A Neutral Density Filter (ND filter) is an optical device used to uniformly attenuate the intensity of incident light without altering its spectral color temperature. Its core function is to flexibly control the amount of light entering the lens in high-brightness environments, accommodating special shooting requirements such as slow shutter speeds for long exposures or large apertures for shallow depth of field.

Figure 1[1]: Neutral Density Filter

I. Application Scenarios
1. Long Exposure Photography
In well-lit environments, if a slow shutter speed is required to achieve long exposure effects (such as capturing silky smooth water, misty clouds, or light trails from traffic), there is still a risk of overexposure even when the ISO and aperture are set to their minimums. An ND filter can attenuate the total amount of light entering the lens by a fixed ratio, achieving safe exposure and completing the long exposure shot without compromising the creative intent of the exposure triangle (aperture, shutter speed, and ISO).

Figure 2: Waterfall scene[2]

Without an ND filter, relying solely on stopping down the aperture would significantly increase the depth of field, ruining the creative intent of a shallow depth of field achieved with a large aperture; relying on post-processing would result in irreversible loss of detail in overexposed areas due to blown-out highlights.

2. Large Aperture and Shallow Depth of Field Photography
When shooting with a large aperture in bright environments, overexposure is likely due to excessive light intake. An ND filter can effectively reduce the amount of light, allowing creators to shoot wide open with large apertures even in strong light, achieving an extremely shallow depth of field. This blurs the background to highlight the subject and enhances the depth and dimensionality of the image.
II. Technical Principles
1. Structure and Mounting
A typical neutral density filter is usually made of optical glass or high-quality plastic as the substrate, mounted in a standard metal or plastic frame. Depending on the mounting specifications, these filters can be screwed in via threads or inserted into a slot to secure them at the front of the lens. Due to their uniform dark gray appearance, they are also commonly referred to as “ND filters.”

Figure 3: Physical image of an ND filter[3]

2. Color Characteristics
The gray base ensures proportional absorption and attenuation across the entire visible light spectrum, unlike color filters that preferentially absorb specific wavelength bands. Using a color filter to attenuate light intensity would cause severe color cast in the image; whereas a neutral density filter only alters the energy intensity of the light, without changing its spectral distribution, thereby ensuring pure image colors and accurate white balance.

3. Optical Density and Transmittance
Optical Density (OD or D) is the core parameter characterizing the attenuation capability of an ND filter. Its definition is mathematically inverse to Transmittance (T): $$ D = \log\left(\frac{1}{T}\right) $$

Where:

• $T$ (Transmittance): Represents the ratio of transmitted light intensity to incident light intensity ($T = I_{out}/I_{in}$), with a value range of $0 \sim 1$.
• $D$ (Optical Density): Represents the degree of light absorption and attenuation; the higher the value, the stronger the attenuation capability.

Example:
If a filter allows 100% of light to pass through, its optical density is 0. If it allows only half of the light to pass through, its optical density is 0.3.
T = 100/100 = 1 → D = log (1) = 0
T = 50/100 = 0.5 → D = log (2) = 0.3

In photography, the optical density of an ND filter is usually directly converted into f-stops to facilitate the adjustment of exposure parameters. An optical density of 0.3 corresponds to 1 f-stop (halving the light intake).

4. Exposure Factor NDX
In engineering and photography practice, in addition to using optical density D to describe attenuation strength, manufacturers often use the Exposure Factor (NDX) to intuitively indicate the light attenuation multiplier. The number in NDX represents the number of times the light intake is halved, such as ND2, ND4, ND10, etc.
Taking ND2 as an example: ND2 indicates that the light intake is attenuated to $1/2$ of the original, allowing only half of the light to reach the sensor, i.e.: $$ T = \frac{1}{2} = 0.5 $$ Substituting $T = 0.5$ into the optical density formula yields:$$ D = \log\left(\frac{1}{0.5}\right) = \log(2) \approx 0.3 $$
Thus, ND2 is equivalent to an optical density of 0.3, which also corresponds to a 1 f-stop attenuation in photography.
Conversion Relationship:

III. Classification and Usage Rules of ND Filters
Neutral density filters are divided into fixed density and variable density categories based on their usage characteristics, with distinct labeling conventions:
1. Fixed Density Filters
These provide a specific attenuation value with clear labeling rules:

• ND + decimal (e.g., ND0.3, ND1.0): Indicates Optical Density
• NDX + integer (e.g., ND4, ND10): Indicates Exposure Factor

For example: ND0.3 (density 0.3, corresponding to 1 f-stop attenuation), ND10 (exposure factor 10, corresponding to 3 1/3 f-stop attenuation).

2. Variable Density Filters
Variable density filters support continuous adjustment of attenuation strength, making them adaptable to different shooting scenarios (especially suitable for traveling with a single versatile filter). They are characterized by the lack of fixed density markings, making it impossible to directly read the specific density value or easily determine the corresponding shutter speed and aperture parameters. Refer to the table below to deduce the required ND filter specification based on the target shutter speed, assisting in adjusting the attenuation level of the variable ND filter for precise exposure.

Filter Stacking Rules
When multiple fixed filters are stacked, parameters are calculated according to corresponding rules, with the total attenuation being the sum or product of each filter:

• Optical Density (D): Add directly
• Exposure Factor (NDX): Multiply directly
• f-stops: Add directly

Example:

• Filter A: Density 1, Exposure Factor: ND X10, f-stops: 3 1/3
• Filter B: Density 2, Exposure Factor: ND X100, f-stops: 6 2/3

When Filter A and B are combined:

• Density D = 1 + 2 = 3
  
• Exposure Factor = 10 * 100 = 1000
  
• f-stops = 3 1/3 + 6 2/3 = 10
  

[1]https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Neutral_density_filter_demonstration.jpg/640px-Neutral_density_filter_demonstration.jpg
[2]https://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Strickland_Falls_Shadows_Lifted.jpg/500px-Strickland_Falls_Shadows_Lifted.jpg
[3]https://en.wikipedia.org/wiki/File:Filtros_ND_Haida_Pro_II_MC,_2017-02-05,_DD_FS.jpg