렌즈의 초점 거리는 무엇인가? 기술적 정의
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.
기술적 정밀도
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.
렌즈 시스템의 초점 거리는 다음 두 가지 역비례 관계를 결정합니다:
- 초점 거리가 길수록 → 시야가 좁아지고 배율이 높아진다
- 초점 거리가 짧을수록 → 시야가 넓어지고 배율이 낮아진다
초점 거리 방정식: 가우시안 광학
기본 초점거리 방정식은 가우스 렌즈 공식을 통해 피사체 거리, 상 거리 및 초점 거리를 서로 연관 짓습니다:
장소: f = 초점 거리, u = 물체까지의 거리, v = 이미지 거리
모든 거리는 주평면으로부터 측정된 것임
광학 엔지니어들에게 초점 거리 방정식은 대개 배율 형태로 나타납니다:
Gives the magnification magnitude. The real image is inverted
Valid for paraxial rays only
초점 거리는 시야에서 어떤 역할을 하나요?
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 = 전체 시야각, d = 센서 치수, f = 유효 초점 거리
Ideal rectilinear projection at a distant focus. Distortion and finite focus distance change the result
초점 거리 계산기 – 필요한 EFL 구하기
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.
센서 포맷 보정
| 서식 | 대각선 길이 (mm) | 폭 × 높이 (mm) | 크롭 팩터 |
|---|---|---|---|
| 1/3인치 | 6.0 | 4.8 × 3.6 | 7.2× |
| 1/2.3인치 | 7.7 | 6.2 × 4.6 | 5.6× |
| 1/2인치 | 8.0 | 6.4 × 4.8 | 5.4× |
| 1/1.8인치 | 8.9 | 7.2 × 5.3 | 4.9× |
| 2/3인치 | 11.0 | 8.8 × 6.6 | 3.9× |
| 1" | 16.0 | 12.8 × 9.6 | 2.7× |
이 표에 포함되지 않은 센서 크기, 픽셀 크기 및 렌즈 호환성에 대해서는 전체 이미지 센서 목록을 참조하십시오.
고급 광학 고려 사항
머신 비전에서의 텔레센트릭성
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-값과 수치 구경의 관계
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.
여기서 D = 입사 동공 직경, NA = 수차 개구율피사 심도와 회절 한계를 결정한다
초점 거리는 시스템 설계에 어떤 영향을 미치나요?
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
- 초점초점거리: H = f²/(Nc) + f
- 회절 한계: 에어리 원반 직경 = 2.44λ(f/#)
- 입사/출사 동공: 비네팅 및 텔레센트릭성을 결정합니다
시야(FOV)에 미치는 왜곡 효과
표준 초점 거리 공식은 왜곡이 전혀 없다고 가정합니다. 실제 렌즈에서는 다음과 같은 현상이 나타납니다:
- 배럴 왜곡: 실제 시야각 > 계산된 시야각 (광각 렌즈에서 흔히 나타나는 현상)
- Pincushion distortion: Actual FOV < calculated (typical in telephoto)
- 어안 투영: 대체 모델(등거리, 입체 투영)이 필요합니다.