![\"Figure](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/07\/Pic461.jpg\")
<\/a>Figure 1. Illustration of the design concept.<\/p><\/div>\n
This is achieved by changing the IDT central frequency\u00a0(increasing it in most cases, while examples of materials exist\u00a0where it needs to be decreased). Slight detuning from discussed\u00a0by Uno optimal conditions used in QARP approach [4] does\u00a0not affect much the resonator performance, while giving the \u00a0flexibility of changing the IDT pitch in such a way, so as to\u00a0avoid narrow or wide gaps between reflectors and the\u00a0transducer.<\/p>\n
In case of a substrate where a NSPUDT conditions are met,\u00a0such as for example langasite (yxlt\/48.5\u00ba\/26.6\u00ba, Euler angles \u03bb,\u00a0\u03b8, \u03bc = 0, 138.5, 26.6\u00ba) the shape of conductance has two equal\u00a0maximums with values that are almost two times lower than\u00a0the single one of a bi-directional IDT. This is the reason why\u00a0resonators with such transducers have somewhat lower\u00a0conductance at resonance and higher relative level of out-ofband\u00a0unwanted signals. Applying the Uno concept and simply\u00a0shifting the IDT frequency up or down increases the relative\u00a0level of unwanted signals.<\/p>\n
<\/a>Figure 2. Example of resonator (solid) and IDT conductance for ordinary\u00a0IDT (top graph) and for linear chirp IDT (lower graph) on a NSPUDT\u00a0substrate.<\/p><\/div>\n
In order to overcome this drawback the unidirectional\u00a0transducer may be in one or another way transformed to\u00a0acquire some features of a bi-directional IDT. Some past\u00a0experience has shown that transducers with pitch variation\u00a0notably chirp transducers at some conditions may acquire
\npronounced unidirectional properties on ordinary substrates [6].<\/p>\n
When a NSPUDT transducer has a linear chirp structure it\u00a0loses to some extent its unidirectional properties and its\u00a0conductance shape becomes more similar to that of a bidirectional\u00a0IDT. Decreasing or increasing the pitch causes the\u00a0left or right peak of the IDT conductance to be enhanced,\u00a0respectively, while the other one is suppressed\u00a0This design concept is illustrated in Fig. 1 where the upper\u00a0layout corresponds to a known approach (for example [5])\u00a0where an additional gap between the IDT and the right reflector\u00a0is used and the lower layout shows how the IDT pitch is\u00a0changed in order to modify the IDT conductance and to\u00a0eliminate the additional gap between the IDT and the reflector,\u00a0thus realizing to some extent the principles of QARP resonators\u00a0[4].<\/p>\n
Figure 2 illustrates the behavior of the IDT conductance\u00a0and of the resulting resonator response for both designs (shown\u00a0in Fig.1) where the relative frequency is the frequency\u00a0normalized to the Bragg frequency of reflectors. Both\u00a0structures are modeled for the reflection phase strictly equal to\u00a090\u00ba most favorable for NSPUDT effect. Note that the reflection\u00a0phase may have different definitions due to arbitrary choice of\u00a0the reflection origin point. Unlike in [5] we will use here the
\nfollowing phase definition [7] \u03c8 = Arg(k12\u03b6* \u2044 \u03b6) = Arg(k12) \u20132Arg(\u03b6). It means that the case of \u03c8 = 0 corresponds to ordinary\u00a0substrate with thin electrodes, \u03c8 = 180\u00ba corresponds to the\u00a0opposite bi-directional case and \u03c8 = \u00b190\u00ba corresponds to\u00a0maximal NSPUDT effect, positive \u03c8 corresponds to larger\u00a0radiation from right side of IDT. The reflectors contain 263\u00a0strips and the transducers have 126 electrodes. The periods of\u00a0the gratings are the same in both cases and correspond to the\u00a0required central frequency that preferably would be in the\u00a0middle of the Bragg band. The pitch of the chirp transducer\u00a0changes from that of the gratings to a value slightly lower (in\u00a0this example by about 0.9%) or slightly larger (in general case). \u00a0As a result the larger maximum of the chirp IDT conductance\u00a0shifts closer to the Bragg frequency of reflectors. The position\u00a0of reflectors is thus slightly adjusted together with the change\u00a0of the IDT dimensions in order to realize QARP-like conditions\u00a0[4].<\/p>\n
Comparison of the responses in Fig.2 demonstrates about\u00a010-15% increase in resonator conductance and some decrease\u00a0of the relative level of unwanted signals. Improvement in\u00a0properties is further enhanced by reduction of scattering from\u00a0the abrupt change in periodicity.<\/p>\n
This design approach becomes seriously more important for \u00a0practical cases of non-ideal natural unidirectivity. As a rule the\u00a0phase of the reflection coefficient depends on the thickness of\u00a0the metal. For langasite the useful value of reflection from the\u00a0gratings is achieved at a thickness of Al above 1% of the\u00a0wavelength. At this thickness the phase \u03c8 reaches 120\u00ba for 50%\u00a0metallization ratio and may be even larger with wider\u00a0electrodes.<\/p>\n
RESONATOR DESIGN METHOD FOR GENERAL CASE<\/h2>\n
As shown in Figure 3 the ideal NSPUDT case with \u03c8 = 90\u00ba\u00a0is rarely achievable in practically interesting cases. In order to\u00a0obtain sufficiently high reflection amplitude one has to choose\u00a0high enough aluminum thickness (2-4% of the wavelength) and\u00a0metallization coefficient (between 50 and 75%). In these\u00a0conditions the reflection phase stabilizes closer to 120\u00ba and this\u00a0feature has to be taken care of in the design, for example as in\u00a0[5].<\/p>\n
<\/a>Figure 3. Amplitude and phase of the reflection coefficient of Al strips with
different metallization ratios on langasite yxlt\/48.5\u00b0\/26.6\u00b0.<\/p><\/div>\n
If the variable electrodes pitch concept described above is\u00a0applied to a substrate with \u03c8 = 0, the symmetry has to be\u00a0observed in order to obtain a QARP-like response. For this case\u00a0the pitch in one half of the IDT goes down linearly and then it\u00a0goes up in the second half. The highest instantaneous frequency\u00a0of the IDT is reached in the middle of the structure, while the\u00a0average frequency satisfies QARP conditions. For the \u03c8 = 180\u00ba\u00a0the inverse relation is true and the IDT frequency is the lowest\u00a0in the centre of the structure. The modeling has shown that the\u00a0resonator performance in these arrangements is very close to\u00a0the performance of the corresponding QARP arrangements.\u00a0But in such cases the variable pitch is usually unnecessary.<\/p>\n
However, in this way we get a useful hint for approaching\u00a0intermediate cases when the reflection coefficient phase is not\u00a00, 180, 90 or 270\u00ba. The intermediate phase difference requires\u00a0intermediate positioning of the lowest (or highest) pitch region\u00a0inside the transducer area.<\/p>\n
<\/a>Figure 4. Electrode pitch versus electrode number for 120\u00b0 reflection phase.<\/p><\/div>\n
This is illustrated in Figure 4, where the details of resonator\u00a0design are shown for \u03c8 = 120\u00b0. The linear change of IDT pitch\u00a0is approximated by 9 sections with 14 electrodes (126\u00a0electrodes total) with constant pitch in each section. The pitch\u00a0of the 263 electrodes identical gratings is equal to 8.077\u00a0microns and the pitch of the IDT section grows from 8.09\u00a0microns in the first section from the left to 8.172 microns in the\u00a0seventh section from the left and falls to 8.144 microns in the\u00a0ninth (last) section. Note that for such small pitch variations\u00a0linear change of the pitch used in this design practically does\u00a0not differ from the linear change of the IDT central frequency\u00a0as in chirp structures, and both variants may be used with\u00a0similar results.<\/p>\n
The position of the IDT pitch maximum in the vicinity of\u00a0the seventh section is realizing to some extent the condition of\u00a0maximum energy transfer discussed in [5]. This position is \u00a0relatively insensitive to slight shifts and the resonator\u00a0performance is not much affected by modifying the linear chirp\u00a0pitch variation by partitioning into sections as in Table 1. A\u00a0reasonable suggestion for the placement of the IDT point where\u00a0the pitch differs the most from the gratings pitch is the<\/a><\/p>\n
\nfollowing:<\/p>\n