I was doing some research for an upcoming guide to long exposure photography, and I was struck by how differently the various filter manufacturers name their neutral density (ND) filters. I had a long phone conversation about filters with a friend this week, and the terminology differences also caused some confusion there as well.

Some manufacturers label their filters using what is called the Optical Density of the filter, whereas some of them use what’s known as the Filter Factor.

F-Stop Reduction | Optical Density | Filter Factor | % transmittance |

0 | 0 | 0 | 100 |

1 | 0.3 | 2 | 50 |

2 | 0.6 | 4 | 25 |

3 | 0.9 | 8 | 12.5 |

4 | 1.2 | 16 | 6.25 |

5 | 1.5 | 32 | 3.125 |

6 | 1.8 | 64 | 1.5625 |

7 | 2.1 | 128 | 0.78125 |

8 | 2.4 | 256 | 0.390625 |

9 | 2.7 | 512 | 0.1953125 |

10 | 3.0 | 1024 (sometimes called ND1000) | 0.09765625 |

11 | 3.3 | 2048 | 0.048828125 |

12 | 3.6 | 4096 | 0.0244140625 |

13 | 3.9 | 8192 | 0.01220703125 |

13 1/3 | 4.0 | 10000 | 0.01 |

14 | 4.2 | 16384 | 0.006103515625 |

15 | 4.5 | 32768 | 0.003051757813 |

16 | 4.8 | 65536 | 0.001525878906 |

16 2/3 | 5.0 | 100000 | 0.001 |

17 | 5.1 | 131072 | 0.0007629394531 |

18 | 5.4 | 262144 | 0.0003814697266 |

19 | 5.7 | 524288 | 0.0001907348633 |

20 | 6 | 1048576 | 0.00009536743164 |

## Filter Factor (ND2, ND4 etc.)

This is simply a representation of the factor by which the neutral density filter reduces the light coming into the lens. For example, an ND filter that reduces light by one stop has a filter factor of 2. A one stop reduction of light is always half the light (see previous tutorial on aperture and f-stop), so the factor by which a one stop neutral density reduces the light, is 2.

The confusion arises because with these smaller filter factor numbers (2,4,8,16), people confuse them the with the reduction of light in f-stops. When they see ND2 on a filter, they think it is a 2-stop ND filter, but actually it’s a 1-stop filter. Similarly, a filter that has ND16 written on the side of it is actually a 4-stop filter and not a 16-stop filter. Please refer to the table above for a full list of common filter factors.

Since the light reduction doubles for each further reduction in f-stop, we can say that where **x** is the reduction in f-stops, **Filter Factor = 2 ^{x}**

**Example**

A 5-stop reduction in light would give us a filter factor of **2 ^{5 }= 2x2x2x2x2 = 32 , **so an ND32 is a 5-stop neutral density filter.

## Transmittance (%)

The transmittance number is actually very rarely mentioned on a filter itself, but you might see it mentioned on some packaging. The reason I included it in the reference table is really because I find it’s just a good visual representation of the light that gets cut by neutral density filters. To many people, when I tell them that a 6-stop ND filter only lets in 1.56% of the light, it’s instantly easier for them to visualize the result of that, as opposed to me just saying it has a filter factor of 64. The % Transmittance also comes into play if you want to mathematically calculate the optical density.

## Optical Density (0.3, 0.6, 0.9 etc.)

These days this seems to be the most common way for manufacturers to represent the amount of light by which their neutral density filter cuts light. An ND0.3 is a 1-stop ND filter, and an ND0.9 is a 3-stop ND filter for example (see the reference table above).

### Optical Density Equation

Being the kind of photo nerd that I am, and also an engineer in a past existence, I couldn’t just trust that these numbers mean what they do, and simply put them into the reference table verbatim! So of course I looked a little deeper and found out how these numbers are achieved.

The formula relating to optical density is: **Fractional Transmittance = 10 ^{-d}**

Where **d **is the optical density we are looking for and **fractional transmittance = (% transmittance/100)**

This now gives us **% transmittance = 10 ^{-d }x100**

But we also know that **% transmittance = (100/Filter Factor)**

So now we can say **100/Filter Factor** = **10 ^{-d }x100**

Additionally, where **X **is the light reduction in f-stops,** **we know from the earlier section that **Filter Factor is = 2 ^{x}**

Now that gives us **100/( 2^{x})** =

**10**

^{-d }x100To solve the equation for **d, **we would get: **d = log _{10} (1/((100/2^{x})/100)**

If you simplify all the stuff in the brackets on the right, you are actually just left with **2 ^{x}**

Therefore we can say that **d = log _{10}**

**2**or

^{x}**d=log**

_{10}Filter FactorUsing this equation we can work out the optical density number using the reduction of light in f-stops.

**Example**

We want a reduction in light of 3-stops and we want to know which filter to pick because they aren’t labelled in stops. We would do -> **d = log _{10} 2^{3} = log_{10 }8 = 0.90308998699**

For simplicity we just say **0.9**! A 3-stop ND filter is also called a 0.9.

Now you know this optical density number it is simply the Log of the factor by which light is decreased!

## Dan… WTF!?

Ok, ok… don’t worry, there’s no need to actually remember all that mathematical stuff. All you really need to know is what is in the table above. I just put all the math there for the small minority that might be curious about it, so please don’t let it scare you off. Neutral density filters are a really important thing for photographers, so it *is* important to know which ones are which, but you can just use the table to memorize the relationships and names.

Hopefully you guys find this useful and it clears up a few things about the naming of neutral density filters.

Fantastic Dan! Great info on how neutral density filters work – this has given me a better understanding, although the math is still as confusing as it was in school!

Thanks man! Glad it was helpful

Oh my dear!

With all this mathematical formulas you have just hit a strong with me! I used to be a math nerd as well in an earlier life… It’s so simple now after you explained the relationship between theses numbers and figures. Why did I need such complicated formulas to understand it the easy way? Thanx, Dan.

Greetings, Andreas.

PS: Hope to meet you in July!

I mean, you stroke a chord…

Haha! I’m glad someone appreciated that part 🙂

As a photographer who flunked out of engineering school, the main thing I want to know is what ND filter do I need to achieve a certain f stop/shutter speed/ISO combination and by your excellent table I see a neat way to turn optical density numbers into f stop reduction numbers. That is, by dividing the optical density number by .3. Or, just ignore the decimals and divide the density number by 3. Now when I see advertisements for filters and wonder what that obscure optical density number means, I have a shortcut to translate that number into meaningful information. I like easy shortcuts like this… and Laplace Transforms. Thanks Dan, for explaining it all and giving us ways to relate the numbers.

That’s great! I’m glad it was helpful 🙂

Seriously. I click, sign up, but I don’t get any pdf!

I wish that all of the internet was totally instantaneous, but it is not. It can sometimes take a few minutes for an email to be delivered to you, and sure enough my system shows that you opened the email with the PDF download 4 minutes after posting this comment.

Dear Dan,

I thank you so much for your help and I am sorry for my previous personal email where I sent you before about downloading problem.

Best wishes.

No problem Osman, glad we got it sorted!

Dan do you know if there are any filters that would fit into the adapters for the EOS R?

The adapter is available with either a built-in polarizer or a built-in ND filter. You can’t fit your own filters into them to my knowledge. –>> https://www.bhphotovideo.com/c/product/1433719-REG/canon_mount_adapter_ef_rf_with.html/BI/20085/KBID/13886/DFF/d10-v21-t1-x915558/SID/DFF

got everything else you wanted me to have, filled in all you requested but totally failed to get what I wanted . the information and nd charts

I have emailed you the download link, Jim. Sorry you had trouble receiving the charts.

My challenge question to you is to do the same photos that you used ND for only with out ND long exposures without ND is that possible to take landscape of water on sunny cloudy day. 90 % sunny.Can it be done..to achive slow water movement of water falls @ high NOON??

Well no, that can’t be done. Put your camera in the lowest ISO setting, and set your lens’ aperture to its smallest setting and the resulting shutter speed will be the longest option you have. On a bright sunny day, that’s not going to even close to being long enough for smoothing out large bodies of water.

Hi Dan, Thanks for your great article. I want to get the best ND for video… No color cast. This took me to the transmission charts for various filters. Doing comparisons between various brands leads me to the conclusion that there is some false marketing going on. There are big discrepancies on transmission results between the different brands. Trouble is, I’m not qualified enough to know who is truthful and who is not. Do you know of an independent lab with trustworthy lab results for the different manufacturer’s ND filters? Thanks so much! 😊

Thank you for explaining this topic so concisely and including the math, it’s just the sort of explanation I was looking for.

You’re welcome!