Fiber-Optical Coupling

Abstract

In modern optical communication systems, it is of the highest importance to transmit as much optical power from the transmitter to the receiver. It seems that future systems will not be that strongly dependent on good optical waveguides . Actually, even after 25 years of existence of low-loss glass fibers , the coupling efficiency remains the biggest concern of the system engineers. In this chapter, the most important principles of the optical coupling that are relevant for engineers working with this topic are discussed.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

4.1 Adjusting Techniques

The adjustment techniques of photonic packaging for fiber–chip coupling can be divided into two fundamentally distinguishable functional groups:

• 1. Active techniques

• 2. Passive techniques.

4.1.1 Active Techniques

The active techniques use the ability to control the adjustment under real electrical excitation conditions of OEICs . By using electrically actuated elements, the adjustment of the coupling is carried out online by micromechanical actuators and fixed permanently after optimization of the optical coupling. This method is very time and labor intensive and therefore relatively expensive. The mechanically active coupling is normally performed by hand. Approaches to automation are tro find only in the area of PC-controlled maximum value search and subsequent fixation of the coupling, e.g., by laser welding techniques (see Chap. 8) or adhesion (see Chap. 9). Both, the stripping and preparation of fiber ends, the insertion of the fibers and the removal of the packages as well as the electromechanical preparation of the inner housing assembly to be completely carried out by hand or in individual production. For smaller quantities or prototypes, this method has of course a privilege, but should be replaced for reasons of cost for high production lots by passive adjusting techniques.

4.1.2 Passive Techniques

Passive adjustment technique means a fiber–chip coupling by pre-alignment grooves that are especially suitable for mass production. The use of high-precision lithography technique s plays a crucial role. There are three methods in this area of application, but these are currently insufficiently used in the telecommunications industry due to the low production numbers:

• 1. LIGA —Lithography/electroplating /impression

• 2. Flip-chip (expensive basic technology)

• 3. Chemical- or laser-etched U-grooves

The LIGA technique (Brück 2001; Ehrfeld et al. 1994) was developed as an evolution of the macroscopic impression techniques from Karlsruhe Institute of Technology, briefly referred to as KIT, and was realized by using a precise preform to manufacture a master, which is able to produce PMMA components with micrometer accuracy. It provides a preform to produce optical waveguides and U-grooves for fiber alignment in the material within one production cycle. Waveguide core material and cladding must then be spun in further process steps on the chip. The glass fibers can then be inserted into the U-grooves and are passively aligned, which is accurately enough for multi-mode applications . For the mass market of the consumer market, this technique seems future-sent.

In flip-chip-bonding technology, the face-down devices are mounted (see Fig. 4.1). This type of contact allows an unpackaged chip assembly that are called “direct-chip-attach” calls (Goodwin et al. 1991; Lau 1995).

Fig. 4.1

Schematic representation of a fiber–chip coupling using the FC-bonding technology for contacting and self-adjustment Fischer (2002b)

By the self-adjustment during the soldering, the flip chip technology can be a way on implementing flip chip technology in photonics. Thus, single-mode fibers can be coupled with waveguides within the required alignment tolerances of submicron distances. Studies by Fischer and Kuhmann (Fischer 2002b) at the Heinrich-Hertz Institute, Berlin, have shown that the permitted tolerances could be met by ±1.0 μm in a single-mode fiber-optic coupling in the lateral direction. It seems possible to replace the complex and costly active coupling of fibers and waveguides with high-frequency optical signal processing by the low-cost mass-market flip chip technology and thus complements the LIGA technique in the passive adjustment range for RF-OEICs.

Small bond geometries provide excellent high-frequency characteristics in the range of frequencies greater than 20 GHz (Karpuzi 2009). Parasitic effects are significantly reduced, and thus, the flip-chip bonded devices are particularly well suited for the optical assembly and interconnection technology. In photonics, the focus is often on the low-loss transmission of high frequencies. The flip-chip-bonding technology is far superior to the conventional wire bonding techniques in the HF range, but also significantly more expensive to technology.

Soldering flux is added to prevent oxide layers so that good connections between the liquid solder and the pads are corresponding. Photonics, however, must be entirely dispensed with the use of fluxes, not to destroy the sensitive optical surfaces of the OEICs . Flux can wet the surface of a diode laser chips, thus changing its emission properties negatively. Therefore, one reduces, during the soldering process, the oxide layers with the addition of formic acid or Hydrogen gas (Kuhmann 1996).

The flux-free process is, in addition to the required positioning accuracy, the main difference between microelectronic and photonic application of this contacting method.

V-grooves can be produced by anisotropic wet etching of silicon. Microsystems technology by wet etching of the silicon wafers (Beyer and Eigler 1996; Menz and Mohr 1997; Steckenborn et al. 1991) is an established process and allows fixation of fibers, fiber arrays, and lenses with very high precision. The FC-bonding technology in combination with high-precision patterned V-grooves represents an inexpensive way to produce self-adjusting electrical and optical connections.

Furthermore, it is possible to structure plastics topologically in the submicron range by laser ablation using Excimer Lasers or Femtosecond Lasers. Instead of working with an expensive master form as in the LIGA process which for prototype devices (Brück 2001) is too expensive to produce, you can ablate the plastic with the help of short-wavelength laser radiation. Suitable laser sources like the excimer laser are working at UV-output wavelengths between 350 and 190 nm. Due to the short wavelengths, the polymer chaines are cracked and peeled off layer by layer, so that a U-groove can be milled. A batch production is possible by the use of masks, which cover the entire wafer. The parts to be removed remain open and can be simultaneously irradiated by excimer laser. This production method is suitable to produce quickly and inexpensively optical waveguides in single-mode technology and U-grooves for the reception of glass fibers (Fischer 2002a; Fischer and Graener 2005).

4.2 Fixation Techniques

Regarding the use of special fixation techniques , many different techniques have been patented by the most part of the packaging companies to prevent the access of outsiders because of secrecy. In the meantime, clamping techniques accomplish for prototyping and simple applications (see Chap. 6). Laser welding is used typically in different shapes for very long-term stable housing and gluing for easy multi-mode applications, but also occasionally for single-mode usage. The adhesives generally have the property of acting in conjunction with hygroscopic water. Therefore, more gastight modules are required for adhesive fixation. Together with a relatively low glass transition temperature, all the adhesives have problems with long-term stability . In the publications that have been summarized in this book, several newly developed different fixation techniques are presented.

In the fixation techniques , we further differentiate the area of single-fiber coupling and the multi-fiber coupling . The multi-fiber technology is significantly more difficult in construction and adjustment of the fiber–chip coupling. It must be simultaneously aligned with each other up to 32 fiber ports and then finally fixed. The fibers are usually present in array form as semi-finished Si with chemically etched V-grooves. Only the sum of all alignment tolerances of the array fibers often results in a deviation of greater than one micron. This complicates the adjustment and can normally produce no optimal coupling results. The alignment tolerances for the straight fiber–chip coupling are longitudinally ±2.5 μm and laterally ±25 μm. However, you can only reach a coupling efficiency of about 10 % for conventional OEICs of InP or GaAs. Only using silica components, coupling efficiencies of over 90 % can be achieved, since the optical fields of both waveguides fit very well with each other (see Chap. 5). The fixation is usually realized in spite of the disadvantages of adhesives in the optical path butt with ended fibers (see Chap. 11).

In order to improve the coupling efficiency to OEICs made of semiconductor materials, mode-field transformers are integrated into the waveguide of the OEICs. These devices are used to achieve a coupling efficiency of about 50 % (see Chaps. 10 and 11), or lens systems are introduced to match the optical fields of both waveguides in the installed optical path.

In modern optical communication systems, it is of crucial importance to send as much optical power of the transmitter to the receiver. It seems possible that future systems will no longer be so dependent on the fiber attenuation , but even after 35 years of existence of low-loss optical fiber, the coupling efficiency remains an utmost concern of system engineers. In this chapter, the basic principles for optical coupling are discussed, which are crucial for photonic engineers.

4.3 Characteristics of a Good Coupling

In an optical coupling system between two waveguides, the power transmitted from one system into another can be described as follows (see Fig. 4.2):

Fig. 4.2

Transmission and reflection between two waveguides

$$T \, = \, P_{2} /P_{ 1}$$

(4.1)

• $P_{1}$ Optical power in system 1

• $P_{2}$ Optical power in system 2

The transmittance $T$ is a number less than unity and is usually expressed as a percentage. Transmission losses occur because of the transition of light through the space between two subsystems. Thereby, the light can be attenuated by the following mechanisms:

• Absorption

• Conversion to phonons and photons, respectively

• Scattering

Sometimes, it is more useful to express transmittance $T$ in dB. One defines then the insertion loss (${\text{IL}}$) as follows:

$${\text{IL }} = \, - 1 0 {\text{ log}}_{ 1 0} T\;{\text{in}}\;{\text{dB}}$$

(4.2)

Corresponding values for the ${\text{IL}}$ between linear and logarithmic values are given in Table 4.1.

Table 4.1 Insertion loss

Other T/IL pairs can be derived in the same way. In addition, the cascade of losses in a transmission system can be summed up to a total power loss, which includes all individual losses. The individual losses may appear discretely (at the interfaces or components) or continuously through the glass fiber section.

4.4 Reflections

Another important parameter in the optical coupling system is the power reflected back into the first component:

$$R \, = \, P_{3} /P_{ 1}$$

(4.3)

• $P_{1}$ Optical power in system

• $P_{3}$ Back-reflected optical power in system

Like $T$, the reflectance $R$ is also a number less than unity and is usually expressed as a percentage. Physically, it means that reflections are caused by the difference of the refractive indices between the optical components and the optical connection elements. The return loss (${\text{RL}}$) can be defined as follows:

$${\text{RL }} = \, - 1 0 {\text{ log}}_{ 1 0} R\;{\text{in}}\;{\text{dB}}$$

(4.4)

The already shown principles valid for ${\text{IL}}$ are also valid for ${\text{RL}}$. High ${\text{IL}}$ is not desirable in fiber optics. It can be improved using the following mechanisms:

• More effective coupling mechanisms

• Stronger light source

• More sensitive detectors

• Optical amplification

On the other hand, it is not so easy to find solutions for reducing the reflections. In optical systems, single and multiple reflections can significantly increase the transmitter’s noise, and degrade the signal quality at the receiver side. In optical amplifiers, the noise figure can be increased and the amplification can be significantly reduced. Some methods for reducing the reflections are as follows:

• Index matching with gel or oil

• Coatings for optical surfaces

• Angular surfaces

• Optical isolators

To achieve the bit error rate of 10⁻⁹ in digital systems, a reflection loss of 40 dB is required for all installed components. In amplitude -modulated analog systems, the target return loss is 55 dB. In wide area networks with bit rates higher than 10 Gbit/s, the value of 55 dB is also sometimes required.

4.5 Mode Fields in Waveguide Structures (Spot Size)

A term often used in coupling calculations is mode-field radius ($w_{0}$). Mode-field diameter (2$w_{0}$) is also known as spot size or focus diameter. The term originates from the calculations of light propagation of cylindrical gas lasers. In glass fibers , it refers to the intensity profile, which has a circular symmetry and can be well approximated by a Gaussian distribution along a radial coordinate $r$. The Gaussian power intensity profile has only a deviation of about one percent to the exact mathematical description, which is realized in the core area by Bessel functions of the first kind and nth order J _n and for the cladding region by an exponentially decaying Hankel function (see for more details Chap. 3), respectively. Therefore, in this chapter, there will be only the description of optical modes by using the Gaussian function.

$$p\left( r \right) = p\left( 0 \right) \cdot { \exp }\left\{ { - 2\left[ {\frac{r}{{w_{0} }}} \right]^{2} } \right\}$$

(4.5)

$w_{0}$ mode-field radius (MFR).

The optical field of a step-index fiber is shown in Fig. 4.3. The normalized refractive index difference is 0.3 %, and the core radius $a$ is 4.25 μm. As already discussed in the chapter about the optical waveguides , the same rules apply for all optical waveguides. The mode-field radius plays the key role in the coupling process between different components, even if the fields under consideration are elliptical or have deviations from the ideal Gaussian profile.

Fig. 4.3

Gaussian distribution of a wave guided through a glass fiber

As already mentioned in Chap. 3, the normalized frequency $V$ can take values between 1.9 $< V <$ 2.4 (Saruwatari and Nawata 1979). The mode-field radius is defined as the distance from the fiber axis where the power density has its maximum value down to the radius value where the power has dropped to $1/e^{2}$ (13.5 %) . It can be noticed that a non-negligible part of the optical power of 13.5 % is guided through the cladding at the wavelength of 1.55 μm. The mathematical relationship between the mode-field radius and the core radius is as follows:

$$w_{0} = a\left[ {0.65 + \frac{1.619}{{V^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }} + \frac{2.879}{{V^{6} }}} \right]$$

(4.6)

with

$$V = \frac{2\pi }{\lambda }a\sqrt {n_{1}^{2} - n_{2}^{2} }$$

(4.7)

$a$ core radius of a waveguide, $\lambda$ wavelength, $n_{1}$ core refractive index , and $n_{2}$ cladding refractive index.

The core radius, given by the change in the refractive index , is always lower than the mode-field radius (4.5 μm at 1300 nm and 5.25 μm at 1550 nm). Another fact is that the large portion of the guided wave is guided outside of the core. This portion can be described by the following mathematical expression:

$$F = { \exp }\left\{ { - 2\left( {\frac{a}{{w_{0} }}} \right)^{2} } \right\}$$

(4.8)

It amounts to about 17 % at 1300 nm and 25 % at 1550 nm. Since a SMF becomes multi-modal at 1240 nm, i.e., there are several modes of propagation, the broadening of the wave into the cladding will be much lower.

As the wave exits the end facet of the waveguide, the mode-field radius widens with the distance $z$ according to the following formula, which is also graphically represented in Fig. 4.4:

Fig. 4.4

Mode-field widening after exiting the waveguide

$$w^{2} \left( z \right) = w_{0}^{2} \left[ {1 + \left( {\frac{z \cdot \lambda }{{2 \cdot n \cdot \pi \cdot w_{0}^{2} }}} \right)^{2} } \right]$$

(4.9)

The spatial distribution of the mode field can also be expressed by the far-field angular distribution. The width of the near-field $w_{0}$ can be converted to the far-field angle (for z > 100 μm) by a simple transformation of Eq. (4.9). The Gaussian profile then yields:

$$\tan \varTheta = \frac{w\left( z \right)}{z} = \left[ {\frac{{\frac{z \cdot \lambda }{{\pi \cdot n \cdot w_{0} }}}}{z}} \right] = \frac{\lambda }{{\pi \cdot w_{0} }}$$

(4.10)

$$w_{0} = \frac{\lambda }{\pi \cdot n \cdot \tan \varTheta }$$

(4.11)

for $w_{0} \ll 1$ and $z \gg 100\,\upmu{\text{m}}$

Equation (5.8) also gives us the dependency of the mode-field widening on the wavelength and on the refractive index $n$ of the medium outside of the waveguide.

The increase in mode-field radius between 1300 and 1550 nm results in the broadening of the far-field angle by approximately 1.3 %.

The sine of the far-field angle θ derived from the Eq. (4.10) is identical to the definition of the numerical aperture NA given in Sect. 3.3.7 and, as already shown, can vary between 0.15 and 0.25 in the case of glass fibers . Typically, far-field angles of the semiconductor lasers result in the numerical apertures that are higher than those of the glass fibers (up to NA = 0.5 or 30°). This further means that it is not possible to achieve the coupling efficiency of more than 15 % from the laser into the glass fiber without mode-field adaptation .

4.6 Coupling Efficiencies

For efficient power transfer between the optical components, the largest possible overlapping of the mode profiles is necessary. The mode-field overlapping of two different Gaussian beams is shown in Fig. 4.5. For single-mode waveguides, the extent of overlapping can be described with simple approximations. Let us define $\kappa_{0}$ as the relationship between mode-field radiuses $w_{0}$ and $w_{1}$ as follows:

Fig. 4.5

Relationship between far and near field

$$\kappa_{0} = \frac{4}{{\left( {\frac{{w_{1} }}{{w_{0} }} + \frac{{w_{0} }}{{w_{1} }}} \right)^{2} }}$$

(4.12)

The loss (in dB) at the interface between two fields can be described according to (Saruwatari and Sugie 1981) as mode-field mismatch (in dB) in the following way:

$$L\left( {\kappa_{0} } \right) = - 10\log \left( {\kappa_{0} } \right)$$

(4.13)

The coupling efficiency between two optimally adjusted waveguide elements can be estimated with this simple equation, thereby, the waveguides can be of any type or shape. The only requirement is that the distribution of the optical fields in the vertical and horizontal plane of propagation is not significantly different. Up to the field ratio of 1:3, the Eq. (4.13) does not introduce a large error and can be reliably used. Otherwise, the overlapping must be separately estimated for both planes of propagation:

$$L\left( {\kappa_{{0\Sigma }} } \right) = \frac{{L\left( {\kappa_{0 \bot } + \kappa_{0 = } } \right)}}{2}$$

(4.14)

If one wants to calculate not only the losses due to the unequal field distributions, but also the additional losses caused by the inaccurate alignment of the waveguides to one another, it is necessary to solve the overlap integral between the two beams with their fundamental modes by means of Hermitian–Gaussian functions (Fig. 4.6). According to Saruwatari (Saruwatari and Nawata 1979) which was in detail discussed in Chap. 3, the portion of coupled beam energy can be expressed as follows:

Fig. 4.6

Coupling between two Gaussian beams with lateral, longitudinal, and angular misalignment

$$\begin{aligned} \eta = & \left| {C_{x0,0} } \right|^{2} \left| {C_{0,0} } \right|^{2} \\ C_{x0,0} = & C_{0,0} {\text{exp}}\left( { - \frac{{jk\theta x_{0} }}{2}} \right) \cdot { \exp }\left[ { - \frac{{x_{0}^{2} }}{q}\left( {\frac{1}{{w_{1}^{2} }} + \frac{jk}{{2R_{1} }} - \frac{jk\theta }{{2x_{0} }}} \right) \cdot \left( {\frac{1}{{w_{0}^{2} }} + \frac{jk}{{2R_{0} }} - \frac{jk\theta }{{2x_{0} }}} \right)} \right]z \\ \end{aligned}$$

(4.15)

$\eta$ is then given by the following expression:

$$\eta = \kappa \;{ \exp }\left( { - \kappa \left\{ {\begin{array}{*{20}c} {\frac{{x_{0} }}{2}\left( {\frac{1}{{w_{1}^{2} }} + \frac{jk}{{2R_{1} }}} \right) + \frac{{x_{0} }}{2}\left( {\frac{1}{{w_{0}^{2} }} + \frac{jk}{{2R_{0} }}} \right) + } \\ { + \frac{{\kappa^{2} }}{8}\left[ {\left( {\theta^{2} w_{1}^{2} - \frac{{2x_{0} \theta w_{1}^{2} }}{{R_{1} }}} \right) + \left( {\theta^{2} w_{0}^{2} - \frac{{2x_{0} \theta w_{0}^{2} }}{{R_{0} }}} \right)} \right]} \\ \end{array} } \right\}} \right)$$

where $\kappa$ stands for:

$$\kappa = \frac{4}{{\left( {w_{1}^{2} w_{0}^{2} \left[ {\frac{1}{4}\left( {\frac{1}{{w_{1}^{2} }} + \frac{1}{{w_{0}^{2} }}} \right)^{2} - \frac{{k^{2} }}{4}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{0} }}} \right)^{2} } \right]} \right)}}$$

(4.16)

Under the assumption that $R_{0} = \infty$ represents the fiber end face and

$$R_{1} = z\left\{ {1 + \left[ {\frac{{\pi w_{1}^{2} }}{\lambda }} \right]^{2} } \right\}$$

The coupling efficiency can be simplified as follows:

$$\eta = \kappa \;{ \exp }\left( { - \kappa \left\{ {\frac{{x_{0} }}{2}\left[ {\frac{1}{{w_{1}^{2} }} + \frac{1}{{w_{0}^{2} }}} \right] + \pi \theta^{2} \left[ {\frac{{\left( {w_{1}^{2} \left( z \right) + w_{0}^{2} } \right)}}{{2\lambda^{2} }}} \right] - \frac{{x_{0} \theta }}{{w_{1} }}} \right\}} \right)$$

with

$$\kappa = \frac{{4w_{1}^{2} w_{0}^{2} }}{{\left[ {w_{1}^{2} + w_{0}^{2} } \right]^{2} + \frac{{\lambda^{2} z^{2} }}{{n\pi^{2} }}}}$$

$$w_{1}^{2} \left( z \right) = w_{1}^{2} \left\{ {1 + \left[ {\frac{\lambda z}{{n\pi^{2} w_{1}^{2} }}} \right]} \right\}$$

where $\lambda$ stands for the wavelength . Under the assumption that $z$ = 0, the coupling efficiency is further simplified to:

$$\eta_{0} = \kappa_{0} \;{ \exp }\left( { - \frac{{2x_{0}^{2} }}{{\left[ {w_{1}^{2} + w_{0}^{2} } \right]^{2} }} - \frac{{2\pi^{2} \theta^{2} w_{1}^{2} w_{0}^{2} }}{{\lambda^{2} w_{1}^{2} + w_{0}^{2} }}} \right)$$

(4.17)

with

$$\kappa_{0} = \frac{4}{{\left[ {\frac{{w_{0} }}{{w_{1} }} + \frac{{w_{1} }}{{w_{0} }}} \right]^{2} }}$$

(4.18)

Analogous to the Eq. (4.2), the transmission loss in dB can be calculated according to the Eq. (4.13).

$$L\left( \eta \right) = - 10\log \left( {\eta_{0} } \right)$$

(4.19)

The exact calculation is relatively complex. Therefore, it is often easier to estimate transmission losses by approximate formulas. A good approximation, with the inaccuracy in the 5 % range, was offered by Mickelson and Basavanhally (1997). However, it is applicable only to the identical single-mode waveguides such as two SMF fibers. Then, the transmission loss (in dB) in case of the lateral misalignment in $x$- or $y$-direction can be approximated as follows:

$$L\left( x \right) \approx 4.343\left[ {\frac{x}{{w_{0} }}} \right]^{2}$$

(4.20)

The losses in the case of longitudinal or angular misalignment can also be estimated with the same approximation:

$$L\left( z \right) \approx 5.3\left[ {\frac{\lambda z}{{10nw_{0}^{2} }}} \right]^{2}$$

(4.21)

$$L\left( \alpha \right) \approx 2.7\left[ {\frac{{w_{0} n\alpha }}{10\lambda }} \right]^{2}$$

(4.22)

Both possibilities presented here for calculating the transmission losses, which include lateral and angular misalignment , show that these losses are independent of each other and that individual losses (in dB) can be easily summed up to result in the total transmission loss. The only requirement is that the spot size of the incoming light beam is located at the fiber surface.

Comparison of Fig. 4.7a, b shows that for the same value of misalignment, the loss due to lateral misalignment is significantly higher than the loss caused by longitudinal misalignment. By comparing the approximate formulas given by Eqs. (4.19) and (4.20), it can be seen that the misalignment in the $z$-direction is reduced by a factor of 10 in the denominator. This is not the case for the misalignment in the $x$-/$y$-direction.

Fig. 4.7

a Transmission loss between two identical waveguides in the presence of lateral misalignment. b Transmission loss between two identical waveguides in the presence of longitudinal misalignment

The loss due to the angular misalignment (Fig. 4.8) rises very quickly for values above 2°. For 10° angular displacements, the additional attenuation is already 10 dB. The misalignment angle of 1.05° results in the additional loss of 1 dB. An overview of discussed loss mechanisms is given in Fig. 4.9. To give the reader the better feeling about the influence of different mismatch mechanisms, the numerical values of different mismatches that result in 1 dB transmission loss are given in Table 4.2. The calculations were performed for a wavelength of 1310 nm and a mode-field diameter of 9 μm for both waveguides (SMF ). The losses are bidirectional and equal and can be easily summed up for small individual contributions.

Fig. 4.8

Transmission loss between two identical waveguides in the presence of angled misalignment

Fig. 4.9

Transmission loss mechanisms between waveguides (here two glass fibers )

Table 4.2 Coupling errors and mismatch/misalignment values (at 1.55 μm) for 1 dB excess loss

In addition to the intrinsic losses, which are caused by mismatch of mode fields or misalignments, extrinsic losses introduce even more attenuation between two waveguides. Dirt or scratches on fiber end faces, cleft or split exit windows have the absorbing or scattering effect on the exiting light, respectively, allowing only a portion of it to reach the opposite waveguide.

4.7 Laser–Fiber Coupling

The radiated light cone of a laser diode is elliptical and has a big aperture, from 20° to 60°, depending on the direction of propagation (Fig. 3.16). An obvious solution to minimize the coupling loss into the glass fiber would be to set two cylindrical lenses one behind the other. They should convert different aperture values in vertical and horizontal direction of propagation into the circular profile of a mono-mode glass fiber with the opening angle of 11.5°. The following imaging systems come into consideration in such optical design:

• Lens systems

  • Cylindrical lenses

  • Spherical lenses

• Graded-index lenses (Grin- or Selfoc^®-lenses)

• Fiber taper (lensed fiber end faces)

Due to the small dimensions of the fiber-optic components, the goal of the design engineer is to minimize the amount of needed optical components. Only when the highest possible transmission values are requested, or a parallel light beam between the OEIC and the fiber is required, for example, to insert the optical isolator because of the reflection loss, a multi-lens system is selected. High costs of the adjustment process make these multi-lens systems very expensive.

In case of coupling a multi-mode fiber with a thin lens (Fig. 4.10), a magnification factor between the object and the image can be calculated in the following way with the conventional geometrical optics:

$$\beta = L^{\prime } /L$$

Fig. 4.10

Geometrical optics mapping

The general thin lens equation sets the focal length in relation to the object and image distance:

$$1/f = 1/x + 1/x^{\prime }$$

(4.23)

Under the assumption of the same media on both sides of the lens, the magnification factor can be calculated by the ratio of the screen distance to object distance:

$$\beta = x^{\prime } /x$$

For the calculation of coupling between multi-mode waveguides, one can map the opening of the fiber core $A$ to the opening of the laser $A^{\prime }$:

$$A {\text{NA}}^{2} = A^{\prime } {\text{NA}}^{\prime 2}$$

(4.24)

The losses in this type of imaging (mapping) can be calculated according to the Eq. (2.23). If the opening and the numerical aperture are known, the respective parameters of the coupled multi-mode waveguide can be calculated.

If the size of the focal point has the same order of magnitude as the wavelength , one cannot use simple geometrical optics imaging any more. Instead, the spreading of a Gaussian beam must be considered. This is the case when, for example, the picture of a single-mode waveguide should be mapped on the laser diode (Fig. 4.11).

Fig. 4.11

Gaussian beam propagation outside of a waveguide

The distance from the smallest beam diameter to the lens corresponds to the focus length and to the value 2$w_{0}$ of the focus diameter. The focus point is defined as the area of the smallest beam diameter, in which the wave front is planar. This corresponds to the focal point at the corresponding focal length F. The illuminated diameter of the lens will be here denoted as D. The depth of focus (DOF ) corresponding to the mode-field diameter can be easily calculated with the following formula if the wavelength and the F-number of the lens are known:

$$\begin{aligned} & F - {\text{number}} = F\# = \left( {{F \mathord{\left/ {\vphantom {F D}} \right. \kern-0pt} D}} \right) = {{2w_{0} } \mathord{\left/ {\vphantom {{2w_{0} } {\left( {{{4\lambda } \mathord{\left/ {\vphantom {{4\lambda }\uppi}} \right. \kern-0pt}\uppi}} \right)}}} \right. \kern-0pt} {\left( {{{4\lambda } \mathord{\left/ {\vphantom {{4\lambda }\uppi}} \right. \kern-0pt}\uppi}} \right)}} = {1 \mathord{\left/ {\vphantom {1 {\left( {2\;{\text{NA}}} \right)}}} \right. \kern-0pt} {\left( {2\;{\text{NA}}} \right)}} \\ & {\text{DOF}} = \left( {{{8\lambda } \mathord{\left/ {\vphantom {{8\lambda }\uppi}} \right. \kern-0pt}\uppi}} \right)\left( {{F \mathord{\left/ {\vphantom {F D}} \right. \kern-0pt} D}} \right)^{2} \\ \end{aligned}$$

Example:

• Lens diameter D = 1 mm,

• Focus length F = 2 mm

• Wavelength λ = 1.55 μm

$$\begin{aligned} & {\text{DOF}} = \left( {{8 \mathord{\left/ {\vphantom {8 {3.14}}} \right. \kern-0pt} {3.14}} \cdot 1.55\; \times \;10^{ - 6} \;{\text{m}}} \right)\left( {{{2\; \times \;10^{ - 3} \;{\text{m}}} \mathord{\left/ {\vphantom {{2\; \times \;10^{ - 3} \;{\text{m}}} {10^{ - 3} \;{\text{m}}}}} \right. \kern-0pt} {10^{ - 3} \;{\text{m}}}}} \right)^{2} = 16\;\upmu{\text{m}} \\ & 2w_{0} = \left( {{4 \mathord{\left/ {\vphantom {4 {3.14}}} \right. \kern-0pt} {3.14}} \cdot 1.55\; \times \;10^{ - 6} \;{\text{m}}} \right)\left( {{{2\; \times \;10^{ - 3} \;{\text{m}}} \mathord{\left/ {\vphantom {{2\; \times \;10^{ - 3} \;{\text{m}}} {10^{ - 3} \;{\text{m}}}}} \right. \kern-0pt} {10^{ - 3} \;{\text{m}}}}} \right)^{2} = 4\;\upmu{\text{m}} \\ \end{aligned}$$

With a 1-mm-diameter lens, it is possible to create a focus diameter of 4 μm and a mode-field diameter of 16 μm at the distance of 2 mm from the lens. For a more detailed calculation of lenses at fiber ends as well as for the Selfoc^®-lenses, please refer to the (Ladany 1993).

An overview of the imaging techniques used in optical modules is shown in Fig. 4.12.

Fig. 4.12

Different coupling techniques for focusing the light of a semiconductor laser into the fiber

In the simplest case, a straight-cut fiber (butt-fiber) is positioned directly in front of the laser. The coupling efficiency that can be thereby achieved is 10–15 % (−10 dB). This solution is typically used in low-priced systems.

The efficiencies between 20 and 50 % can be achieved by means of ball lenses with radii between 0.3 and 3 mm. In the commercial laser modules with TO-3 housings , such lenses are usually already built in. This allows the fiber to be cost-effectively placed in the right position in front of the lens.

Selfoc^®- or Grin-lenses are cylindrical parts , which are constructed similarly to glass fibers with core and cladding structure. In addition, the refractive index of the glass in the core of the lens is adjusted along the direction of propagation in such a way that the light takes a nonlinear path within the lens, as shown in Fig. 4.13. In order to produce a parallel beam between the laser diode and fiber, two identical Grin-lenses should be placed one behind the other, so that the parallel light beam is formed in the space between them.

Fig. 4.13

Beam propagation within Selfoc^® lens

In order to reduce the adjustment costs, the number of adjusting elements should be reduced. An effective way to achieve this is to realize an imaging element directly on the glass fiber ’s end face. This is known as a fiber taper. One solution can be made by means of casting resin, which has a refractive index adapted to the fiber (n = 1.446). The fiber is briefly dipped in the resin, so that a half-ball lens is formed on its end face once it is pulled out. The coupling efficiency of up to 30 % can be achieved. Another solution is to melt a ball of glass to the end ofthe fiber using a splicing machine with electrodes which heaten the fiber and the glass ball.

A more complex variant of the taper with a cone shape can be produced by wet etching, by mechanical grinding (sanding), from the fused material, or by laser ablation. Here, the cone shape acts as a lens. If the fiber end is warmed up to the melting point and then additionally pulled, a semi-spherical lens is formed (Fig. 4.14). The coupling efficiency of this most often used solution is between 30 and 70 %. If two different radii are polished on the vertical and horizontal plain on the fiber end face, the coupling efficiency can be increased to even more than 90 %. However, such tapers are very expensive (Fig. 4.15).

Fig. 4.14

Taper with core and rejuvenating cladding

Fig. 4.15

Taper with different taper angles

When coupling with a taper , in one longitudinal displacement point, the coupling efficiency from the laser into the fiber is the highest. In this case, the distance between the laser and fiber taper depends on the chosen radius of the taper.

Rule of thumb: The smaller the half-sphere radius of the taper, the smaller is the distance from the laser and the greater is the coupling efficiency . The coupling efficiencies of three different fiber ends [small lens, big lens, and straight-cut fiber (Butt-fiber )] are compared in Fig. 4.16. It can be seen that a taper with a 10 μm radius has the highest coupling efficiency. However, the danger of having a contact between the chip and the taper is huge and consequently the risk of damaging the laser diode with the fiber component. Distances between the OEIC and the fiber taper end are typical from 5 and 15 μm.

Fig. 4.16

Comparison of the coupling efficiency of tapers with different radii and butt coupling

It should be additionally considered that the spatial sensitivity significantly increases with the higher coupling efficiency , as can be seen in Fig. 4.17. The sensitivity to the temperature changes and vibrations is in that case also increased. At high coupling efficiencies the mechanical stability of the whole structure is crucial to the long-term stability of the module.

Fig. 4.17

Efficiency of the laser-to-fiber coupling for axial misalignment of a taper with 10 μm radius

Another important aspect is the avoidance of back reflections into the transmitter diode. Planar surfaces (e.g., fiber end faces) perpendicular to the optical axis behave very unfavorably in this respect with about 4 % reflection at the air-to-glass transitions. One could try to compensate the refractive index change between the OEIC (e.g., InP n = 3.3) and air (n = 1) by cutting the fiber end with an angle of about 7°, or by coating the fiber surface with a material similar to those used in camera lenses or glasses. In optical connector couplings, an index matching gel can be used. It has the refractive index of 1.5 and is applied on the connectors’ end faces.

Further on, it is of crucial importance that the system is mechanically stable with respect to vibrations and thermal influences. In industrial applications, both the fiber tapers and lens systems are used for fiber–chip coupling.

4.8 Waveguide Taper

A way to realize mechanical insensitive coupling modules is the insertion of waveguide structures in the semiconductor material to adjust the mode fields of the laser to the glass fiber . These structures are called waveguide tapers or mode-field expanders. Some realization options are listed below. One can make a difference between a lateral (thick) taper and a vertical (wide) taper (Honecker et al. 2002). The combination of two of these would be ideal to optimally adjust the field of a single-mode glass fiber.

For the simplicity, only a single vertical taper is typically realized during the production of a semiconductor. The lateral taper is technologically more difficult to produce and is still rarely commercially available. Anyway, the coupling efficiency significantly increases in comparison with the butt-fiber coupling. The comparison of coupling efficiencies between butt and taper coupling is shown in Fig. 4.18. The coupling efficiency would be different for horizontally and vertically polarized light. A waveguide taper has a gain of more than 7 dB over butt-fiber coupling . Therefore, a coupling efficiency of more than 50 % can be expected.

Fig. 4.18

Vertical and horizontal waveguide taper

4.9 Mode-Field Measurement Methods

To predict an effective optical coupling between various optical components, it is necessary to exactly determine the mode fields of the OEICs, as shown in Sect. 3.3. If the mode field or the spot size is known, the overlap integral of the Gaussian beams can be used to predict the coupling efficiency . Here the comparison of the fields of active OEICs (such as lasers) with the fields of tapered single-mode fibers  is of particular interest, since such coupling arrangements are frequently used in real systems. During that time, two methods for measuring the spot size have been established. These are as follows:

• Near-field method ,

• Far-field method .

Other methods have also been developed (Andersen 1984), ITU-T G652 (Keil 1984). These are however beyond the scope of this book.

There are several methods to characterize the optical mode field of photonic ICs. The most used method is the so-called far-field method. This method provides a good resolution of the field in a large distance in comparison with the wavelength of the used light. Additionally, the setup is easy to realize with standard electromechanical parts such as rotating stages. In this case, the field is scanned by a small photodiode, rotating around the output side of the waveguide. The EIA standard, RS-455-47, describes the measurement procedure to qualify optical fibers. The resolution of the method has its limits by the scanning steps and the active field of the photodiode. You must keep in mind that in the far field, a smearing of the fine contour of the near field will appear. The setup needs a large room around the DUT , because the detector motion takes wide space.

To get more detailed information about the field distribution at the end facet of the waveguide, an additional method is to use a microscope objective to magnify the spot to a camera system. This technique is called in literature near-field observation. The resolution is limited by the diffraction limit of the used light of the exciting optical source, which is fed through the tested waveguide. At 1550 nm, the resolution is limited to 2 µm in both lateral dimensions. For typical optical fields of waveguides of laser diodes, the resolution is insufficient to get a detailed knowledge of the optical field parameters. The resolution is also adulterated by the lens due to aberrations , which broaden the focus on the focus plane, and the DUT must be fixed in micrometer precision in front of the setup. This is a time-consuming procedure limiting the near-field technique to laboratory use. Both techniques are wasteful due to time-consuming measurement and positioning times. These facts are focusing the field analysis on random sample survey and disqualify these techniques basically for high-output industrial use.

4.9.1 Near-Field Method

The near-field measurement method can be used for high-accuracy measurement of the mode-field radii $w_{x}$ and $w_{y}$ of a waveguide under investigation. This method is described in DIN standards (DIN EN 60793-1-45:2004-07 and DIN 58002:2001-12). The advantage of this measurement technique is high measurement resolution. However, high precision is typically followed by high equipment costs. As shown in Fig. 4.19, the near-field intensity distribution exiting the waveguide is enlarged by means of an objective and displayed on the image plane. The coordinates X and Y have been shown here in accordance with ISO/CD 11807-1. On the image plane, the intensity distribution is spatially resolved and measured by means of a photodetector. High magnification allows very precise estimation of the light distribution in the waveguide of OEICs. An example of such a measurement on a SMF fiber is shown in Fig. 4.19.

Fig. 4.19

Near-field measurement setup

The measuring system consists of a light source used for the excitation of OEICs, an imaging system (preferably a microscope objective ), and a receiver system, which is usually a Vidicon video camera tube with electronic image processing (Fig. 4.20).

Fig. 4.20

Near-field of a single-mode fiber (SMF ) measured at 1550 nm

The complete measurement setup should be installed so that all components are aligned on the same optical axis. The intensity of the light source must be adjusted so that the full dynamic range of the camera system can be used, typically with 8-bit resolution.

If the optical attenuators are used in order not to overexcite the camera, they must attenuate the light homogenously over the whole scanning area. When characterizing the passive optical components, the waveguide can be illuminated externally, so that leaky modes, scattered light, or irradiated light are avoided, and the waveguide is excited by a single mode. The center wavelength and the spectral bandwidth of the light source must thereby be known. If the component is polarization dependent, the angle of polarization must be kept constant during the whole measurement process. In addition, the temperature of the device under test should be stabilized and recorded during the measurement.

The numerical aperture of the imaging system should be at least three times larger than the numerical aperture of the component to be measured. Otherwise, the measurement accuracy will be severely reduced. The magnification factors of used lenses should be known. In addition, the lenses should have the anti-reflective coating to reduce the reflections and interferences. It is also recommended that the lenses are corrected for the wavelength range of interest in order to avoid the distortion of the near-field image. The lateral extent of the detector should be chosen so that the image of the waveguide corresponds to at least four times the 1/e ² spot size.

An infrared Vidicon photodetector system is commonly used in the praxis. It is known to show strong deviations of local sensitivity and intensity dynamics. Consequently, the following points are of particular significance when installing the measurement system:

• Isotropy, spatial invariance

• Nonlinearity lower than 5 %

• Variation of sensitivity of less than 2 %

• Background variations of less than 1 %

• Signal-to-noise ratio of less than 20

• Spatial resolution greater then 10 times 1/e ² spot size

• Digital analysis with more than 6 bits

The measurement should be only performed when the light source with the coupling system, the objective and the detector system have reached the steady-state operation over time. This should prevent the additional intensity fluctuations to reduce the measurement accuracy.

4.9.2 Median-Field Method

The drawbacks of the near- and far-field techniques are evidently. Therefore, at the Harz University, a new setup to measure the optical field without any additional optics or micrometer-precise adjustment of the DUT was developed. The new technique is called median-field method. Here, an easy optical setup is combined with a mass production compatible measurement cycle. A closer description of this conversion and the measurement setup can be found in Fischer and Windel (2004).

The intensity is recorded metrologically by a small photodiode in a 2-axis scanning setup, using a distance r in the range of 2–3 mm from DUT to photodiode, which is sketched in Fig. 4.21. The spacing r can be seized on use of a 3-axis system, starting from zero spacing at the DUT to 3 mm longitudinal traverse path. It is constant during the measurement. To get comparative data to the far-field method, a conversion of the measured intensities must be made. The spacing of the photodiode from the longitudinal axis no longer directly aligns the photodiode to the test object. It would have to be turned, in order to keep the effective receipt surface constant, hereby the distance of the photodiode to the DUT is changing. Both factors affect the effective surface area of the measuring receiver device, and therefore, a distance coefficient must be introduced .

Fig. 4.21

Median-field measurement setup

The distance coefficient is specifically calculated for each detection point and stored in an online table. The surface of the receiver photodiode is assumed as rectangular. In order to achieve the conversion from Cartesian coordinates to hemispherical plane, the corner points are first considered as vectors, as shown in Fig. 4.21.

By normalizing the vectors on the length of the beam that reaches the center of the rectangle, one obtains the effective area of the illuminated surface, which is depicted in Fig. 4.22.

Fig. 4.22

Reception plane and DUT

The smaller distance of the photodiode reduces the plane of the surface at the Z-axis. It must also considered that the surface has to be rotated, which becomes an irregular square in the aquisition plane, and the surface is calculated according to the equation for the general square:

$$A = \frac{1}{2} * d_{1} * d_{2} * \sin \angle \left( {d_{1} ,d_{2} } \right)$$

(4.25)

d _1/2 are the diagonals between the corner points of the measuring plane. Now, the surface can be calculated by:

$$A_{\text{EN}} = \frac{1}{2} * \left| {\overrightarrow {{A_{\text{EN}} C_{\text{EN}} }} } \right| * \left| {\overrightarrow {{B_{\text{EN}} D_{\text{EN}} }} } \right| * \sin \angle \left( {\overrightarrow {{A_{\text{EN}} C_{\text{EN}} }} ,\overrightarrow {{B_{\text{EN}} D_{\text{EN}} }} } \right)$$

(4.26)

The difference of a plane area and the semispherical one is shown in Fig. 4.23. To calculate the area of the surface on the spherical surface, the vectors of the corner points of the receiving diode are standardized on the radius of the sphere. The radius corresponds direct to the distance of the receiver diode from the DUT on the z-axis. It stretches to an anomalous square, whose surface can be determined as follows:

Fig. 4.23

Difference of plane areas of flat measurement surface and spherical surface

$$A_{K} = \frac{1}{2} * \left| {\overrightarrow {{A_{K} C_{K} }} } \right| * \left| {\overrightarrow {{B_{K} D_{K} }} } \right| * \sin \angle \left( {\overrightarrow {{A_{K} C_{K} }} ,\overrightarrow {{B_{K} D_{K} }} } \right)$$

(4.27)

For the determination of the distance coefficient , both surfaces must be kept in relationship. The distance factor κ depends on the coordinates, where the receiver diode is working in the acquisition plane

$$\kappa \left( {{\text{distance}}\_\,{\text{faktor}}} \right) = \frac{{A_{\text{EN}} }}{{A_{K} }}$$

(4.28)

The intensities collected at the respective coordinates must be calibrated using the distance factor κ, in order to get the proper intensity values.

4.9.2.1 Automated Acquisition

Automation of the median-field acquisition is realized using a 6-axis-motion system from Physic Instruments (PI) F-604. The motion of the photodiode is controlled by a Labview program. It also calculates the distance coefficient κ . A virtual instrument (VI) was developed, which controls the PI system and processes the measuring data. The user front end is depicted in Fig. 4.24 (Windel 2006).

Fig. 4.24

Labview VI user front end for automated field acquisition

One can see at the red-framed part the user interface to print in the file name for the storage of the measuring data. Further on, the user is asked to type the incrementation steps size in µm and the count of measurement steps per line (“steps”). These inputs are necessary for the automated acquisition of the intensity in connection of the VI drivers developed by PI.

Within the yellow-bordered part of the surface on the right upper side, one can see the measured intensities converted with the distance coefficient , the maximum intensity and their coordinates are indicated in the table. In the blue frame, the optical field widths in x- and y- direction, determined by the fitting process, are indicated. The intensity field and the result of the fitting process are shown in the two graphs.

4.9.2.2 Measurement Setup

To check the accuracy of the method, a SMF manufactured after ITU-G.652 was measured. This reference measurement was realized with the following measurement setup, shown in Fig. 4.25.

Fig. 4.25

Schematic measurement setup

Fig. 4.26

Photograph of setup

As optical source, the Anritsu MU951001A integrated laser set was used. The module consists of two laser diodes with at 1310 and 1550 nm, which are alternatively changeable. The detector was a photodiode from RCA with 50 µm active surface. The devices were installed on a motion controller system of Physics Instruments and controlled by an additional PC, which serves only as interface for the PC where the recorded data processing and calculating are performed  (Fig. 4.26).

4.9.2.3 Results

A standard single-mode fiber was subsequently measured to analyze with different measurement points and measuring point intervals and the accuracy of the method at two wavelengths (1550 and 1310 nm). No significant discrepancy was detected comparing all variations. In Fig. 4.27, the optical mode-field diameters determined are depicted. A value of 9 µm was expected according to the ITU-G.652 standard. From the results, it can be seen that the values are mainly too small without distance factor. The results show values at 1310 nm of 2w ₀ ≈ 7.4 µm and 2w ₀ ≈ 8 µm were at 1550 nm excitation wavelength. The deviation based on the fact that the distance between DUT and sensor is not accurate enough measured.

Fig. 4.27

Comparison of mode fields with different detectors and wavelengths with and without distance calibration

In order to use this new median-field method, it is important to calibrate the whole measuring setup on the distance from the DUT to the photoreceiver. The help of highly accurate distance sensors could realize this determination. An easier approach is to make use of standardized mode-field width of a SMF at 1310 nm. The mode field can be measured at the beginning of the measurements using 1310 nm excitation source fed into the SMF. The calibration factor for the distance can easily be calculated. To get a highly accurate distance factor F _K , a mode-field calibrated fiber from the National Institute of Standardization should be used conforming with the ITU-Standard at λ = 1310 nm:

$$F_{K} = 1. 2 1 9 4 5$$

The whole time for the complete scan of the field was less than one minute, which is a substantial improvement in relation to traditional far- and near-field measuring procedures. In order to reduce the absolute measurement times further, the use of a CCD camera is necessary. This setup has the advantage that the measurement of the mode field can be realized in only one step and measurement times will reduce less than one second.

The median-field method is ideal to measure the spot size of single-mode optical components. In comparison with the classical far-field and near-field method, the new method is ideal for use in automated systems, e.g., at the production of lasers, where the lasers must be characterized very quickly on bar. The measurement time can be reduced in combination with additionally current-power characterizations.

Using automated concepts with Labview programming, the median-field method shows high potential for automated high-speed and high precise mode-field measurement in agreement with ITU Recommendation G.652.

4.9.3 Far-Field Method

As just shown, it is relatively expensive to build up the near-field measurement system correctly. A more simple solution is offered by the far-field measurement method (Fig. 4.28). As already shown in Sect. 4.3, Eq. (4.8), it is possible to estimate the near-field distribution of light based on the angular distribution of the far field by means of a simple approximation. There is a direct relationship between the spot size and the tangent of a far-field angle . Analogous to the mode-field diameter of the near field, the far-field angle is defined as the angle from the axis where the intensity of light has dropped to 1/e ² of its maximal value.

Fig. 4.28

Far-field measurement setup

The measurement setup is similar to the one for the near-field, except of the measured OEIC /waveguide and the photodetector which can be circularly moved across two axes around the OEIC. The detector first circularly scans across the first angular axis. After that, the light intensity of the chip is scanned across the second angular axis. This is how the intensity profile of the emitted light is obtained. The angular distribution of light can be converted to the mode-field diameter of the waveguide by means of the following formula:

$$w_{0} = \frac{\lambda }{\pi \cdot n \cdot \tan \varTheta }$$

(4.29)

Like in the case of the near-field measurement, the difference between maximum and minimum measured intensities must be at least 64 digital levels. Since commercially available photodiodes are typically used as detectors, it can be assumed that they are very homogenous and linear. The actuator should have the angular resolution of less than 0.5°, in order to provide high resolution of the measurement. Scattered light is avoided by means of a phase-sensitive detection with a lock-in amplifier .

Comparison of the two measurement systems (near- and far-field) shows only a small difference of less than 10 % because the near-field measurement has better resolution than the direct calculation if the spot size based on the BPM program. A typical far-field distribution of a drawn fiber taper is shown in Fig. 4.29. A Gaussian approximation is also shown in the figure with a dashed line. It can be noticed that the measured angular distribution is relatively noisy.

Fig. 4.29

Far-field measurement curve and Gaussian approximation

It can also be seen that both curves match quite well at the value of 1/e ². Since the measured curve is slightly asymmetrical, the mean of the two values has to be used in this case. The far-field angle equals 17° on the left side and 14° on the right side, resulting in a mean value of 31°/2 = 15.5°. By substituting this value in the Eq. (4.27), a corresponding mode-field radius is obtained:

$$w_{0} - {{1.55\;\upmu{\text{m}}} \mathord{\left/ {\vphantom {{1.55\;\upmu{\text{m}}} {\left( {3.14 \cdot 1 \cdot \tan \;\left( {15,5^\circ } \right)} \right)}}} \right. \kern-0pt} {\left( {3.14 \cdot 1 \cdot \tan \;\left( {15,5^\circ } \right)} \right)}} = 1.78\;\upmu{\text{m}}$$

This is a typical value for fiber tapers with radii around 10 μm.

4.10 Summary

The optical coupling between different optical components requires low coupling losses and low reflections. In most cases, the geometrical optics cannot be used. Instead, the wave analysis should be applied. The optical modes of the components (laser, fiber, waveguide) are usually described with a Gaussian distribution, and the coupling efficiencies can be calculated by means of the overlap integrals . The biggest problem represents the mechanical alignment of the components.