レンズの焦点距離とは？ 技術的な定義
Focal length is the distance from the lens's rear principal plane to the rear focal point when the lens is focused at infinity, measured in air (refractive index n=1). Together with the sensor dimension, this fundamental optical parameter sets the angular field of view. Together with object distance, it sets magnification.
技術的な精度
For thick lens systems, focal length is specifically measured from the rear principal plane H' to the rear focal point F'. For thin lenses where thickness t << f, the principal planes coincide at the lens center, simplifying to the elementary definition.
レンズシステムの焦点距離は、次の2つの逆比例関係を決定します：
- 焦点距離が長いほど → 視野が狭くなり、倍率が高くなる
- 焦点距離が短いほど → 視野が広くなり、倍率が低くなる
焦点距離の式：ガウス光学
基本的な焦点距離の式は、ガウスレンズの公式を通じて、被写体距離、像距離、および焦点 距離を関連づけています：
場所： f = 焦点距離、 u = 物体までの距離、 v = 画像距離
すべての距離は主平面からの距離として測定されています
光学エンジニアにとって、焦点距離の式は、より一般的には倍率の 形で表される：
Gives the magnification magnitude. The real image is inverted
Valid for paraxial rays only
画角において、焦点距離はどのような役割を果たすのでしょうか？
Focal length and sensor size determine the ideal field of view at a distant focus. For rectilinear lenses, the angular field of view follows:
θ = 全視野角、 d = センサーの寸法、 f = 有効焦点距離
Ideal rectilinear projection at a distant focus. Distortion and finite focus distance change the result
焦点距離計算ツール – 必要なEFLを算出
EFL vs BFL vs FFL: Which Focal Length Is Which?
Three lengths share the name and cause most datasheet confusion:
Effective focal length (EFL) is the distance from the rear principal plane H′ to the rear focal point with the lens focused at infinity. It is the optical quantity that sets field of view and magnification, and the principal plane it starts from can sit in front of, inside, or behind the physical barrel. Commonlands specs every lens by its EFL.
Back focal length (BFL) is the distance from the vertex of the last glass surface to the rear focal point. It is an optical distance that helps assess clearance. Usable mechanical space also depends on the rear barrel, focus distance, and filter or sensor stack. Telephoto designs have BFL < EFL. Retrofocus designs have BFL > EFL.
Front focal length (FFL) is the distance from the vertex of the first glass surface to the front focal point. Together with EFL, BFL, and the defined vertex references, it can locate the principal planes in the same surrounding medium.
Related but distinct: flange focal distance is a mount standard (17.526mm for C-mount), not a property of the lens design.
Worked Example: CIL059 on an OX08B40 Sensor
The Commonlands CIL059 has a 5.9mm EFL, and the Omnivision OX08B40 on its compatibility list has an 8.064mm-wide active area. At a 2m working distance:
Angular check: θ = 2 × arctan(8.064 / (2 × 5.9)) ≈ 69° horizontal
Measured on this pairing: 71° horizontal, 79° diagonal – the lens's −4% barrel distortion maps slightly more scene onto the sensor than the zero-distortion estimate predicts
Focal-Length Ratio Calculator
This tool reports f2/f1, the ratio of angular image scales for distant objects. It approximates the magnification ratio at the same distant object distance. At finite distance, compare m = f/(u-f) for each refocused thin lens instead.
センサーフォーマットの補正
| 書式 | 対角線（mm） | 幅 × 高さ (mm) | クロップファクター |
|---|---|---|---|
| 1/3インチ | 6.0 | 4.8 × 3.6 | 7.2× |
| 1/2.3インチ | 7.7 | 6.2 × 4.6 | 5.6× |
| 1/2インチ | 8.0 | 6.4 × 4.8 | 5.4× |
| 1/1.8インチ | 8.9 | 7.2 × 5.3 | 4.9× |
| 2/3インチ | 11.0 | 8.8 × 6.6 | 3.9× |
| 1" | 16.0 | 12.8 × 9.6 | 2.7× |
この表に記載されていないセンサーの寸法、画素サイズ、およびレンズの互換性については、イメージセンサーの完全な一覧をご覧ください。
高度な光学上の考慮事項
マシンビジョンにおけるテレセントリック性
Object-space and bi-telecentric lenses approximately preserve magnification as object distance changes, with residual telecentricity error. Image-space telecentricity alone does not provide that property. Object-space telecentricity places the entrance pupil at infinity, while image-space telecentricity places the exit pupil at infinity, ensuring chief rays are parallel to the optical axis.
F値と開口数の関係
Here D is the entrance-pupil diameter. At infinity focus, working f-number approaches nominal f-number. At finite magnification, use the working f-number, including the lens's pupil magnification, when relating aperture to image-space NA.
ここで、D は入射瞳径、NA は開口数である。被写界深度と回折限界を決定する
焦点距離はシステム設計にどのような影響を与えるのか？
When choosing focal length, also check:
- Depth of Field: DOF ≈ 2Nc(u²/f²) at low magnification and focus distances well below hyperfocal. Here N is f-number and c is the allowed geometric blur diameter. Use near and far limit equations near hyperfocal focus
- ハイパーフォーカル距離：H = f²/(Nc) + f
- 回折限界：エアリー円盤の直径 = 2.44λ(f/#)
- 入射・出射瞳：ヴィネット効果とテレセントリック性を決定する
視野（FOV）に対する歪みの影響
標準的な焦点距離の式は、歪みがゼロであることを前提としています。実際のレンズには、次のような特徴が見られます：
- 樽型歪み：実際の画角 > 計算値（広角レンズではよくある現象）
- Pincushion distortion: Actual FOV < calculated (typical in telephoto)
- 魚眼投影：別のモデル（等距離投影、ステレオ投影）が必要