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When we want to inject the output signal from a certain optical instrument into a second optical system  via fiber couplers, we need to take some caution in order to avoid that the signal suffers, even heavy, losses.Each optical instrument is characterized by an input entrance and by its "Numerical Aperture" (N.A.). The input entrance can be, for example, the entrance pupil of an objective or the entrance slit of a monochromator or an optical fiber core (onto the transversal section of the fiber).
The numerical aperture (N.A.) is the sine of the halfof the acceptance or emission angle (in the case that the entrance or the exit of the instrument are in air or in vacuum).
The F# is the ratio between the focal length F and the entrance pupil diameter D (or equivalently, the distance between the principal optics axis and the images plane); hence F# = F / D.
It follows:
Hence, if we know only theF#,the N.A. can be simply derived by using the following formula:
Note that atan(F#/2) is the entrance (or exit) semiangle of the instrument.
The N.A. can also be approximated calculated by the formula:
for optical fibers:
Fig.23 N.A.of an optical fiber.
Fig.24– Optical fiber core and cladding.
Fig.25 N.A. of a microscope objective.
Fig.26 N.A. of a photographic objective.
In some cases, like in the optical fibers and in themicroscope objectives, the N.A. is indicated; while in photographic and cinematographic objectives is provided the F# . These two values are a different indications of the same angle. The N.A. is referred to the sine of the semiangle; while the F# refers to the ratio between the focal length and entrance pupil diameter; hence the F# is related to the entire angle.
Note that in the optical fibers the same N.A. isapplied to both the exit and the entrance (unless the case of tapered optical fibers or with integrated lenses).
In order to avoid any signal losses, in addition to the alignment between the two optical instruments, it is necessary to make the characteristics of the first instrument output signal suitablefor the characteristics of the second instrument.
The product of the dimension of the input entrance (considering separately each dimension ) by the acceptance angle (or semiangle) is constant for all the conjugate plans.
EXAMPLE OF COUPLING BETWEEN A LIGHTSOURCE, AN OPTICALFIBER AND A MONOCHROMATOR.
Let us consider a monochromator with F#=4 (i.e. an entrance angle of 7.125°). The source is a surface with a Lambertian emission over a semiangle of 180°. The fiber has a N.A. of 0.5.
An optical system must form the image of the area of the source to be analyzed (or of the entire source) within the diameter of the fiber core. If the source image is larger than the fiber core,the exceeding part is lost. The maximum angle with which the image is formed must not exceed the one indicated by the fiber N.A.(the N.A. is the sine of the acceptance or output semiangle). If the angle exceeds the maximum angle, the exceeding part is lost. If the part of the source of which an image is wanted, is larger than the fiber core, the image must be reduced and the angle, under whichis collected the energy of the source, must be equal to thefiber acceptance angle reduced by the magnification ratio (if the source image is smaller than the source, then theangleunder which the energy is collected is smaller see Fig.26). The fiber output has the same diameter of the core and the maximum angle defined by the N.A. (whatever is the entrance angle).
If we want to inject the fiber output into the monochromator, we can proceed with the same reasoning: the image must be at maximum equal to the slits thickness and the semiangle must be the maximum one accepted by the monochromator, which is defined by its F#:
The radiation exceeding the maximum angle (even for a misalignment effect), goes out from theoptics of the monochromator: it does not contribute to the signal and can increase the noise.
Usually, the image on the entrance slit has a small diameter, as well the monochromator output. When the image is enlarged in order to obtain longer rows as output, part of the energy is lost. If we want to elongate the image by means of cylindrical (or toric) optics, the risk is to exceed the maximum angle in a direction perpendicular to the slit, with consequent lost of energy.
Fig.27– Coupling Scheme.
In some cases it is possible to use optical fibers bundles, that when they are ordered at the output along, are able to reduce the losses (see Fig.28).
Fig.28
Alternatively the fiber can be directly coupled to the source without any other intermediate optics. Naturally the distance fibersource must be enough to allow the fiber to collect the maximum possible energy (Fig.29).
Fig.29
EXAMPLE OF COUPLING BETWEEN TWO OBJECTIVE IN “PARALLEL LIGHT”
When we want to transport an image, for example, from a screen to a detector is possible to proceed by couplingtwo objectives: the screen is placed on the focal plane of the first objective while the detector is placed on the focal plane of the second objective. In order to obtain the wanted magnification, the two objectives can have different focal lengths. In Fig.30 is shown the coupling between two high quality objectives placedone close to the other.
Fig.30  Coupling between two objectives.
As can be seen from Fig.29, part of the rays emitted along thediagonal of the image,it cannot be cross the two objectives, because it subtends an angle too wide with respect to the optical axis.
This occurs since the exit pupil of the first objective does not coincide with the entrance pupil of the second one.The commonly called “vignetting” effect is shown in Fig. 31.
Fig. 31
The “tandem” objectives must be projected at the their origin in order to minimize these effects. Their functioning is totally similar to the enlargers with the advantage of a high brightness.
We offer a wide range of products for coupling between optical systems  see our Micro optics  beem transformation system (BTS) and our fiber couplers
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