Phase Mask FAQs

What is Chirp?

Calculating the Required Phase Mask Chirp for Dispersion Compensation Applications

Calculating the Required Length of a Chirped Phase Mask

Chromatic Dispersion: Introduction and Mathematical Description

Cleaning Apodized Phase Masks

Cleaning Unapodized Phase Masks

The ITU Grid Channel Spacings: Converting from Frequency Channel Spacings to Wavelength Channel Spacings

Measuring the Effective Index of Refraction of a Fiber for FBG Fabrication

What is Chirp?

Short answer: Chirp is the slope of the function of phase mask period versus position.

Long answer: StockerYale's phase masks can either have a uniform period (constant over the entire length of the phase mask) or have a linearly chirped period function. The chirp is simply the slope of the period function, as shown in the figure below.

Why is the word "chirp" used to describe this type of phase mask and the associated FBG? The answer is this: The use of the word chirp in optics is a reference the concept of chirp in common sounds. If a sound is chirped, its frequency increases from beginning to end. Similarly, if a phase mask is chirped, its spatial frequency increases from beginning to end of the grating.

Calculating the required phase mask chirp for dispersion compensation applications

The short answer to this question is this: The phase mask chirp ch^(PM) required for the fabrication of dispersion compensating FBGs can be calculated from:

ch^(PM) =	2

	cD

where ch^(PM) is in nm/cm, c (the speed of light in the vacuum) is in cm/ps, and where D (the dispersion of the FBG) is in ps/nm. There is a simple inverse relationship between the dispersion of the FBG and the chirp.

Now for the long answer:

The dispersion D of an optical communications component is defined to be the slope of the function that relates the group time delay τ to the free-space wavelength of the light λ:

D =	∂τ (λ)

	∂λ

The most practical units for D are picoseconds per nanometer.

For many purposes, it is sufficient to consider τ to be a linear function of λ, at least locally, in which case D is a constant in the spectral region under consideration. Then D can be calculated from the rise over run of τ(λ)

D =	τ_L - τ_S

	λ_L - λ_S

where, as shown in Figure 1, λ_L and λ_S are respectively the longer and shorter of the two wavelengths used in the calculation.

Let us calculate the dispersion that arises from a linearly chirped FBG. Our strategy will be to calculate λ_L, λ_S, τ_L and τ_S, and to plug them into the rise-over-run definition of D. Consider a device with length L, central period Λ^(FBG) and chirp ch^(FBG). The period function of the FBG is the simple linear expression

Λ^(FBG)(x) = Λ^(FBG) + x • ch^(FBG)

where x is the distance along the phase mask. The center of the phase mask is at x = 0. The period at one end of the FBG is

Λ_S^(FBG) = Λ^(FBG) -	L	• ch^(FBG)

	2

and at the other end it is

Λ_L^(FBG) = Λ^(FBG) +	L	• ch^(FBG)

	2

Light is reflected back by that part of the FBG for which the Bragg condition is satisfied. Assume that light enters the FBG from the short-period end of the device, as shown in Figure 2. This light is reflected back immediately upon entering if its wavelength corresponds to

λ_S = 2n_eff Λ_S^(FBG) = 2n_eff ( Λ^(FBG) − ^L/₂ • ch^(FBG) )

and it is reflected back at the other (far) end of the FBG if its wavelength corresponds to

λ_L = 2n_eff Λ_L^(FBG) = 2n_eff ( Λ^(FBG) − ^L/₂ • ch^(FBG) )

While the light that is reflected at the near end of the FBG is undelayed (τ_S = 0) as it emerges from the grating, the light that is reflected at the far end of the FBG requires a time of

τ_L =	2L

	ν

before it emerges reflected from the FBG. The factor of 2 takes into account that fact that the light must travel back and forth through the FBG. Here, ν is the group velocity (as opposed to phase velocity), since we are interested in the transmission and reflection of information-carrying signal pulses. Since the group velocity and phase velocity are quite close, we can relax the distinction for the purposes of this calculation. The above formula for τ_L can therefore be rewritten as

τ_L =	2L

	c / n_eff

Having calculated λ_L, λ_S, τ_L and τ_S, we now have all the pieces necessary to assemble our calculation of D:

D =	τ_L − τ_S	=	1

	λ_L − λ_S		c ch^(FBG)

Note that the parameters L and n_eff have cancelled from the result. Inverting the above formula, and recalling that the period and chirp of the FBG are exactly one half of the period and chirp of the phase mask used in fabrication, we obtain

ch_(PM) =	2

	cD

which is the formula reported at the beginning of this FAQ.

Note that the speed of light in units convenient for this calculation is c = 3.0 x 10^-2 cm/ps

Figures for this FAQ

Group time delay versus free-space wavelength

Figure 1. The figure shows group time delay τ plotted against free-space wavelength λ, for some optical communications component. The dispersion of the device is the slope of τ(λ), as in

D =	τ_L − τ_S

	λ_L − λ_S

Figure 2. For concreteness, suppose an FBG of length L and suppose that the signal enters the FBG from the short-period end. Light of wavelength λ is reflected back by the FBG at the point where λ = 2n_eff Λ^(FBG)(x). It turns out that the total delay time for light that makes the full round-trip in the FBG is τ = 2n_eff ^L/_c.

Calculating the required length of a chirped phase mask

To calculate the length of the phase mask, one needs to know

the bandwidth over which the FBG is required to function properly, Δλ
the chirp of the phase mask, ch^(PM)
the effective phase index of the core mode of the fiber in question, n_eff

(In the specification of the bandwidth Δλ, λ refers to free-space wavelength.) The required length of the phase mask (and of FBG itself) is given by

L =	Δλ

	n_eff ch^(PM)

How is this formula derived? We start with the basic equation for FBGs, namely the familiar relationship between the Bragg wavelength λ_BR and the period of the FBG Λ^(FBG). This equation is

λ_BR = 2n_eff Λ^(FBG)

Since the period of the FBG is exactly one half the period of the phase mask that is used in recording ---

Λ^(FBG) =	1	Λ^(PM)

	2

--- it follows that the relationship between the Bragg wavelength and the period of the phase mask is

λ_BR = 2n_eff Λ^(PM)

From this equation it follows that the long and short extremes of the phase mask period are connected to the bandwidth of the FBG by

Δλ = n_eff (Λ_L - Λ_S)

Now since the period of the phase mask at any point is given by the simple linear expression

Δ(x)^(PM) = Λ_c + x ch^(PM)

it is easy to see that the difference Λ_L - Λ_S is equal to L ch^(PM). Therefore, we conclude finally that

Δλ = n_eff L ch^(PM)

Solving for L, we have

L =	Δλ

	n_eff ch^(PM)

L is in centimeters, Δλ is in nanometers, and ch^(PM) is in nanometers per centimeter.

Chromatic Dispersion: Introduction and Mathematical Description

Introduction to Chromatic Dispersion

Pulses carrying digital signals have a tendency to spread out as they propagate through optical fiber. This broadening degrades the performance of a communication system and is due to a widespread feature of the propagation of electromagnetic signals, called dispersion. In Figure 1 is shown the broadening of a digital pulse due to dispersion.

At the outset, it is important to distinguish between two types of dispersion: these are chromatic dispersion and modal dispersion. The type that we will discuss in this FAQ is chromatic. Roughly stated, chromatic dispersion is the phenomenon in which the speed of light is noticeably different for different frequencies. In this case, if a modulation signal is carried on a superposition of carrier waves of various frequencies, it eventually blurs and information is lost. Since the signal sources available today are not perfectly monochromatic, some amount of chromatic dispersion is present.

As was just mentioned, apart from the chromatic variety, the other category of dispersion is modal dispersion. Modal dispersion has to do with the fact that, at a given light frequency or wavelength, the waves can evolve from one mode to another as they propagate through the fiber or waveguide. In general, even in single-mode fiber, several distinct modes can be present. An example of this effect is polarization mode dispersion (PMD). Polarization mode dispersion arises in waveguides which are not perfectly cylindrical (due to, say, imperfections in the fiber manufacturing process) and which therefore display some amount of uncontrolled birefringence. In these circumstances, light in one polarization mode might have a speed different from that of light in another polarization mode. The result is that signal pulses that contain light in more than one polarization mode tend to broaden as they propagate through the system, with each mode traveling at its own pace. If this broadening is too severe, the bit error rate can become unacceptably high.

Let us return to our discussion of chromatic dispersion. Chromatic dispersion can be divided into two effects: material dispersion and waveguide dispersion. Material dispersion is the dispersive effect that one observes while studying the propagation of light waves in a large isotropic volume of bulk material; the speed of the light waves in the medium understudy varies as a function of frequency. On the other hand, waveguide dispersion arises due to the geometric configuration of the waveguide. For example, consider two different waveguide geometries (say a cylindrical optical fiber on the one hand and a rectangular guide in an integrated optical circuit on the other hand). Even if the materials in question are very similar, the dispersive effects in one waveguide geometry will in general be different from those in the other configuration. That is, at the same frequency, there is a difference between the speeds of the two waves.

The various categories of dispersion (material chromatic, waveguide chromatic, and modal) are organized schematically in Figure 2.

Mathematical Description of Chromatic Dispersion

In the preceding section, we spoke of chromatic dispersion as being the phenomenon by which light of different wavelengths or frequencies propagates with different speeds through a waveguide or material. We will now bring some more precision to this idea.

Both types of chromatic dispersion -- material and waveguide -- can be understood with the same general formalism. A nice way to understand the mathematics of dispersion is to start with the familiar concept of amplitude modulation (AM). By this, we mean the modulation of a high-frequency carrier wave by varying its amplitude with some low-frequency signal. In the absence of modulation, the carrier wave's amplitude C is given at any point in time and position by the sinusoid

C(t,z) = sin(ωt - β(ω)z)

We are to imagine that the wave is emitted at a point z = 0 and that we can receive it at some distant point, z. The units of ω are s^-1 and the units of β are m^-1. The parameter β is the propagation function. It is a function of frequency, as in β(ω), and the explicit expression for β as a function of ω is called the dispersion relation. In the case of light propagating through free-space, the dispersion relation is simply

β(ω) =		ω

		c

so that the carrier wave is described by sin(ω(t - ^z/_c)).

We now amplitude-modulate the carrier by (slowly) varying its amplitude as it is emitted. Let us choose to modulate with a sinusoidal signal of frequency μ, cos(μt). The frequency of the modulating signal is much smaller than the frequency of the carrier. That is,

μ << ω

At the source, z = 0, the modulated wave is given by the product of the modulation and the carrier:

(t,z = 0) = cos(μt) • sin(ωt)

This can be re-expressed as a sum of two pure sinusoidal functions:

(t,z = 0) = ½sin((ω - μ)t) + ½sin((ω + μ)t)

where we have used some basic trigonometric identities to pass from one line to the other.

Since each sinusoidal frequency component propagates independently (according to Maxwell's equations), the modulated wave looks like this at some point z:

(t,z = 0) = ½sin((ω - μ)t - β(ω - μ)z) + ½sin((ω + μ)t - β(ω + μ)z)

where we have simply inserted the propagation function in each argument. We now assume that the dispersion relation can be expressed with some accuracy as a Taylor expansion about ω:

β(ω ± μ) = β(ω) ± μβ′(ω)

(The appropriateness of the Taylor expansion hinges on the modulation frequency being much smaller than the carrier frequency.) Putting the expansions into the formula for (t,z) and applying the trigonometric identities again, one obtains the simple two-factor form

(t,z) = cos(μ(t - β′(ω)z)) • sin(ωt - β(ω)z)

This final form reveals that at some distance z away from the source, the wave looks like a modulated carrier wave, with the modulation signal having a frequency μ traveling along at a speed (β′(ω))^-1. This can be seen by comparing the first factor to the generic form

cos(μ(t - ^z/_ν ))

where ν is the velocity of a sinusoidal wave. The modulated carrier wave for a carrier of a single frequency ω is shown in Figure 3.

The above analysis is directly analogous to the case in optical telecommunications. The carrier corresponds to the light itself, and the amplitude modulation signal corresponds to the pulses that carry the digital information.

Now, as we said earlier, the frequencies of the sources in optical telecommunications are not perfectly monochromatic. So we must consider what happens if, rather than just one single carrier frequency ω, there is actually a spectrum of carrier frequencies. For simplicity, let's say that there are several discrete carrier frequencies, ω₁, ω₂, ω₃, etc., and that each carrier wave has the same amplitude. Then, the modulated amplitude received at some point z would be just the sum of several contributions, each with a different carrier frequency

= Σ_jcos(μ(t − β′ (ω_j)z)) • sin(ω_jt − β(ω_j)z)

We see that the modulation signal is proportional to

Σ_jcos(μ(t - β′(ω_j)z))

and, reading off directly, we note that for each carrier frequency ω_j, the modulation signal travels at a speed given by (β′ (ω_j)) ^-1. This information allows us to now make a key observation regarding dispersion theory: if (β′ (ω_j)) takes a slightly different value for each value of ω_j, then each carrier wave will carry the modulation signal at a slightly different speed. A signal which starts out being a perfect cosine at the source will eventually become blurred at some distance z. The modulated transmission will be a sum of various cosines, all traveling along at somewhat different speeds and all out of phase with one another. This effect, which is obviously undesirable in telecommunications, is chromatic dispersion. If, on the other hand, (β′ (ω_j)) is constant over the spectral range that is being used for transmission, then each carrier wave transports the modulation wave at exactly the same speed, and no blurring occurs.

And so, by considering the familiar situation of amplitude modulation of a carrier signal, we have come upon the heart of the mathematical description of chromatic dispersion. The theory hinges on the dispersion relation β(ω) and on its properties --- in particular, on its derivatives. The dispersion relation describes the chromatic dispersion that a modulation signal undergoes as it is carried on a spectrum of carrier waves through a waveguide such as an optical fiber, or for that matter, any medium other than free-space.

As we were saying above, there is no chromatic dispersion when

β′(ω) = constant

So, the amount of dispersion can be quantified by the second derivative of the dispersion relation, β″(ω). If β″(ω) = 0 in the spectral region of interest, then there is no dispersion.

We have seen that the velocity with which a modulation signal is carried on a carrier wave of frequency ω is given by (β′(ω))^-1. This velocity is called group velocity, ν_g:

ν_g(ω) =	{	dβ(ω)	}	-1

		dω

The inverse of group velocity is formally known as the group time delay

τ_g =		dβ(ω)

		dω

The group time delay is the derivative of the dispersion function. Another way of stating the condition for zero dispersion is therefore

τ_g (ω) = constant

That is, in the case of zero dispersion, the group time delay for a signal is the same, regardless of the frequency of the carrier. A way to quantify chromatic dispersion is therefore via the quantity

dτ_g	=	d²β

dω		d²ω

which expresses deviations away from the case τ_g(ω) = constant. However, since instrumentation generally allows us to measure not the frequency of light but rather its wavelength, it is customary to define the quantity D (chromatic dispersion) as the derivative of group time delay with respect to wavelength:

D =	dτ_g

	dλ

where λ is the free-space wavelength of the carrier.

In passing, we note that since the free-space wavelength λ and the frequency of the light ω are straightforwardly related by the universal wave equation, which is "velocity equals frequency times wavelength." In optics, this is

c =	ω	λ

	2π

Thus, one can easily relate the two quantities	dτ_g	and	dτ_g	. This is done as follows:

	dλ		dω

dτ_g	=	dτ_g	dω	=	dτ_g	{	-	2πc	}

dλ		dω	dλ		dω			λ²

where we have used the chain rule. In general, the derivative of any quantity X with respect to frequency is related to the derivative with respect to free-space wavelength by

ω	dX	=	-λ	dX

	dω			dλ

In the above discussion, the units of group time delay are seconds per meter. The total time delay of a component or optical fiber is found by multiplying the time delay by the length of the device in question, which gives the time delay of a specific passive component in seconds. Similarly, in the logic of the above discussion, the units of dispersion D are seconds per meter per meter, so that the dispersion due to a specific passive component is given is seconds per meter.

Most frequently, one uses the following practical units. For group time delay one uses picoseconds per kilometer. For dispersion, the most practical units are often picoseconds per nanometer per kilometer

Dispersion in Materials and Waveguides versus the Absence of Dispersion in Free Space

Let us equate the expression for a sinusoidal wave in free space to the basic definition for the dispersion relation

sin(ω(t - ^z/_c)) = sin(ωt - β(ω)z)

We see that the dispersion relation for a wave propagating free space is

β(ω)	=	ω

		c

This is a linear relation. In a dispersive material, the dispersion relation is some generally nonlinear function. The inverse of the derivative of the dispersion relation is the group velocity which gives the speed with which a modulation signal travels on a carrier wave of frequency ω. In Figure 4 are shown the linear dispersion relation for propagation of light in free-space and the nonlinear dispersion relation for the propagation of light in some dispersive material.

Looking again at the expression for the modulated wave (derived previously)

(t,z) = Σ_jcos(μ(t − β′ (ω_j)z)) • sin(ω_jt − β(ω_j)z)

we see that, if the medium is free-space, the result reduces to

(t, z) = cos(μ(t − z/c)) • Σ_jsin(ω_j(t − z/c))

This describes the propagation of a sum of a spectrum of carrier waves (of various frequencies ω_j and each one of velocity c) all modulated by the same signal of frequency μ. The modulation signal also travels with velocity exactly equal to c. This is to say that in the idealization of the non-dispersive medium of free-space, there is no chromatic dispersion, and what is more, the group velocity of a signal carried by light is identical to the velocity of the carrier.

It is possible to achieve zero-dispersion in media other than free-space. The necessary condition is simply that the dispersion relation must be linear in the spectral region that is being used for transmission.

Group Velocity versus Phase Velocity, Group Index versus Phase Index

In reading introductory texts on such topics as dispersion and wave propagation in waveguides, one is faced with the task of sorting out the definitions of "group" quantities and "phase" quantities. The difference can perhaps be most easily understood by considering the expression describing a modulated wave

(t,z) = cos(μ(t - β′(ω)z)) • sin(ωt - β(ω)z)

This expression is a product of two factors: the carrier factor sin(ωt - β(ω)z), and the signal factor cos (μ(t - β′(ω)z)). The basic idea to retain in regard to the difference between "phase" and "group" is that the quantities in the carrier factor are "phase quantities" and those in the signal factor are "group quantities."

It is important to get the distinction ironed out in one's mind. Phase quantities pertain to the movement of the peaks and valleys of the electric and magnetic fields of the carrier wave. Phase velocity can be thought of in the following way: Suppose that one could somehow "photograph" the movement of the peaks of the fields, by taking a series of snap shots in rapid succession; then by knowing the time interval between the images and by measuring the progress of the peaks, one could calculate the phase velocity. (Similarly, the wavelength of the carrier is also a phase quantity. It is simply the distance between peaks in the electric field, and in principle it could be measured off from our imaginary snap shots of the electric field amplitude.) In Figure 5, we illustrate the idea of taking an imaginary sequence of snap-shops of the electric field in order to measure "phase quantities" such as the phase velocity and carrier wavelength. The phase index of refraction, n(ω), is defined to be ratio of the speed of light in free-space to the phase velocity ν(ω):

n(ω)	=	c

		ν(ω)

The following equation displays several useful relations between the various "phase quantities":

n(ω)	=	c	=	2πc

		ν(ω)		ωλ(ω)
	=	cβ(ω)

		ω

These relations hinge on the definition of the phase index and on the universal wave equation. They can be used to rewrite the carrier factor of the modulated wave, as may be required for the comprehension of various points.

In the preceding paragraph we discussed the phase quantities. We now turn to the group quantities. Group velocity, as we have seen, is the speed at which useable signals can be carried on the carrier wave. It is the speed at which a deliberately initiated disturbance on a sinusoidal carrier (such as in AM radio transmission) can propagate from an emitter to a receiver. When talking about group velocity, we are referring only to the speed of propagation of signals whose frequencies (such as μ in our present discussion) are much smaller than the carrier frequencies ω. In practical situations in engineering, the group velocity is slower than the phase velocity. Analogously to the phase index of refraction, we also define the group index of refraction, N(ω). It is the ratio of the speed of light in free-space to the group velocity ν_g(ω):

N(ω)	=	c	=	c	dβ(ω)

		ν_g(ω)			dω

We have seen that the condition for zero dispersion is

D	=	d	dβ	=	0

		dλ	dω

Thus, another way of expressing the situation of zero dispersion is

dN	=	0

dλ

In the theory and fabrication of FBGs, one often sees the standard equation

λ_BR = 2n_eff Λ^(FBG)

which relates the Bragg wavelength (given as a free-space wavelength) and period of the FBG. The parameter n_eff is the index of the core mode of the fiber. One might wonder: Is n_eff the phase index or the group index?

The answer can be found by first realizing that a more intuitive manner of writing this relation would be

(λ_BR)^infiber = 2Λ^(FBG)

where (λ_BR)^infiber refers to the wavelength of the electric field measured not in free-space but right inside the fiber. Using the universal wave equation to relate various phase quantities, one can easily convince oneself that

(λ_BR)^{free space}	=	c	=	n(ω)

(λ_BR)(ω))^infiber		ν(ω)

Putting this information into (λ_BR)^infiber = 2Λ^(FBG) brings us to conclude

(λ_BR)^{free
space} = 2n(ω)Λ^(FBG)

We have recovered the standard equation for the Bragg wavelength of an FBG, and in so doing, we have discovered that n_eff appearing in the standard equation is actually a phase index.

Figures for this FAQ

Figure 1 - The broadening of a digital pulse due to dispersion

The drawing shows the broadening of a digital pulse due to dispersion. The digital pulse is a modulation in the amplitude of a carrier wave (light).

Figure 2 - Various types of dispersion

The various types of dispersion are displayed here. Dispersion is divided into chromatic and modal. Chromatic dispersion is further sub-divided into two types: material and waveguide.

Figure 3 - A sinusoidal carrier wave modulated by a sinusoidal signal

The figure shows a carrier wave (high frequency mono-chromatic sinusoidal wave) modulated by a slowly varying sinusoidal wave.

Figure 4 - Dispersion Relations

In this figure we see the linear dispersion relation for propagation of light in free-space in contrast to the nonlinear dispersion relation for the propagation of light in some dispersive material. For the case of light in free-space, the slope of the straight line is simply 1/c, and the function passes through the origin.

Figure 5 - The concept of phase velocity

Phase velocity expresses the speed of the movement of the peaks and valleys of the electric and magnetic fields. If one could somehow "photograph" the movement of the peaks of the fields by taking a series of snap shots in rapid succession, then, by knowing the time interval between the images and by measuring the progress of the peaks, one could calculate the phase velocity. In a dispersive medium, one finds that the speed at which a modulation signal travels is generally different from (slower than) the phase velocity of the carrier light.

Cleaning StockerYale's Apodized Phase Masks

Place apodized phase mask in "Piranha" acid solution, for 5 minutes. The Piranha formula consists of 1 part by volume of H₂O₂ and 6 parts by volume of H₂SO₄. The concentration of the peroxide should be 30%, and the concentration of the sulfuric acid should be 49%. The presence of the peroxide causes the desired effect that the solution heats to 70º C. It is possible to do the cleaning using sulfuric acid only; in that case, it is preferable to heat the acid to 70º C. The reason for the temperature specification is that the process is much faster at 70º C than at room temperature.
Rinse in deionized water for 10 minutes.
Rinse phase mask in isopropanol for 5 minutes. Some workers have used an acetone bath for this step, but we find that isopropanol dries more slowly and therefore leaves less residue.
Blow dry with nitrogen or clean air.

Notes

If a more effective cleaning procedure is required, StockerYale offers a phase mask cleaning service.
Concerning the isopropanol bath, do not use the ultrasonic agitation for apodized phase masks. Over time, the ultrasound might damage the apodization profile.
The optimum concentration of the peroxide is 30% and the optimum concentration of the sulfuric acid is 49%, for both apodized and unapodized phase masks.
The Piranha composition for apodized phase masks is H₂O₂:H₂SO₄ (1:6). For unapodized phase masks, this composition is 1:4.

Cleaning StockerYale's Unapodized Phase Masks

Place (unapodized) phase mask in "Piranha" acid solution, for 10 minutes. The Piranha formula consists of 1 part by volume of H₂O₂ and 4 parts by volume of H₂SO₄. The concentration of the peroxide should be 30%, and the concentration of the sulfuric acid should be 49%. The presence of the peroxide causes the solution to heat to 70º C. It is possible to do the cleaning using sulfuric acid only; in that case, it is preferable to heat the acid to 70º C. The reason for the temperature specification is that the process is much faster at 70º C than at room temperature.
Rinse in deionized water for 10 minutes.
Place phase mask in isopropanol ultrasonic bath, 200 W, high frequency (40 KHz) for 10 minutes. Some workers have used an acetone bath for this step, but we find that isopropanol dries more slowly and therefore leaves less residue.
Blow dry with nitrogen or clean air.

Notes

If a more effective cleaning procedure is required, StockerYale offers a phase mask cleaning service.
Concerning the isopropanol bath, do not use the ultrasonic agitation for apodized phase masks. Over time, the ultrasound might damage the apodization profile.
c) The optimum concentration of the peroxide is 30% and the optimum concentration of the sulfuric acid is 49%, for both apodized and unapodized masks.
The Piranha composition for upapodized phase masks is H₂O₂:H₂SO₄ (1:4). For apodized phase masks, this composition is 1:6.

The ITU Grid Channel Spacings: Converting from Frequency Channel Spacings to Wavelength Channel Spacings

The ITU grid involves channel spacings given in terms of frequency. The spacings currently of interest are 50 and 100 GHz. However, in the fabrication of FBGs and in the purchasing of phase masks, it is more convenient to talk in terms of wavelength. We provide here an explanation of how to convert back and forth from frequency to wavelength intervals.

The relationship between frequency intervals (such as 50 GHz and 100 GHz) and wavelength intervals is

δλ = -8.01 x 10^-3 δƒ

with δλ in nm and δƒ in GHz. So, for a 50 GHz spacing the wavelength spacing is 0.4 nm, and for a 100 GHz frequency interval, the corresponding wavelength interval is 0.8 nm.

The rest of this FAQ is devoted to deriving and justifying the above formula. We start with the universal wave equation for any wave (sound, light, etc.):

velocity = frequency • wavelength

where frequency is in cycles per second (as opposed to radians per second), velocity is in meters and wavelength is in meters. In optics, we write

c = ƒ λ

where c is the speed of light in the vacuum, and where λ is the free-space wavelength. We may easily calculate the relationship between small intervals in ƒ and small intervals in λ. Taking the above equation and solving for λ, we have

λ =	c

	ƒ

from which we can calculate

δλ =	∂λ	δƒ

	∂ƒ

δλ =	c	δƒ

	ƒ ²

Now, since ƒ = ^c/_λ, we can write this latter equation in terms of wavelength

δλ =	λ²	δƒ

	c

Taking λ to be 1550 nm, c to be 3 x 10¹⁷ nm/s, we have

δλ = −	(1550)²	δƒ

	3 x 10¹⁷

= −8.01 x 10^-12 δƒ

with δλ in nanometers and δƒ in Hertz.

Thus, for a frequency interval of 50 GHz (50 x 10⁹ Hz), we have a wavelength interval of

δλ = −8.01 x 10^-12 • (50 x 10⁹)

= 0.4 nm

and, similarly, for a frequency interval of 100 GHz, we have a wavelength interval of 0.8 nm.

Measuring the effective index of refraction of a fiber for FBG fabrication

When one sets out to order a phase mask for FBG fabrication, in order to specify the correct phase mask period one must know the index of refraction of the fiber in which the FBG will be written. More precisely, the index of refraction that one needs is the effective core mode phase index (and not the group index), denoted by n_eff. One needs to know this index with a certain precision because it is closely related to the most important design specification of the FBG, namely the Bragg wavelength λ_BR. This relation is given by the familiar formula

λ_BR	= 2n_eff Λ^(FBG) = n_eff Λ^(PM)

where Λ^(FBG) is the period of the FBG, Λ^(PM) is the period of the phase mask used in fabrication, and n_eff is the effective phase index of the core mode of the fiber in question.

Method I

One can measure the effective index n_eff of the core mode of a fiber using two simple methods, both involving phase masks which must be already possessed by the FBG manufacturer. In the first method, n_eff is determined by writing a very weak grating with a single laser pulse. If one knows the phase mask period Λ and the Bragg wavelength λ_BR of the resulting FBG, one can then obtain the index from the basic relation n_eff = ^{^λ_BR}/_{_Λ}.

What precision can one expect in such measurements? The period of a phase mask is known to within ± 0.02 nm on a typical period of 1060 nm. With contemporary OSAs, the Bragg wavelength of the FBG can be measured with a precision of ± 0.010 nm at a wavelength of roughly 1550 nm. Thus, adding the relative precisions, one finds that the effective index can be derived to within 3 parts in 10⁵.

Method II

The effective mode index can also be measured using a second method, which hinges on comparing the Bragg wavelengths of two weak FBGs. The idea is to write two gratings using the same phase mask (of period Λ), one grating in the fiber under measurement, the other in some standard fiber for which n_eff is known to high precision. Since for both gratings, the quantity ^{^λ_BR}/_{_{n_eff}} is equal to Λ, we have

n_eff = (n_eff )_ref	λ_BR

	(λ_BR)_ref

This method has the advantage of being independent of the phase mask period Λ, and of depending only on the resolution of the OSA used to determine the Bragg wavelengths of the two FBGs. Adding the relative precisions, one sees that the effective index can be derived to within 2 parts in 10⁵.

The reason why these measurements must be performed with the weakest possible gratings is that, otherwise, the dc component of the grating index modulation contributes non-negligibly to the effective index that is measured.

The FBG DC Index Modulation and its Contribution to n_eff

When one records an FBG, of course the fiber index receives an ac modulation, but there is also a dc modulation, and this modulation contributes to the index n_eff. Assuming that the zeroth order is perfectly nullified by the phase mask, then the dc modulation corresponds to one half of the peak-to-peak modulation, as shown in Figure 1. The ac peak-to-peak modulation in FBGs is commonly of order 10^-4 or 10^-3, and the dc modulation is therefore of the same order of magnitude. This dc contribution is not non-negligible, and turns out to be a source of uncertainty in regard to the correct value of n_eff to take. This in turn bears upon the calculation of the required phase mask period. These questions must be resolved for each application.

Figures for this FAQ

Figure 1. In the recording of an FBG, the fiber index receives both an ac modulation and a dc modulation. The latter contributes to the index n_eff. If the diffractive efficiency of the phase mask is ideal, the dc increase corresponds to exactly one half of the peak-to-peak ac modulation. The ac modulation in FBGs is commonly of order 10^-4 or 10^-3, so the dc modulation is of the same order of magnitude and is therefore a significant contribution to n_eff.