The numerical aperture with respect to a point
P depends
on the half-angle
θ of the maximum cone of light that can
enter or exit the lens.
In
optics, the
numerical
aperture (
NA) of an optical system is a
dimensionless number that
characterizes the range of angles over which the system can accept
or emit light. The exact definition of the term varies slightly
between different areas of optics.
General optics
In most areas of optics, and especially in
microscopy, the numerical aperture of an optical
system such as an
objective lens is
defined by
- \mathrm{NA} = n \sin \theta\;
where
n is the
index of
refraction of the medium in which the lens is working (1.0 for
air, 1.33 for pure
water,
and up to 1.56 for
oils), and
θ is the
half-angle of the maximum cone of light that can enter or exit the
lens. In general, this is the angle of the real
marginal ray in the system. The
angular aperture of the lens is
approximately twice this value (within the
paraxial approximation). The NA is
generally measured with respect to a particular object or image
point and will vary as that point is moved.
In microscopy, NA is important because it indicates the
resolving power of a lens. The size of the
finest detail that can be resolved is proportional to λ/NA, where λ
is the
wavelength of the light. A lens
with a larger numerical aperture will be able to visualize finer
details than a lens with a smaller numerical aperture. Lenses with
larger numerical apertures also collect more light and will
generally provide a brighter image.
Numerical aperture is used to define the "pit size" in
optical disc formats.
Numerical aperture versus f-number
Numerical aperture is not typically used in
photography. Instead, the angular aperture of a
lens (or an imaging mirror) is
expressed by the
f-number, written or N,
which is defined as the ratio of the
focal
length to the diameter of the
entrance pupil:
- \ N = f/D
This ratio is related to the image-space numerical aperture when
the lens is focused at infinity. Based on the diagram at right, the
image-space numerical aperture of the lens is:
- \mathrm{NA_i} = n \sin \theta = n \sin \arctan \frac{D}{2f}
\approx n \frac {D}{2f}
- thus N \approx \frac{1}{2\;\mathrm{NA_i}}, assuming normal use
in air (n=1).
The approximation holds when the numerical aperture is small, and
it is nearly exact even at large numerical apertures for
well-corrected camera lenses. For numerical apertures less than
about 0.5 (f-numbers greater than about 1) the divergence between
the approximation and the full expression is less than 10%. Beyond
this, the approximation breaks down. As Rudolf Kingslake explains,
"It is a common error to suppose that the ratio [D/2f ] is
actually equal to \tan \theta, and not \sin \theta ... The tangent
would, of course, be correct if the principal planes were really
plane. However, the complete theory of the
Abbe sine condition shows that if a lens
is corrected for coma and spherical aberration, as all good
photographic objectives must be, the second principal plane becomes
a portion of a sphere of radius
f centered about the focal
point, ..." In this sense, the traditional thin-lens definition and
illustration of f-number is misleading, and defining it in terms of
numerical aperture may be more meaningful.
"Working" or "effective" f-number
The f-number describes the light-gathering ability of the lens in
the case where the marginal rays on the object side are parallel to
the axis of the lens. This case is commonly encountered in
photography, where objects being photographed are often far from
the camera. When the object is not distant from the lens, however,
the image is no longer formed in the lens's
focal plane, and the f-number no longer
accurately describes the light-gathering ability of the lens or the
image-side numerical aperture. In this case, the numerical aperture
is related to what is sometimes called the "
working f-number" or "effective f-number."
The working f-number is defined by modifying the relation above,
taking into account the magnification from object to image:
- \frac{1}{2 \mathrm{NA_i}} = N_\mathrm{w} = (1-m)\, N,
where N_\mathrm{w} is the working f-number, m is the lens's
magnification for an object a
particular distance away, and the NA is defined in terms of the
angle of the marginal ray as before. The magnification here is
typically negative; in photography, the factor is sometimes written
as 1 +
m, where
m represents the
absolute value of the magnification; in
either case, the correction factor is 1 or greater.
The two equalities in the equation above are each taken by various
authors as the definition of working f-number, as the cited sources
illustrate. They are not necessarily both exact, but are often
treated as if they are. The actual situation is more complicated —
as Allen R. Greenleaf explains, "Illuminance varies inversely as
the square of the distance between the exit pupil of the lens and
the position of the plate or film. Because the position of the exit
pupil usually is unknown to the user of a lens, the rear conjugate
focal distance is used instead; the resultant theoretical error so
introduced is insignificant with most types of photographic
lenses."
Conversely, the object-side numerical aperture is related to the
f-number by way of the magnification (tending to zero for a distant
object):
- \frac{1}{2 \mathrm{NA_o}} = \frac{m-1}{m}\, N.
Laser physics
In
laser physics, the numerical
aperture is defined slightly differently. Laser beams spread out as
they propagate, but slowly. Far away from the narrowest part of the
beam, the spread is roughly linear with distance—the laser beam
forms a cone of light in the "far field". The same relation gives
the NA,
- \mathrm{NA} = n \sin \theta,\;
but
θ is defined differently. Laser beams typically do not
have sharp edges like the cone of light that passes through the
aperture of a lens does. Instead, the
irradiance falls off gradually away from
the center of the beam. It is very common for the beam to have a
Gaussian profile. Laser physicists
typically choose to make
θ the
divergence of the
beam: the
far-field angle between the
propagation direction and the distance from the beam axis for which
the irradiance drops to 1/e
^{2} times the wavefront total
irradiance. The NA of a Gaussian laser beam is then related to its
minimum spot size by
- \mathrm{NA}\simeq \frac{2 \lambda_0}{\pi D},
where λ
_{0} is the
vacuum
wavelength of the light, and
D is the diameter of the
beam at its narrowest spot, measured between the 1/e
^{2}
irradiance points ("Full width at e
^{−2} maximum"). Note
that this means that a laser beam that is focused to a small spot
will spread out quickly as it moves away from the focus, while a
large-diameter laser beam can stay roughly the same size over a
very long distance.
Fiber optics
Multimode optical fiber will
only propagate light that enters the fiber within a certain cone,
known as the
acceptance cone of the
fiber. The half-angle of this cone is called the
acceptance angle,
θ_{max}.
For
step-index multimode fiber,
the acceptance angle is determined only by the indices of
refraction:
- n \sin \theta_\max = \sqrt{n_1^2 - n_2^2},
where
n_{1} is the refractive index of the fiber
core, and
n_{2} is the refractive index of the
cladding.
When a light ray is incident from a medium of
refractive index n to the core of index
n_{1},
Snell's law at
medium-core interface gives
- n\sin\theta_i = n_1\sin\theta_r.\
From the above figure and using trigonometry, we get :
- \sin\theta_{r} = \sin\left({90^\circ} - \theta_{c} \right) =
\cos\theta_{c}\
where \theta_{c} = \sin^{-1} \frac{n_{2}}{n_{1}}is the
critical angle for
total internal reflection,
since
Substituting for sin θ
_{r} in Snell's law we
get:
- \frac{n}{n_{1}}\sin\theta_{i} = \cos\theta_{c}.
By squaring both sides
- \frac{n^{2}}{n_{1}^{2}}\sin^{2}\theta_{i} = \cos ^{2}\theta_{c}
= 1 - \sin^{2}\theta_{c} = 1 - \frac{n_{2}^{2}}{n_{1}^{2}}.
Thus,
- n \sin \theta_{i} = \sqrt{n_1^2 - n_2^2},
from where the formula given above follows.
This has the same form as the numerical aperture in other optical
systems, so it has become common to
define the NA of any
type of fiber to be
- \mathrm{NA} = \sqrt{n_1^2 - n_2^2},
where
n_{1} is the refractive index along the
central axis of the fiber. Note that when this definition is used,
the connection between the NA and the acceptance angle of the fiber
becomes only an approximation. In particular, manufacturers often
quote "NA" for
single-mode fiber
based on this formula, even though the acceptance angle for
single-mode fiber is quite different and cannot be determined from
the indices of refraction alone.
The number of bound
modes, the
mode volume, is related to the
normalized frequency and thus to the
NA.
In multimode fibers, the term
equilibrium numerical
aperture is sometimes used. This refers to the numerical
aperture with respect to the extreme exit angle of a
ray emerging from a fiber in which
equilibrium mode
distribution has been established.
See also
References
External links