1. A Review of the Resolving Power of Optical Instruments
Luke Charbonneau
Department of Physics, University of Colorado at Boulder, 2000 Colorado Avenue, Boulder, Colorado 80309-0390, USA
luke.charbonneau@colorado.edu
1. Introduction
Optical instruments have allowed scientists from a multitude of fields to make seminal discoveries about the
fundamental properties of nature. Prior to the advent of telescopes, microscopes and cameras, great scientists argued
incessantly over pivotal theories which these image-enhancing instruments helped prove to be incontrovertible truth:
the heliocentric solar system, the “building-blocks” of life – cells and the explanation of solar eclipses, to name a
few. Modern-day optical instruments now allow scientists to not only enhance the images of natural phenomenon,
but to also closely analyze the fundamental properties of light along a wide swath of the electromagnetic spectrum.
This unprecedented ability to profile the interaction of light with samples allows scientists today to probe even
deeper into the unsolved mysteries of nature. However, the capability of optical instruments to yield meaningful
information to their operators is ultimately limited by the physical properties of electromagnetic waves. A detailed
review of these limitations imposed by the nature of light on optical instruments is discussed here.
2. Historical Context
The very first archeological evidence of manmade optics yet discovered is a crude, convex lens known as the
“Nimrud Lens”, which was ground and polished from quartz in modern-day Iraq during the Neo-Assyrian period (8th
century BCE) [1]. Although little is known about its usage, it potentially represents the first known attempt to create
an optical device capable of processing light waves for viewing. Around the turn of the 3rd
century BCE, Euclid of
Alexandria developed some of the first ideas regarding geometrical optics, including the assertion that light rays
travel in straight lines and diverge from the eye to infinity [2]. Several hundred years later, Claudius Ptolemy built
on these ideas and wrote about the propagation of light between different media – thereby describing the first, albeit
ultimately incorrect, diffraction theory. It was not until the 9th
century CE that Iraqi philosopher Abu Yusuf Ya’qub
ibn Ishaq al-Kindi developed a refraction theory which was mathematically equivalent to Snell’s law [2].
In 1690, Dutch physicist Christiaan Huygens published his seminal work “Traitė de la Lumiere” (Treatise
on Light), wherein he postulates that every point in space which experiences a luminous disturbance becomes the
source of a spherical wave [3]. Notably, the first functioning telescopes (later improved upon by Johannes Kepler
and Sir Isaac Newton) and microscopes were invented about one hundred years earlier by Dutch lens makers. Over a
century later, French physicist Augustin-Jean Fresnel used Huygens’ description in combination with his theory of
interference to create the famous Huygens-Fresnel principle - which explains diffraction phenomena by assuming
that light is a wave. Later, during the 19th
century, German physicist Gustav Kirchhoff used Green’s theorem in
conjunction with the Huygens-Fresnel principle to develop a more generalized theory for the propagation of light as
a wave. However, it was not until the early 20th
century that Albert Einstein postulated the existence of discrete
quanta of light energy in particle form, known as photons, to elucidate the physics behind the photoelectric effect.

2. 3. Diffraction Theory
3.1 Huygens-Fresnel Principle
The Huygens-Fresnel principle is an approximation of the more rigorous Kirchhoff diffraction formula, but because
of the complicated mathematics involved with the latter, the Huygens-Fresnel principle remains the most useful for
the description of the majority of optical instrument phenomena [4]. The principle describes every point along a
light wave front to be the center of a secondary disturbance, which bring about spherical “wavelets”, where the wave
front at any later time can be described as the envelope of these wavelets [4]. This principle is illustrated below
using the “Fresnel zone construction”:
where S is the instantaneous position of a spherical, single-wavelength (monochromatic) wave front of radius r0,
originating from a point source at P0 and where P is the evaluation point. With the periodic time factor removed, the
wave-front at a point Q is represented by Aeikr0
/r0, where A is an amplitude.
The Huygens-Fresnel principle asserts that the amplitude of the secondary waves is maximized in the
original direction of propagation and quickly decreases as the direction of the wave front becomes perpendicular to
the original direction of propagation. Therefore, the luminous disturbance at point P is described by equation (1) [4]:
𝑈(𝑃) =
𝐴𝑒 𝑖𝑘𝑟0
𝑟0
∬
𝑒 𝑖𝑘𝑠
𝑠
𝐾(𝜒)𝑑𝑆
𝑆′
(1)
where s = QP, 𝐾(𝜒) describes the variation of the amplitude of the secondary waves with respect to direction
(known as the “inclination factor”) and χ is the angle of diffraction. The Huygens-Fresnel description of light
propagation in free space agreed with experiment and was well-received by the contemporary scientific community,
which ultimately enabled the wave theory of light to become widely accepted over the corpuscular theory of light
[4].
3.2 Kirchhoff Diffraction
Gustav Kirchhoff later proved (by invoking Green’s theorem and the Helmholtz equation for a monochromatic wave
in a vacuum) that the above Huygens-Fresnel principle was an approximation of an integral theorem known as the
“integral theorem of Helmholtz and Kirchhoff”, equation (2) [4]:
𝑈(𝑃) =
1
4𝜋
∬ [𝑈
𝜕
𝜕𝑛
(
𝑒 𝑖𝑘𝑠
𝑠
) −
𝑒 𝑖𝑘𝑠
𝑠
𝜕𝑈
𝜕𝑛
] 𝑑𝑆 (2)
𝑆
where S is a closed surface bounding a volume, n is the inward normal to S, U is light wave’s amplitude with respect
to its position, and s is the distance from point P to the instantaneous position of the wave.
Kirchhoff’s integral theorem (2) is relatively complex, but fortunately can be reduced to a significantly
simpler form with a few approximations. To derive this formulation, first consider an opaque screen between point
Figure 1 [4]: The “Fresnel zone construction”.

3. P0 and P, with a slit which is large compared to the wavelength of
the monochromatic wave, but small compared to 𝑃0 𝑃⃗⃗⃗⃗⃗⃗⃗ . Let A be the
surface of the slit, B the portion of the surface on the unilluminated
side of the screen and C a portion of a large sphere with radius R,
centered around P. Together, A, B and C form a closed surface
(illustrated in figure 2) over which the Kirchhoff integral (2) may
be taken [4]:
𝑈(𝑃) =
1
4𝜋
[∬
𝐴
+ ∬
𝐵
+ ∬
𝐶
] [𝑈
𝜕
𝜕𝑛
(
𝑒 𝑖𝑘𝑠
𝑠
) −
𝑒 𝑖𝑘𝑠
𝑠
𝜕𝑈
𝜕𝑛
] 𝑑𝑆 (3)
Now it can be approximated that everywhere on A, except close to
the edges of the slit, U and
𝜕𝑈
𝜕𝑛
will not vary much from their values
when the screen is absent. Furthermore, it is reasonable to assume
that these same quantities will be approximately zero on B. These
assumptions led Kirchhoff to define the following boundary
conditions [4]:
𝑂𝑛 𝐴: 𝑈 = 𝑈 𝑖
,
𝜕𝑈
𝜕𝑛
=
𝜕𝑈 𝑖
𝜕𝑛
𝑎𝑛𝑑 𝑂𝑛 𝐵: 𝑈 = 0,
𝜕𝑈
𝜕𝑛
= 0 (4)
where: 𝑈(𝑖)
=
𝐴𝑒 𝑖𝑘𝑟
𝑟
,
𝜕𝑈(𝑖)
𝜕𝑛
=
𝐴𝑒 𝑖𝑘𝑟
𝑟
[𝑖𝑘 −
1
𝑟
]cos(𝑛, 𝑟) (5)
Equations (4) and (5) are known as “Kirchhoff’s boundary
conditions”. If it is further assumed that a portion W of an incoming
wave front fills the aperture together with a portion of C which is a
cone with a vertex at P0, neglecting the in the normal derivatives the
1/r and 1/s terms, setting cos(n, r0) = 1 on W, setting χ = π – (r0, s)
and making the radius of curvature of the incoming wave as
sufficiently large, then equation (6) is obtained [4]:
𝑈(𝑃) = −
𝑖
2𝜆
𝐴𝑒 𝑖𝑘𝑟0
𝑟0
∬
𝑒 𝑖𝑘𝑠
𝑠
(1 + cos[𝜒])
𝑊′
𝑑𝑆 (6)
which provides an explicit formula for the inclination factor in
equation (1) [4]:
𝐾(𝜒) = −
𝑖
2𝜆
(1 + cos[𝜒]) (7)
which agrees with Fresnel’s assumption that the inclination factor’s magnitude is maximized at χ = 0, but shows that
Fresnel’s assumption that K(𝜒 =
𝜋
2
) = 0 is incorrect.
3.3 Fraunhofer and Fresnel Diffraction
The most general form of the “Fresnel-Kirchhoff diffraction integral” (equation 6) is [4]:
𝑈(𝑃) = −
𝐴𝑖
2𝜆
∬
𝑒 𝑖𝑘(𝑟+𝑠)
𝑟𝑠
(cos(𝑛, 𝑟) − cos(𝑛, 𝑠))
𝐴
𝑑𝑆 (8)
where A is the aperture and the parameters r and s were defined in figure 2.
If the distances of P0 and P from the diffracting screen are much
larger than the width of the aperture, A, then the following useful
approximation of (8) may be made [4]:
𝑈(𝑃) ≈ −
𝐴𝑖
𝜆
𝑐𝑜𝑠𝛿
𝑟′ 𝑠′
∬ 𝑒 𝑖𝑘(𝑟+𝑠)
𝐴
𝑑𝑆 (9)
where δ is the angle between the line from 𝑃0 𝑃̅̅̅̅̅ and the normal to
the screen, and r’ and s’ are the distances of P0 and P from the
origin, respectively. This approximation can be made due to the
Figure 2 [4]: Illustration for the derivation of the
Fresnel-Kirchhoff diffraction formula.
Figure 3 [4]: Illustration for equation (6).
Figure 4 [4]: Illustration for Fraunhofer and Fresnel diffraction.

4. fact that the term (cos(n, r) – cos(n, s)) will not vary by much over the aperture. If (ε, µ) are the coordinates of a
point Q in the aperture (figure 4), P0 = (x0, y0, z0) and P = (x, y, z), then using elementary geometry, expressions for
r, r’
, s and s’
can be written as [4]:
𝑟2
= 𝑟′2
− 2(𝑥0 𝜀 + 𝑦0 𝜇) + 𝜀2
+ 𝜇2
𝑎𝑛𝑑 𝑠2
= 𝑠′2
− 2(𝑥𝜀 + 𝑦𝜇) + 𝜀2
+ 𝜇2
(10)
which can be expanded as a power series in ε/r’, μ/r’, ε/s’ and μ/s’ due to the approximation that the dimension of
the aperture is small when compared to r’ and s’. Therefore, using (9), U(P) can be written as [4]:
𝑈(𝑃) = −
𝑖𝑐𝑜𝑠𝛿
𝜆
𝐴𝑒 𝑖𝑘(𝑟′+𝑠′)
𝑟′𝑠′
∬ 𝑒 𝑖𝑘𝑓(𝜀,𝜇)
𝑑𝜀𝑑𝜇 (11)
𝐴
where: 𝑓(𝜀, 𝜇) = −
𝑥0 𝜀+𝑦0 𝜇
𝑟′ −
𝑥𝜀+𝑦𝜇
𝑠′ +
𝜀2+𝜇2
2𝑟′ +
𝜀2+𝜇2
2𝑠′ −
(𝑥0 𝜀+𝑦0 𝜇)2
2𝑟′3 −
(𝑥𝜀+𝑦𝜇)2
2𝑠′3
If the quadratic and higher order terms in 𝑓(𝜀, 𝜇) can be neglected, i.e.: in the far field, then the above
expression is known as “Fraunhofer diffraction”. In the more complex case, where the higher order terms are not
negligible, the expression is known as “Fresnel diffraction”. Another way of thinking about the difference between
the two cases is as follows: for Fraunhofer diffraction, the incoming wave will be from a distant source, so the wave
front at the aperture will essentially be a plane wave. But, for Fresnel diffraction, the radiation incident on the
aperture will be from a nearby source, so it will still have significant curvature. In the former case, the phase will
vary linearly across the aperture [5] – simplifying the mathematics by making the quadratic and higher order terms
negligible. Fortunately, for many common situations, including for the focusing of light with a well-corrected lens
[4], the more simple Fraunhofer diffraction is appropriate and can often yield an analytic solution. Two common
examples of Fraunhofer diffraction are explained in the successive sections.
3.3.1 Rectangular Aperture
For a rectangular aperture of dimensions (a, b) in the y-z plane with
Fraunhofer diffraction conditions (far-field), the following is the
expression for the diffracted electric field incoming along the x-axis [5]:
𝐸 =
𝐶
𝑅
(∫ 𝑒
𝑖𝑘𝑦𝑌
𝑅 𝑑𝑦
+
𝑏
2
−
𝑏
2
) (∫ 𝑒
𝑖𝑘𝑧𝑍
𝑅 𝑑𝑧
+
𝑎
2
−
𝑎
2
) (12)
where C is a constant related to the source strength, Y is the y-position
in the far field, Z is the z-position in the far field and R is the distance
from the center of the aperture to the observation point (situation
illustrated in figure 5). This relation then leads directly to the expression
for the intensity at the observation point in the far field [5]:
𝐼(𝑌, 𝑍) = 𝐼𝑜(
sin(
1
2𝑅
𝑘𝑏𝑌)
1
2𝑅
𝑘𝑏𝑌
)2
(
sin(
1
2𝑅
𝑘𝑎𝑍)
1
2𝑅
𝑘𝑎𝑍
)2
(13)
where I0 is the initial intensity of the incoming wave.
3.3.2 Circular Aperture
For Fraunhofer diffraction of a circular aperture of diameter D, the
intensity is derived via integration of Bessel functions; the result for
the irradiance pattern of such a situation is reproduced below [6]:
𝐼 = 𝐼0 |
2𝐽1[(
𝜋𝐷
𝜆
) sin(𝜃)]
(
𝜋𝐷
𝜆
) sin(𝜃)
|
2
(14)
where J1 is a Bessel function of the first kind and θ the angular
radius from the pattern’s maximum. The diffraction pattern for a circular
Figure 5 [5]: Rectangular aperture, Fraunhofer diffraction.
Figure 6 [7]: Airy pattern from two perspectives.

5. aperture is known as an “Airy pattern” (figure 6), which will play an important role in section 4.
4. Diffraction in Optical Devices
The ability of an optical device to resolve close together parts of an image is called the “resolving power” [8]. The
resolving power is limited by a number of factors, including the quality of the device, how well the device is aligned
and the environment that the device is being used in. However, ultimately, the resolving power of all optical devices
is limited by diffraction. Even with an optical device which is perfect in every way, including ideal alignment and
the absence of aberrations (which usually necessitates an increased cost), it can still only reach a maximum
resolving power which is known as the “diffraction limit”. For the purposes of this review, all optical devices
discussed are assumed to be limited by diffraction only, that is, have no aberrations, perfect alignment and flawless
construction.
The diffraction limit is determined by a measure known as the “Rayleigh criterion”. The Rayleigh criterion
states that it is only possible to resolve two separate light sources with any optical device if their Airy pattern peaks
are no closer together than the radius of their Airy disks, which is defined as the distance from the peak of the Airy
pattern (see section 3.3.2 and figure 6) to the first minimum. The Airy pattern occurs for almost all optical devices
such as telescopes, microscopes, lenses and cameras, because almost all of these devices have circular apertures
which cause the Fraunhofer diffraction described in the previous section. An illustration of the Rayleigh criterion is
shown below in figure 7:
The leftmost figure in figure 7 clearly has two resolvable light sources, as their Airy disks do not overlap at all. This
indicates that the optical device is not yet diffraction limited. In the center figure, the optical device is now near the
diffraction limit as the two light sources overlap by slightly less than their Airy disk radii – indicating that the two
sources can still be resolved. In the rightmost figure, the two light sources now overlap by more than their Airy disk
radii, which indicates that the two sources can no longer be resolved and the optical device is now diffraction
limited.
4.1 Image-Forming Systems
Perhaps the most ubiquitous optical devices are those which are intended to form images for easier viewing by
utilizing a lens or system of lenses. This class of optical devices includes telescopes, microscopes and cameras.
Although there are now many different variations of these image-forming devices in the modern era, the ability of
these devices to resolve images is restrained by the fundamental limit placed on them by diffraction.
Figure 7 [9]: An illustration of the Rayleigh criterion.

6. 4.1.1 Telescopes
The purpose of any telescope, regardless of size, design or location, is to collect light from far away sources and
form an image which can be viewed by the human eye (or analyzed by a computer). If a telescope with a circular
aperture of radius a and diameter D is considered, then equation (14) may be modified to determine the angular
resolution of the telescope [10]:
𝐼 = 𝐼0 |
2𝐽1[(
𝜋𝐷
𝜆
) sin(𝜃)]
(
𝜋𝐷
𝜆
) sin(𝜃)
|
2
= 𝐼0 |
2𝐽1[(
2𝜋𝑎
𝜆
) sin(𝜃)]
(
2𝜋𝑎
𝜆
) sin(𝜃)
|
2
≈ 𝐼0 |
2𝐽1[𝑘𝑎𝜃]
𝑘𝑎𝜃
|
2
(15)
where θ is the angle between the axis from the center of the aperture to the imaging point, which is assumed to be
small by invoking Fraunhofer diffraction, therefore the small angle approximation can be made: sin(θ) ≈ θ.
Thus, the minimums of I occur when J1[kaθ] is minimized, which occurs when kaθ ≈ 3.83, 7.02 [10]. Therefore the
minimums occur when:
𝜃 =
0.61𝜆
𝑎
,
1.12𝜆
𝑎
, … (16)
which implies that the limit of the angular resolution of a telescope is:
Δ𝜃 =
0.61𝜆
𝑎
=
1.22𝜆
𝐷
(17)
due to the definition of the Rayleigh criterion – that the central maximum of one sources’ image is at the first
minimum of the second sources’ image [10]. Furthermore, the radius of the central disk is [10]:
r′ = 𝑓Δ𝜃 =
1.22𝜆𝑓
𝐷
(18)
where f is the focal length of the telescope’s objective (light-collecting lens). This shows that increasing the
diameter of the telescope’s objective yields smaller Airy disks in the image plane – allowing for a smaller angular
resolution limit, and therefore, a higher resolving power. This is the reason why the most powerful telescopes have
very large diameters.
4.1.2 Microscopes
A conventional microscope serves the purpose of collecting
light from a nearby, small object in order to provide a
magnified image of the object to the user. The most common
type of (visible light) microscope in use today is the
compound microscope. A compound microscope (figure 8)
works by collecting light diffracted off of a sample in an
objective lens, which has a small focal length and aperture
[11]. The object lens forms a real, inverted image, which is
then magnified for viewing by a lens referred to as the “eye-
piece”. The final image formed by the eyepiece is inverted and
imaginary. In 1873, Ernst Abbe found the diffraction limit of a
microscope to be [12]:
𝑑 =
𝜆
2𝑛𝑠𝑖𝑛𝜃
≈
𝜆
2
(19)
where n is the refractive index that the light from the sample
travels through to the objective lens and d is the minimum radius
that the light can be converged to (note: nsinθ is often referred to
as the “numerical aperture” (NA)). Abbe’s equation indicates
that using light with a smaller wavelength to illuminate the sample allows for a higher resolution, which is an
important concept for electron microscopes (discussed in section 5.1). Furthermore, according to (19), increasing the
refractive index of the medium between the sample and the objective lens allows for a higher resolution – which is
often achieved by immersing the sample and objective lens in certain types of oil.
Figure 8 [11]: Simplified diagram of a compound
microscope.

7. 4.1.3 Cameras
The ever-increasing depth of camera technology, including the relatively recent realm of digital photography, makes
it difficult to define a universal limitation of a camera’s resolving power. Therefore, the following discussion will be
restricted to standard film cameras.
Although the field of photography has many subtleties, the determining factor for the resolving power of a
camera is the “f-number”, N, of the lens aperture used, which is defined as:
𝑁 =
𝑓
𝐷
(20)
where D is the diameter of the effective aperture and f is the focal length of the camera lens. Although the ultimate
resolution of a camera depends on a number of other factors, including the size of the camera’s sensor and exposure
time, in general it is true that a smaller f-number (i.e.: a larger aperture diameter) will lead to a smaller Airy radius
and therefore, a higher resolving power - at the expense of the depth of field [13].
4.2 Diffraction Gratings
Although most optical devices have traditionally been used to form images via a system of lenses to be analyzed by
eye, often it is advantageous to instead analyze the spectrum of light from a source (or collection of sources). A
diffraction grating is an optical device which imposes some periodic variation of phase, amplitude, or a combination
of both on an incident light wave. A diffraction grading can be used in tandem with microscopes, telescopes,
cameras and other devices to yield information about the spectral, or frequency-domain, composition of an incident
light wave. This ability of diffraction gratings to elucidate the spectral components of light forms the basis for a vital
collection of research techniques known as optical spectroscopy.
4.2.1 Theory of Diffraction Gratings
To simplify the discussion of the theory of diffraction gratings, a planar transmission grating will be considered
here. For plane-wave light normally incident on this type of diffraction grating, the grating may be considered as a
collection of uniformly separated, single slit apertures, described in the far-field by the mechanics of Fraunhofer
diffraction. If the separation of these slits of width d, is L, then à la equations (13 and 14) multiplied by a term for
the periodic “emitters” (slits), the intensity of the transmitted light is described by [14]:
𝐼 = 𝐼𝑜
𝑠𝑖𝑛2
(𝑁
𝜋𝐿𝑠𝑖𝑛𝜃
𝜆
)
𝑠𝑖𝑛2(
𝜋𝐿𝑠𝑖𝑛𝜃
𝜆
)
𝑠𝑖𝑛𝑐2
(
𝜋𝑑𝑠𝑖𝑛𝜃
𝜆
) (21)
where I0 is the intensity of the incoming wave, N is the number of illuminated, single slit apertures and θ is the angle
of refraction for each slit. This pattern is a result of interference minimums and maximums between the light which
passes through each individual slit. Furthermore, for non-monochromatic light, each spectral component, i.e.
wavelength, of light will be dispersed by the grating at a unique angle of diffraction, as described by [14]:
𝑠𝑖𝑛𝜃 𝑚 = 𝑚
𝜆
𝐿
(22)
where m is an integer that represents each primary maximum of the resulting interference pattern. In this case, each
order, m, corresponds to a different wavelength, allowing for the separation of the spectral components of the
incident wave. The dispersive power of a diffraction grating, D, is then [14]:
𝐷 = (
𝑑𝜃 𝑚
𝑑𝜆
) =
𝑚
𝐿𝑐𝑜𝑠𝜃 𝑚
(23)
Therefore, the dispersive power of a diffraction grating is inversely proportional to the slip spacing, that is, the finer
the grating, the higher its dispersive power.

8. The above shows that dispersive gratings can be
used to determine the spectral composition of an incident
light wave. To define the chromatic resolving power of a
give diffraction grating, the Rayleigh criterion can be used
(figure 9). The minimum for λ in figure 9 is [14]:
Δ𝜃 =
𝜆
𝑁𝐿𝑐𝑜𝑠𝜃 𝑚
(24)
and likewise, the minimum for λ + Δ λ is [14]:
Δ𝜃 =
mΔ𝜆
𝐿𝑐𝑜𝑠𝜃 𝑚
(25)
Equating (24) and (25) allows the definition of the
chromatic resolving power, R, as [14]:
R =
𝜆
Δ𝜆
= 𝑁𝑚 (26)
which shows that the chromatic resolving power of a diffraction grating is greater for a larger number of slits or
grooves (for a fixed illuminated grating length). This property allows diffraction gratings to be instrumental in
optical spectrometers. (Note: Δλ in (26) is the resolution (i.e. smallest measurable wavelength difference) of the
grating, whereas R is the resolving power.)
4.4 Comparison of Lens-based and Diffraction Grating-based Optical Instruments
In general, when one wants to create an image viewable by the human eye from a small or distant light source, then
a conventional instrument, such as a compound microscope or refracting telescope, respectively, is most appropriate.
However, if one wants to instead analyze the spectral components of the same sources, a diffraction grating
combined with a spectrograph is more appropriate. In terms of resolution, both approaches perform better when used
with smaller wavelengths of light (see equations 18, 19 and 26). However, the resolution of the diffraction grating,
at least when considering only geometric optics, appears to be limited only by the ability to manufacture gratings
which are closely ruled, i.e. have many grooves per mm (see equation 26). However, when considering real light,
the natural divergence of a beam and the distribution of energy throughout the spectrum also limits the achievable
resolution of a diffraction grating [15].
In practice, diffraction gratings are almost always used in tandem with lenses to create devices such as
optical spectrometers. Basic optical spectrometers essentially focus light from one section of a reflected wave’s
chromatically-separated spectrum, which is beforehand scattered off of an object of interest, to be measured. The
resulting shift in frequency, phase and amplitude can lead the user to determine what the light was scattered off of.
Therefore, for most optical instruments which utilize diffraction gratings, lenses are also integral to the design and
the limitations of both classes of devices must be considered. Two notable exceptions are holograms and compact
disks (CDs), which involve only diffraction gratings and no lenses.
5. Modern Optical Instruments
Many modern optical instruments (particularly those designed for fundamental research in the physical sciences) are
designed to mitigate, or even circumvent, the limitations described in the previous section. Two interesting examples
of this class of devices are briefly described in this section.
5.1 Electron Microscopy
Electron microscopes are divided into two main categories: transmission electron microscopes (TEM) and scanning
electron microscopes (SEM) [16]. For TEM, a high-voltage electron beam is focused onto the sample by magnetic
lenses which then scan over the entire area of the sample [16]. The height of the sample at each point in the TEM’s
plane is determined by measuring the partial transmission of a highly energetic (~200 kV) electron beam through to
the bottom of the microscope’s stage. This measurement can be performed via a photographic plate, florescent
screen or a CCD camera. SEM work in a similar manner to TEM, but secondary electrons emitted from the surface
of the sample by excitation from the beam are measured (albeit, with a lesser resolving power) [16]. Therefore,
Figure 9 [14]: Chromatic resolving power of a
diffraction grating for two wavelengths.

9. regardless of the exact type, electron microscopes outperform
their visible light counterparts, resolution-wise, by utilizing the
very short wavelength of an energetic electron beam (~ 3 pm at
200 kV vs. hundreds of nm for visible light). This allows for a
correspondingly much lower diffraction limit and higher
resolving power, as indicated by equation (19) of section 4.1.2.
5.2 Fluorescence Microscopy (STED microscopy)
Fluorescence microscopy uses the properties of fluorophores
(fluorescent chemical compounds) within samples to achieve
resolutions beyond the Abbe diffraction limit for microscopes
(equation 19). An important variant of fluorescence
microscopy, known as “Stimulated Emission Depletion” or
STED, is able to surpass this diffraction limit to yield a
resolution on the order of 100 nm for visible light [17]. STED
uses two synchronized, pulsed lasers (figure 10): one to excite
the fluorophores and the other to keep the electron from
returning to its normal ground state – causing a red-shift in the
emitted fluorescence. This red-shifted fluorescence is then
isolated and focused into a detector.
STED is able to achieve a sub-Abbe diffraction limit
resolution due to the non-linear relationship between the
intensity of the STED (766 nm light in figure 10) beam and the
residual population of the fluorescent state [17]. This leads to a
modified version of the microscope diffraction limit (equation
19) [18]:
𝑑 ≈
𝜆
2𝑁𝐴√1 + 𝐼/𝐼𝑠𝑎𝑡
(27)
where Isat is the saturation beam intensity.
Summary
Diffraction theory places a fundamental limit on the ability of optical instruments (and their constituent optical
components) to resolve the finer details of images. This limit applies, in different ways, to both devices which utilize
lenses and diffraction gratings. Therefore, in order for modern devices to achieve greater resolving power than
conventional devices, researchers must constantly look for ways to push the diffraction limit, or to circumvent the
limit altogether. Regardless of how high-resolution optical instruments will continue to progress, there is little doubt
that these instruments will continue to play a large role in enabling new discoveries in the physical sciences.
Figure 10 [17]: STED experimental setup.

10. References
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microscope.html
[12] Boundless. “Limits of Resolution and Circular Apertures.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved Apr. 2016 from
https://www.boundless.com/physics/textbooks/624/vision-and-optical-instruments-25/other-optical-instruments-173/limits-of-resolution-and-
circular-aperatures-630-6094/
[13] Tutorials: Depth of Field. (n.d.). Retrieved April, 2016, from http://www.cambridgeincolour.com/tutorials/depth-of-field.htm
[14] Diffraction grating. (2009, January 6). Retrieved April, 2016, from
http://www.cts.iitkgp.ernet.in/home/phy1/copy/lect_notes/node60.html#fig:diff.4b
[15] Paschotta, R. (n.d.). Diffraction Gratings. Retrieved April, 2016, from https://www.rp-photonics.com/diffraction_gratings.html
[16] What is Electron Microscopy? (n.d.). Retrieved April, 2016, from https://www.jic.ac.uk/microscopy/intro_EM.html
[17] Klar, T. A., Jakobs, S., Dyba, M., Egner, A., & Hell, S. W. (2000). Fluorescence microscopy with diffraction resolution barrier broken by
stimulated emission. Proceedings of the National Academy of Sciences, 97(15), 8206-8210. doi:10.1073/pnas.97.15.8206
[18] STED. (n.d.). Retrieved April, 2016, from http://www.abberior.com/knowledge/microscopy-tutorials/sted/