エンベロープ（包絡）関数

envelope function

高分解能法での像のコントラストには位相コントラスト伝達関数が重要な役割を果たすが、電子線のエネルギーの違いによる（対物レンズの）色収差、入射電子線の開き角、対物レンズの安定度によって、散乱角とともに位相コントラストは減衰する。この減衰を与える関数を包絡関数という。包絡関数が事実上ゼロになるところをインフォーメーションリミットという。

envelope function⇒図
加速電圧200kVにおけるエンベロープ関数（包絡関数）の例 (黒線)。横軸は空間周波数、縦軸は試料の情報が電顕像にどのように伝達されるかを示す。包絡関数の値が1（あるいは-1）に近いほど、試料の構造の情報が結像に多く寄与する。包絡関数が零に近づくにつれて試料の構造情報は欠落してゆく。
包絡関数は色収差や照射角広がり等の影響によって決まる干渉性の度合いを表す。色収差や照射角広がりが大きい場合は、散乱波の干渉性は空間周波数の増加とともに早く劣化し、包絡関数は早く零に近づく。実際の位相コントラスト伝達関数は、灰色線で示すように包絡関数が掛かったもので、空間周波数の増加とともに像強度への散乱波の寄与が減少する。

The phase-contrast transfer function plays an important role for the resolution of an HREM image contrast. Another important factor is the envelope function. The phase contrast is damped with increase of scattering angle due to chromatic aberration of the objective lens caused by energy difference in the electron beam, the divergence angle of the electron beam, stability of the objective lens, etc. The function that expresses the damping is called "envelope function." The spatial frequency at which the envelope function becomes practically 0 (zero) is called "information limit."

envelope_function

Example of an envelope function (black line) at an accelerating voltage of 200 kV. The horizontal axis stands for the spatial frequency and the amount of information on the crystal structure of a specimen transferred to a TEM image. The closer the value of the envelope function to 1 (or –1), the more structural information on the specimen contributes to form a TEM image. The closer the envelope function to 0 (zero), the more structural information is lost.
The envelope function expresses the degree of coherence of the scattered waves, which is determined by chromatic aberration, angular spread of an incident electron beam, etc. When the chromatic aberration or angular spread is large, the degree of coherence of scattered waves deteriorates quickly with increase of spatial frequency, or the envelope function attenuates quickly to 0. The phase-contrast transfer function (PCTF) should incorporate the effect of the envelope function, then resulting to an actual PCTF as indicated by a gray line. This PCTF shows that the contribution of the scattered waves to the image formation damps with the spatial frequency.