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With film cameras filters have traditionally been used to modify both the spectral
content (color) and intensity of light, as well as generating special effects like soft
focus. Digital cameras operate somewhat differently with respect to color though. Color
modification can easily be done "in camera", and that's what you're doing when
you set white balance. So while for film warming and cooling filters, or filters which
convert fluorescent light to look like daylight, may be required, digital can achieve the
same effects by internal manipulation of the digital data.
So the first type of filter you really don't need for digital is a color modification
filter. These would be warming filters like the Hoya 81 series or the Tiffen 812, cooling
filters like the 82 series and fluorescent-daylight conversion filters.
So which filters do you need for digital? Well, there are probably 6 main
A polarizing filter
A UV filter
A Neutral Density filter
A graduated Neutral Density filter
Filters for Infrared Effects
Special Effect Filters
You can't digitally simulate the effect of a polarizer. It can darken blue skies,
increase contrast of partly reflecting subjects (like leaves and flowers), and reduce
reflections from non metallic surfaces like glass and water. I'd put a polarizing filter
#1 on my list of "must have" digital filters.
Note that all DSLRs will require a circular polarizer. Digicams generally don't care
and are quite happy with linear polarizers. Their effect on the image is exactly the same,
but for various technical reasons involving the partly reflecting reflex mirror in DSLRs
they need a circular polarizer to avoid metering errors (and possible potential problems
UV filters block ultraviolet light, which can cause a bluish cast in images shot where
there's a lot of UV (e.g. high altitudes). However most people use UV filters to protect
the front element of their lens as much as to block UV. I'm not going to take sides in the
debate over whether of not it's a good idea to use a UV filter all the time. Certainly
using a low quality filter can degrade image quality, but using a good filter shouldn't
really have any noticeable adverse effects on the image.
Neutral Density Filters
Neutral density filters simply absorb light of all wavelengths. A
0.6D filter absorbs 2 stops of light and a 0.9D filter absorbs three stops of light. The
major application is to allow the use of slower shutter speeds or wider apertures than
would otherwise possible. For example in bright sunlight at ISO 100 (the lowest speed on
most DSLRs, though some will go to ISO 50 and others won't go below ISO 200), the shutter
speed at f16 (the smallest aperture you should probably use on an APS-C DSLR to avoid
diffraction blurring) is 1/100s.
If you want a "flowing water" effect when photographing a waterfall, you
won't get it at 1/100s. With a 0.9D (3-stop) ND filter you can reduce the shutter speed to
1/12s, which will start to give you that "silky" look for the water. At the
other end of the range, if you wanted to shoot at f1.4 in bright sunlight to minimize your
depth of field, even at ISO 100 you'd have to use a shutter speed of 1/12500s, which is
faster than most DSLRs allow. By adding an ND filter you could drop that to 1/4000s or
1/2000s which most DLSRs do have.
Note that a polarizer will act as about a 2 stop ND filter, so if you have a polarizer,
get a 3 or 4 stop ND so you have more options.
Graduated Neutral Density Filters
Grad ND filters are used when the dynamic range of a scene exceeds the capabilities of
the camera (film or digital) to record detail in both the brightest and darkest part of
the scene. The go from clear on one side to a neutral .6D (2 stop) or 0.9D (3 stop)
neutral filter on the other. A typical use is to record detail in a bright sky (or
mountains) while simultaneously recording detail in a foreground in shadow. You have to
hope that the dark/light transition is a straight line though, since that's what a grad ND
filter is designed to cope with.
Grad ND filters are available with "soft" and "hard" transitions
between the two halves and they are available in a number of sizes with a variety of
different holders. The filters are typically much larger than the lens diameter to allow
then to be positioned with the transition at any angle and any position in the frame.
Typical sizes might be 84mm x 120mm (84mm is the width of a standard Cokin filter holder).
You can also get round grad ND filters which screw into a lens like a normal filter,
but unless you always want your dark/light transition running across the exact center of
the frame, they may not be ideal.
The above shows the effect of an ND grad. On the left is the "straight" shot.
On the right is the effect of an ND grad with the transition across the horizon and the ND
part over the upper half of the image.
Note that you can duplicate the effects of ND filters by making multiple exposures and
combining them in software. However that means taking multiple images and if you have a
moving subject that could lead to problems. For landscape work though (which is where an
ND grad is most often used), the digital stitching of multiple images taken with different
exposures may be just as effective (even maybe more effective), though it probably
requires more effort than just taking the shot using a filter.
Just about all digital cameras use a filter in front of the sensor which allows visible
light to pass but blocks almost all (but not quite all) of the infrared. This is needed
because typical sensors are very sensitive to infrared and if it was allowed to reach the
sensor it could mess up both exposure and color balance. Since you can't see in IR, you
don't need (or want) IR response for general purpose photography.
However if you want the special IR effects (black sky and white foliage) you can use a
filter such as the Hoya R72, which blocks all of the visible light while allowing IR to
pass though it. Though the camera's IR blocking filter will remove most of this IR light,
some will get through, and since the visible light is now blocked, a long exposure will
record an image in IR light. In most cases the image will need a long exposure at wide
aperture and high ISO settings and so the image will be noisy. There are companies who
will remove the IR blocking filters from the camera to enable much better IR response, but
this isn't cheap (typically around $500) and will void any warranty on the camera as well
as making it tricky to use for normal photography.
Special Effect Filters
The effects of most of these can actually be duplicated digitally. You can do very good
soft filter effects for example. Maybe there are a few effects like "starbursts"
or multiple images that might be easier to do with a filter than in software, but
personally, if I really wanted such effects, I'd probably go the software route since the
effects are more controllable.
Why filters (and other optical elements) need to be anti-reflection coated
Below is a diagram of what happens when a ray of light passes through a filter:
Light comes in as a ray  and hits the filter surface at point [a]. Most of the light
is refracted to point [b], but a small amount is reflected as ray . This is light
reflected from the filter surface and it does no harm except to make the ray passing
though the lens slightly dimmer.
At point [b] a small amount of the light is reflected back to point [c] but most is
refracted and emerges from the filter as ray . This is the main light ray, the one that
we want and it's where most of the light goes.
Most of the light reflected back to point [c] is refracted and emerges from the front
of the filter as ray  and goes off into space, again doing no harm, but a small
amount is reflected back to point [d].
So far, no problems, but it's here where we see the first hint of trouble. Some of the
light reflected from [c] to [d] gets reflected back again, but most of it emerges from the
filter along ray  and heads off towards the camera where it isn't wanted.
The process repeats (see the green rays) and so you get a number in increasingly faint
rays heading back into the camera. All we want is ray , the other rays just cause
problems. The same thing happens at every glass/air interface in the lens, so you can see
that by the time the light has passed though all the elements in the lens (usually between
about 6 and 20), there are all sorts of unwanted rays bouncing around inside the lens.
With digital sensors, there's also light which refects off the sensor and may eventually
end up coming out of the front of the lens, as well as bouncing around and ending up
headed back to the sensor again!
By using anti-reflection coatings on all air/glass surfaces the intensity of all these
extra rays can be greatly diminished. The main effect of these extra rays is to reduce
image contrast by causing flare. There are so many reflections that the light essentially
gets "scrambled" and so doesn't form an image. It's very low level light, but it
can be enough to noticeably drop contrast.
With no anti reflection coating at all, about 4% of the light gets reflected at each
glass/air interface. This means that ray  has about 92% of the original intensity of
ray  and that ray  has about 0.15% of the intensity of ray . Not much, but enough
to cause problems. If the lens elements were not coated, you'd lose 1/2 the light after
passing though 8 elements!
For a single layer antireflection coating, reflection can be reduced from 4% to 2.5%.
Multilayer coatings reduce reflection even more. A typical multicoated surface might only
have 1.5% reflection and the best coatings, like the 12 layer coatings on the Hoya Pro 1
series filters, lower reflection losses to around 0.15% per surface.
So assuming you want a polarizer, UV or ND filter, which one should you buy? Actually
it's more of which one shouldn't you buy. You shouldn't buy an uncoated filter or a
monocoated filter if you can avoid it for the reasons described above and you shouldn't
buy the cheapest filter you can find since it's important that the glass be flat and the
sides be parallel - and that's not something you're likely to find in the cheapest
filters. I've been happy with Tiffen filters, though they are sometimes hard to find in
multicoated versions.. I've also been happy with Hoya filters. Hoya is probably the major
manufacturer of multicoated and supermulticoated filters at reasonable prices. Their
filters are very good and their Super Multicoated Pro 1 series have the lowest
reflectivity coatings of just about any filter on the market. B+W and Heliopan also make
good filters, though they tend to be more expensive then equivalent Hoya filters.
Glass ND grads are available in a number of sizes and from a number of suppliers,
for example Singh-Ray (84mm x 120mm) and Tiffen (3"x3" and 3"x 4")
Hi-Tech and Cokin make "optical plastic" filters which are considerably
cheaper, but which are more susceptible to scratching. The Cokin filters are
"graduated grey" and may not be fully neutral. The square or rectangular ND
grads attach to lenses using a custom adapter (the Cokin adapters are inexpensive and
IR filters are available which transmit different wavelengths of IR. Probably the most
useful for DSLR work is the Hoya R72 which blocks just about all visible light but
transmits near IR light above 720nm which will (with some attenuation) pass through the
camera's IR blocking filter. There are stronger IR filters like the 87C, but these work
better with IR film than DSLRs since they only transmit longer wavelength IR which is
totally blocked by the digital camera's built in IR blocking filter.