               
                 2.67 
                 0 
                 1.70 
                 −0.73959 
                 −17.5961 
               
               
                 2.67 
                 0 
                 1.75 
                 −0.75644 
                 −17.1398 
               
               
                 2.67 
                 0 
                 1.80 
                 −0.77298 
                 −16.719 
               
               
                 2.67 
                 0 
                 1.85 
                 −0.7892 
                 −16.3292 
               
               
                 2.67 
                 0 
                 1.90 
                 −0.80514 
                 −15.9666 
               
               
                 2.67 
                 0 
                 1.95 
                 −0.8208 
                 −15.6281 
               
               
                 2.67 
                 0 
                 2.00 
                 −0.8362 
                 −15.311 
               
               
                 2.67 
                 0.025 π 
                 1.60 
                 −0.6712 
                 −18.6126 
               
               
                 2.67 
                 0.025 π 
                 1.65 
                 −0.68881 
                 −18.0671 
               
               
                 2.67 
                 0.025 π 
                 1.70 
                 −0.70606 
                 −17.5698 
               
               
                 2.67 
                 0.025 π 
                 1.75 
                 −0.72297 
                 −17.1135 
               
               
                 2.67 
                 0.025 π 
                 1.80 
                 −0.73955 
                 −16.6928 
               
               
                 2.67 
                 0.025 π 
                 1.85 
                 −0.75583 
                 −16.303 
               
               
                 2.67 
                 0.025 π 
                 1.90 
                 −0.77182 
                 −15.9404 
               
               
                 2.67 
                 0.025 π 
                 1.95 
                 −0.78753 
                 −15.602 
               
               
                 2.67 
                 0.025 π 
                 2.00 
                 −0.80297 
                 −15.285 
               
               
                 2.67 
                 0.05 π 
                 1.60 
                 −0.64155 
                 −18.5329 
               
               
                 2.67 
                 0.05 π 
                 1.65 
                 −0.65932 
                 −17.9876 
               
               
                 2.67 
                 0.05 π 
                 1.70 
                 −0.67674 
                 −17.4904 
               
               
                 2.67 
                 0.05 π 
                 1.75 
                 −0.6938 
                 −17.0343 
               
               
                 2.67 
                 0.05 π 
                 1.80 
                 −0.71054 
                 −16.6137 
               
               
                 2.67 
                 0.05 π 
                 1.85 
                 −0.72697 
                 −16.2241 
               
               
                 2.67 
                 0.05 π 
                 1.90 
                 −0.7431 
                 −15.8617 
               
               
                 2.67 
                 0.05 π 
                 1.95 
                 −0.75896 
                 −15.5234 
               
               
                 2.67 
                 0.05 π 
                 2.00 
                 −0.77455 
                 −15.2066 
               
               
                 2.67 
                 0.075 π 
                 1.60 
                 −0.61652 
                 −18.3991 
               
               
                 2.67 
                 0.075 π 
                 1.65 
                 −0.63367 
                 −17.8541 
               
               
                 2.67 
                 0.075 π 
                 1.70 
                 −0.65136 
                 −17.3571 
               
               
                 2.67 
                 0.075 π 
                 1.75 
                 −0.66869 
                 −16.9013 
               
               
                 2.67 
                 0.075 π 
                 1.80 
                 −0.68569 
                 −16.481 
               
               
                 2.67 
                 0.075 π 
                 1.85 
                 −0.70238 
                 −16.0917 
               
               
                 2.67 
                 0.075 π 
                 1.90 
                 −0.71877 
                 −15.7295 
               
               
                 2.67 
                 0.075 π 
                 1.95 
                 −0.73488 
                 −15.3915 
               
               
                 2.67 
                 0.075 π 
                 2.00 
                 −0.75071 
                 −15.075 
               
               
                 2.67 
                 0.1 π 
                 1.60 
                 −0.59326 
                 −18.2094 
               
               
                 2.67 
                 0.1 π 
                 1.65 
                 −0.61172 
                 −17.6648 
               
               
                 2.67 
                 0.1 π 
                 1.70 
                 −0.6298 
                 −17.1682 
               
               
                 2.67 
                 0.1 π 
                 1.75 
                 −0.64752 
                 −16.7128 
               
               
                 2.67 
                 0.1 π 
                 1.80 
                 −0.6649 
                 −16.2929 
               
               
                 2.67 
                 0.1 π 
                 1.85 
                 −0.68197 
                 −15.904 
               
               
                 2.67 
                 0.1 π 
                 1.90 
                 −0.69872 
                 −15.423 
               
               
                 2.67 
                 0.1 π 
                 1.95 
                 −0.71519 
                 −15.2047 
               
               
                 2.67 
                 0.1 π 
                 2.00 
                 −0.73138 
                 −14.8886 
               
               
                 2.67 
                 0.125 π 
                 1.60 
                 −0.57422 
                 −17.9615 
               
               
                 2.67 
                 0.125 π 
                 1.65 
                 −0.59343 
                 −17.4174 
               
               
                 2.67 
                 0.125 π 
                 1.70 
                 −0.61204 
                 −16.9214 
               
               
                 2.67 
                 0.125 π 
                 1.75 
                 −0.63028 
                 −16.4665 
               
               
                 2.67 
                 0.125 π 
                 1.80 
                 −0.64818 
                 −16.0471 
               
               
                 2.67 
                 0.125 π 
                 1.85 
                 −0.66574 
                 −15.6588 
               
               
                 2.67 
                 0.125 π 
                 1.90 
                 −0.68299 
                 −15.2977 
               
               
                 2.67 
                 0.125 π 
                 1.95 
                 −0.69994 
                 −14.9607 
               
               
                 2.67 
                 0.125 π 
                 2.00 
                 −0.71661 
                 −14.6452 
               
               
                 2.5 
                 −0.05 π 
                 1.60 
                 −0.75396 
                 −14.2231 
               
               
                 2.5 
                 −0.05 π 
                 1.65 
                 −0.77163 
                 −14.0227 
               
               
                 2.5 
                 −0.05 π 
                 1.70 
                 −0.78894 
                 −13.8258 
               
               
                 2.5 
                 −0.05 π 
                 1.75 
                 −0.8059 
                 −13.6331 
               
               
                 2.5 
                 −0.05 π 
                 1.80 
                 −0.82253 
                 −13.445 
               
               
                 2.5 
                 −0.05 π 
                 1.85 
                 −0.83884 
                 −13.2617 
               
               
                 2.5 
                 −0.05 π 
                 1.90 
                 −0.85486 
                 −13.0834 
               
               
                 2.5 
                 −0.05 π 
                 1.95 
                 −0.8706 
                 −12.9101 
               
               
                 2.5 
                 −0.05 π 
                 2.00 
                 −0.88607 
                 −12.7418 
               
               
                 2.5 
                 0 
                 1.60 
                 −0.66763 
                 −18.639 
               
               
                 2.5 
                 0 
                 1.65 
                 −0.69509 
                 −18.0935 
               
               
                 2.5 
                 0 
                 1.70 
                 −0.71218 
                 −17.5961 
               
               
                 2.5 
                 0 
                 1.75 
                 −0.72893 
                 −17.1398 
               
               
                 2.5 
                 0 
                 1.80 
                 −0.74536 
                 −16.719 
               
               
                 2.5 
                 0 
                 1.85 
                 −0.76147 
                 −16.3292 
               
               
                 2.5 
                 0 
                 1.90 
                 −0.7773 
                 −15.9666 
               
               
                 2.5 
                 0 
                 1.95 
                 −0.79284 
                 −15.6281 
               
               
                 2.5 
                 0 
                 2.00 
                 −0.80812 
                 −15.311 
               
               
                 2.5 
                 0.05 π 
                 1.60 
                 −0.6182 
                 −18.5329 
               
               
                 2.5 
                 0.05 π 
                 1.65 
                 −0.63587 
                 −17.9876 
               
               
                 2.5 
                 0.05 π 
                 1.70 
                 −0.65318 
                 −17.4904 
               
               
                 2.5 
                 0.05 π 
                 1.75 
                 −0.67014 
                 −17.0343 
               
               
                 2.5 
                 0.05 π 
                 1.80 
                 −0.68677 
                 −16.6137 
               
               
                 2.5 
                 0.05 π 
                 1.85 
                 −0.70308 
                 −16.2241 
               
               
                 2.5 
                 0.05 π 
                 1.90 
                 −0.71911 
                 −15.8617 
               
               
                 2.5 
                 0.05 π 
                 1.95 
                 −0.73484 
                 −15.5234 
               
               
                 2.5 
                 0.05 π 
                 2.00 
                 −0.75032 
                 −15.2066 
               
               
                 2.5 
                 0.1 π 
                 1.60 
                 −0.57325 
                 −18.2094 
               
               
                 2.5 
                 0.1 π 
                 1.65 
                 −0.59161 
                 −17.6648 
               
               
                 2.5 
                 0.1 π 
                 1.70 
                 −0.60959 
                 −17.1682 
               
               
                 2.5 
                 0.1 π 
                 1.75 
                 −0.62719 
                 −16.7128 
               
               
                 2.5 
                 0.1 π 
                 1.80 
                 −0.64446 
                 −16.2929 
               
               
                 2.5 
                 0.1 π 
                 1.85 
                 −0.66141 
                 −15.904 
               
               
                 2.5 
                 0.1 π 
                 1.90 
                 −0.67805 
                 −15.423 
               
               
                 2.5 
                 0.1 π 
                 1.95 
                 −0.69439 
                 −15.2047 
               
               
                 2.5 
                 0.1 π 
                 2.00 
                 −0.71046 
                 −14.8886 
               
               
                 2.0 
                 −0.05 π 
                 1.60 
                 −0.63977 
                 −14.2231 
               
               
                 2.0 
                 −0.05 π 
                 1.65 
                 −0.65708 
                 −14.0227 
               
               
                 2.0 
                 −0.05 π 
                 1.70 
                 −0.67402 
                 −13.8258 
               
               
                 2.0 
                 −0.05 π 
                 1.75 
                 −0.69059 
                 −13.6331 
               
               
                 2.0 
                 −0.05 π 
                 1.80 
                 −0.70682 
                 −13.445 
               
               
                 2.0 
                 −0.05 π 
                 1.85 
                 −0.72274 
                 −13.2617 
               
               
                 2.0 
                 −0.05 π 
                 1.90 
                 −0.73835 
                 −13.0834 
               
               
                 2.0 
                 −0.05 π 
                 1.95 
                 −0.75367 
                 −12.9101 
               
               
                 2.0 
                 −0.05 π 
                 2.00 
                 −0.76872 
                 −12.7418 
               
               
                 2.0 
                 0 
                 1.60 
                 −0.57975 
                 −18.639 
               
               
                 2.0 
                 0 
                 1.65 
                 −0.59685 
                 −18.0935 
               
               
                 2.0 
                 0 
                 1.70 
                 −0.61357 
                 −17.5961 
               
               
                 2.0 
                 0 
                 1.75 
                 −0.62994 
                 −17.1398 
               
               
                 2.0 
                 0 
                 1.80 
                 −0.64597 
                 −16.719 
               
               
                 2.0 
                 0 
                 1.85 
                 −0.66169 
                 −16.3292 
               
               
                 2.0 
                 0 
                 1.90 
                 −0.67711 
                 −15.9666 
               
               
                 2.0 
                 0 
                 1.95 
                 −0.69224 
                 −15.6281 
               
               
                 2.0 
                 0 
                 2.00 
                 −0.70711 
                 −15.311 
               
               
                 2.0 
                 0.05 π 
                 1.60 
                 −0.53418 
                 −18.5329 
               
               
                 2.0 
                 0.05 π 
                 1.65 
                 −0.55149 
                 −17.9876 
               
               
                 2.0 
                 0.05 π 
                 1.70 
                 −0.56843 
                 −17.4904 
               
               
                 2.0 
                 0.05 π 
                 1.75 
                 −0.585 
                 −17.0343 
               
               
                 2.0 
                 0.05 π 
                 1.80 
                 −0.60123 
                 −16.6137 
               
               
                 2.0 
                 0.05 π 
                 1.85 
                 −0.61715 
                 −16.2241 
               
               
                 2.0 
                 0.05 π 
                 1.90 
                 −0.63276 
                 −15.8617 
               
               
                 2.0 
                 0.05 π 
                 1.95 
                 −0.64808 
                 −15.5234 
               
               
                 2.0 
                 0.05 π 
                 2.00 
                 −0.66313 
                 −15.2066 
               
               
                 2.0 
                 0.1 π 
                 1.60 
                 −0.50128 
                 −18.2094 
               
               
                 2.0 
                 0.1 π 
                 1.65 
                 −0.51926 
                 −17.6648 
               
               
                 2.0 
                 0.1 π 
                 1.70 
                 −0.53684 
                 −17.1682 
               
               
                 2.0 
                 0.1 π 
                 1.75 
                 −0.55405 
                 −16.7128 
               
               
                 2.0 
                 0.1 π 
                 1.80 
                 −0.57091 
                 −16.2929 
               
               
                 2.0 
                 0.1 π 
                 1.85 
                 −0.58744 
                 −15.904 
               
               
                 2.0 
                 0.1 π 
                 1.90 
                 −0.60365 
                 −15.423 
               
               
                 2.0 
                 0.1 π 
                 1.95 
                 −0.61956 
                 −15.2047 
               
               
                 2.0 
                 0.1 π 
                 2.00 
                 −0.63519 
                 −14.8886 
               
               
                   
               
             
          
         
       
     
     From the above analysis and data, it is apparent that a variable chirp modulator may be constructed in accordance with the invention in which the electrode length ratio, the bias and the power ratio may be varied to influence both the chirp and the extinction or On/off ratio. In accordance with the principles of the invention, a variable chirp Mach Zehnder Interferometer type modulator is obtained by selecting an initial modulator design with uneven modulation between the two modulator waveguide arms by selecting a modulation ratio of 0.25, for example, to achieve a chirp parameter of α=−0.6, as a baseline. 
     The chirp parameter value can then be adjusted from the baseline by: 
     1. Changing the power ratio γ=P 1 /(P 1 +P 2 ) between the two arms of the modulator. The power ratio may be changed with a tunable attenuator in one arm of the modulator; or 
     2. Moving the bias point away from quadrature by applying an offset DC bias; or 
     3. Combining both 2. and 3. to obtain a larger chirp range. 
     Advantageously, a chirp modulator in accordance with the invention can be realized with an X-cut or Z-cut substrate. In addition, a chirp value of −0.7 is achieved with an adjustable chirp range of up to ±0.25 or more with an extinction ratio of better than 15 dB. The impact of V π can be compensated with longer electrodes in the modulator. 
     In an implementation of a modulator in which an on/off ratio of −14 dB is desired, and the power between the two waveguide arms is not balanced, the power ratio x=4/9, and 9/4. Setting the value for negative chirp at −0.9, a length ratio of            L   e2       L   e1       =         l   2       l   1       =   2.61                            
     is obtained. If it is desired to obtain the same phase shift for both a configuration of equal length electrodes and un-balanced length electrodes, the following calculations can be made. For equal length electrodes φ 0 =2KL 0  and for the unbalanced electrode case φ=2Kl 1+K(l   2 −l 1 )=3.61Kl 1 . Equating the two cases yields l 1 =0.554L 0  and l 2 =1.446L 0 . In this embodiment, the electrode length of the longer electrode is approximately 45% longer than the equal length electrodes, whereas, for balanced power, the electrode length of the longer electrode is about 55% greater that the equal length electrodes. 
     Utilizing the foregoing analysis, other embodiments of the invention have been developed. 
     Turning now to FIG. 10, a modulator  1010  in accordance with the principles of the invention is shown. Integrated optic modulator  1010  includes an optical waveguide  1012  that is split by an optical splitter  1014  into two waveguide arms  1016 ,  1018 . Optical waveguide arms  1016 ,  1018  are rejoined with an optical coupler  1020 . FIG. 11 shows modulator  1010  in cross section taken along lines  11 — 11 . The modulator  1010  is formed on a substrate  1001  that is lithium niobate. As will be appreciated by those skilled in the art, substrate  1001  may be of other material including, but not limited to GaAs. Optical waveguide arms  1016 ,  1018  are formed in substrate  1001 . Electrodes  1022 ,  1024 ,  1026  are formed on substrate  1001  proximate optical waveguide arms  1016 ,  1018 . The distance between electrode  1022  and electrode  1024  is the same as the distance between electrode  1024  and portions  1027  of electrode  1026 , providing even gaps  1028 ,  1030  between electrode pairs. Electrodes  1022  and  1024  are of substantially equal effective length, l 2 , proximate waveguide arm  1016 . Electrode  1026  is configured so as to provide a different effective electrode length, l 1 , proximate waveguide arm  1018 . The particular configuration shown in FIGS. 10 and 11 is such that electrode  1026  includes a plurality of portions  1027 , numbering two in the specific embodiment shown, that have a combined length of l 1  proximate waveguide arm  1018 . The remaining portion or portions  1029  of electrode  1026  are spaced apart from waveguide arm  1018  so as to have minimal modulation effect. A variable attenuator  1040  is formed in waveguide arm  1018  and includes electrodes  1041 ,  1042 . The variable attenuator  1040  is utilized to provide unequal power splitting in the two waveguide arms  1016 ,  1018 . Bias electrodes  1050 ,  1051 ,  1052  are also provided. 
     Turning now to FIG. 12, a modulator  1210  in accordance with the principles of the invention is shown. Integrated optic modulator  1210  includes an optical waveguide  1212  that is split by an optical splitter or tunable coupler  1214  into two waveguide arms  1216 ,  1218 . Optical waveguide arms  1216 ,  1218  are rejoined with an optical coupler  1220 . FIG. 13 shows modulator  1210  in cross section taken along lines  13 — 13  and FIG. 14 shows modulator  1210  in cross section taken along lines  14 — 14 . Modulator  1210  is formed on a substrate  1201  that is lithium niobate. As will be appreciated by those skilled in the art, substrate  1201  may be of other material including, but not limited to GaAs. Optical waveguide arms  1216 ,  1218  are formed in substrate  1201 . Electrodes  1222 ,  1224 ,  1226  are formed on substrate  1201  proximate optical waveguide arms  1216 ,  1218 . The distance or gap  1228  between electrode the electrode pair comprising electrode  1222  and electrode  1224  is the same as the distance or gap  1230  between the electrode pair comprising electrode  1224  and electrode  1226 . Electrodes  1222  and  1224  are of substantially equal effective length, l 2 , proximate waveguide arm  1216 . Electrode  1226  and optical waveguide arm  1218  are cooperatively configured so as to provide a different effective electrode length, l 1 , proximate waveguide arm  1218 . The particular configuration shown in FIGS. 12,  13  and  14  is such that waveguide arm  1218  includes a first portion  1219  that is disposed between electrodes  1224 ,  1226  and a second portion  1221  that is offset from portion  1219  and disposed out of the area of substrate  1201  that is affected by a modulation voltage impressed across electrodes  1226 ,  1128 . The amount of offset x1 between portion  1219  and portion  1221  provided by offset portion  1223  is matched in waveguide arm  1216  by portion  1215 . The offset portion  1215  is provided in waveguide arm  1216  to provide for equal optical path lengths in the two waveguide arms  1216  and  1218 . A variable attenuator  1240  is formed in waveguide arm  1218  and includes electrodes  1241 ,  1242 . Variable attenuator  1240  is utilized to provide unequal power splitting in the two waveguide arms  1216 ,  1218 . Bias electrodes  1250 ,  1252 ,  1254  are also provided and utilized to provide bias. Also shown if FIG. 12 is a variable coupler  1270  comprising electrodes  1271 ,  1272 ,  1273 . Typically either variable coupler  1270  or variable attenuator  1240  is used to control the power split between waveguide arms  1216 ,  1218 . 
     The invention has been described in terms of several embodiments. It will be apparent to those skilled in the art that various changes and modifications may be made to the embodiments shown and described without departing from the spirit or scope of the invention. It is intended that the invention includes all such changes and modifications and other changes and modifications that are not specifically mentioned. It is further intended that the invention not be limited in scope to the embodiments shown and described, but that the invention is limited in scope only by the claims appended hereto.