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Quarter Wave Plates and Half Wave Plates

Wave Plates, also known as Retarders, are essential optical components made of birefringent materials that are designed to manipulate the polarization state of light. A waveplate alters the phase relationship between orthogonal components of polarized light, introducing a specific phase retardance. This phase difference changes the polarization state of the light without affecting the light intensities. The two most common types are Quarter Wave Plates and Half Wave Plates. The “quarter” and “half” in the names refer to the fractional wavelength of the phase retardance. The primitive difference between a quarter wave plate and a half wave plate is that a quarter wave plate introduces a retardation of a quarter wavelength between the two orthogonal polarization components, whilst a half waveplate introduces a retardation of half of a wavelength. In this article, we will help you to understand quarter wave plates and half wave plates, guide you through the working principle of quarter waveplates and half waveplates, how quarter waveplates and half waveplates change the polarization states, and explain the uses of quarter wavelength plates and half wavelength plates.

waveplates and retarders

Figure 1. Waveplates from Shalom EO


What is a Waveplate?

A Wave plate/Waveplate, also known as a Retarder, operates on the principle of birefringence—birefringence is characteristic of certain optically non-isotropic materials where the refractive index is dependent on the polarization and propagation direction of light (there are also exceptions like Fresnel Romb Retarders which operate based on the principle of Total Internal Reflection/TIR). This unique characteristic enables wave plates to introduce a phase difference between orthogonal components of polarized light.

Waveplates can be birefringent crystal waveplates like quartz waveplates, mgf2 waveplates, or made of a combination of two different birefringent materials, or birefringent Polymer Waveplates (Which are also available in Shalom EO). When light enters these birefringent materials, the refraction depends on the polarization direction. This is because light, regarded as being composed of oscillating electric fields and magnetic fields, interacts with the asymmetric micro-structures of the birefringent materials in different manners. Incident light hitting the optical surface of the material at non-normal angles will be resolved into two components with orthogonal polarization directions. The two components then encounter two distinct refractive indices: the Ordinary Refractive Index no, and the Extraordinary Refractive Index ne.

From the two distinct refractive indices, no and ne arise the two principal optical axes of a wave plate, the Fast Axis and the Slow Axis. The axis with the lower refractive index where light polarized in this direction propagates at a higher speed is the fast axis, and the axis with the higher refractive index where light polarized in this direction propagates at a slower speed is the slow axis.

The difference in traveling speeds gives rise to the phase difference, or so-called Phase Retardance/Retardation between two orthogonal polarization components, the phase retardation leads to the resultant change in the polarization state.

Interaction of Light within a Wave Plate

As mentioned above, when a wave of light enters a wave plate, it splits into two orthogonal components: one aligned with the fast axis and the other with the slow axis. These two components travel at different speeds due to the birefringence, resulting in a phase shift between them.

The phase retardation depends on:
-The thickness of the wave plate (d);
-The birefringence (Δn=ne−no);
-The wavelength of the light (λ)

Therefore, the Phase Difference δ of can be calculated using the formula:
δ=(2π*Δn*d)/λ

The Phase Difference δ here in the context of waveplates is called “Retardance” or “Retardation”, and the retardation value of a waveplate is usually expressed as a fractional of wavelength or waves, or written in radians. For example, a quarter wave/wavelength=λ/4, half wave/wavelength=λ/2, octadic wave/wavelength=λ/8, and so on.  The retardation values of waveplates are the most fundamental parameters when it comes to choosing a waveplate or retarder, the retardation value modifies the superposition of the two components, altering the polarization state of the outgoing light. Depending on the phase shift, the light can emerge with a new polarization state, such as linear, circular, or elliptical polarization.

From the formula, it can be deduced that by carefully engineering the thickness of the waveplate and utilizing certain materials with specific birefringence properties, one can obtain the desired retardation value at the wavelength of interest, and hence manipulate the polarization state of light.

There is an additional point worth noting: waveplates can be divided into multiple order waveplates and zero order waveplates. A multiple order waveplate made of a single birefringent material generates the desired phase retardation plus the phase shifts of several excess integral wavelengths. A zero order waveplate, on the other hand, consisting of two waveplates made of the same birefringent material with their optical axis perpendicular to each other, is designed to be thinner and produce precisely the desired phase retardation without the excessive wavelengths. For example, a multiple order quarter wave plate introduces in total one quarter and several integers of wavelength retardation, whilst a zero order quarter wave plate introduces exactly a retardation of one-quarter wavelength between the fast and slow components. In comparison, a multiple order quarter wave plate is thicker and easier to handle, but its retardation will shift when wavelength varies or temperature changes. A zero order waveplates provide more stable retardation corresponding to spectral variation. 

There are also true zero order wave plate, achromatic and super achromatic waveplates, you can click here to to learn more about Different Types of Waveplates.

Understanding Quarter Wave Plates and Half Wave Plates

The most frequently used types of waveplates are Quarter Wave Plates and Half Wave Plates, because users can pile several quarter waveplates and half waveplates to achieve a wide range of phase shifts (retardations) that are either greater or smaller than what a single waveplate can provide.

Quarter Wave Plate
A Quarter Wave Plate, or Quarter Waveplate is a waveplate that introduces a phase retardation of  90° (or λ/4, a quarter wavelength) between the two orthogonal polarization components of the incoming light (i.e. the polarization components aligned with fast axis and the polarization components aligned with the slow axis). 

The primary function of a quarter waveplate is converting linear polarization into circular polarization and vice versa. Note that to convert linear polarization into circular polarization, the polarization plane of the incident linearly polarized light must be oriented at an angle of 45° to the slow axis or the fast axis of the quarter wave plates, if not, the resultant polarization will be elliptical instead of circular.

The process of how a quarter waveplate converts linear polarization into circular polarization can be simpler to explain if we perceive the polarization components of light as electric vectors. 

First, let us look at what is circular polarization. In electrodynamics, the electric field vector describes the strength and direction of the electric field at a given point in space. For circular polarization, the tip of the electric field vector traces a circle in a plane perpendicular to the direction of wave propagation as time progresses. The phase difference between the two perpendicular components determines the orientation of the electric field vector at any moment. 

If we follow the tip of the resulting electric field vector formed from the superposition of the fast and slow components, we observe that it describes a helix. This happens because the wave consists of two perpendicular components of the electric field that are equal in amplitude but have a 90° phase difference. This helical motion is the result of the constant phase difference of one-quarter wavelength (lamda/4) between the two orthogonal polarization components.

Below is how to use a quarter wave plate to convert linear polarization into circular polarization:
Begin with placing a polarizer in the path of the beam. Rotate the polarizer to discover the plane of polarization of the incident light. This establishes a reference for aligning the waveplate. Position the quarter wave plate between the light source and the polarizer. Rotate the quarter waveplate around the beam axis to find the orientation where the polarizer blocks the most light. At this position, the optical axis of the waveplate is aligned with the plane of polarization. Rotate the quarter-waveplate by 45 from the extinction position. At this angle, the quarter waveplate introduces a phase retardation of one-quarter wavelength (lambda/4), converting the linear polarization into circular polarization. 

quarter waveplate circular polarization

Figure 2. Conversion of linear polarization into circular polarization using quarter waveplates

Half Wave Plate: Rotating Polarization States

A Half Wave Plate or a Half Waveplate is a waveplate that introduces a phase difference of 180° (or half a wavelength) between the fast and slow polarization components of light passing through it. This phase retardation is the result of the different refractive indices no and ne of the birefringent material, enabling precise control over the polarization state of light.

A half waveplate can be used to rotate the polarization direction of linear polarization.

When a linearly polarized wave is incident on a half wave plate with its polarization plane at an angle θ respective to the fast axis of the plate, the light beam undergoes a 180° phase shift between its fast and slow components due to the half-waveplate. As a result, the polarization direction of the wave is rotated by 2θ. The output wave remains linearly polarized but now at an angle 2θ relative to the fast axis.

Below is a pratical example: If a linearly polarized wave is incident on a half-waveplate with a polarization direction at θ=30° relative to the fast axis, the output wave will be linearly polarized at an angle of 60° relative to the fast axis (since 2×30°=60°). 

A common application of half wave plates is the rotation of vertical polarization planes into horizontal polarization planes for lasers. Most large ion lasers are of vertical polarization, and particularly when the laser system is too bulky to be spun, the half-wave plate offers a convenient approach to change the polarization plane into horizontal polarization. One can just use a half waveplate oriented such that its fast (or slow) axis is at a 45° angle to the vertical polarization, the output light will be rotated by 2x45°=90°.

For circularly polarized light, a half-wave plate can also effectively change its handedness. Specifically, if right-handed circularly polarized (RHCP) light passes through a half-wave plate, it will emerge as left-handed circularly polarized (LHCP) light, and vice versa. This transformation occurs because the half-wave plate shifts one component of the electric field vector relative to the other by half a wavelength, effectively reversing the direction of rotation of the electric field vector.


Important Factors to Consider When Choosing a Quarter Wave plate and Half Wave Plate

1. Wavelength

The wavelengths of the incident light are crucial in determining the performance of the quarter wave plates and the half wave plates. The thickness and the birefringence of the material used to fabricate waveplates are designed for specific wavelengths. Due to chromatic dispersion (click here to learn more about what is Chromatic Dispersion), incident light of different wavelengths will encounter different refractive indices, and as the phase shift is derived from the difference of propagation velocities due to the two refractive indices no and ne, the phase shift the waveplates introduce is wavelength-dependent. In most cases, waveplates are optimized for a specific design wavelength. Using a waveplate at a wavelength far from its design wavelength can lead to imperfect phase shifts and degrade functionalities. For example, a quarter waveplate with a design wavelength of 405nm will not introduce the desired one-quarter wavelength retardation at other wavelengths besides 405nm. 

However, the wavelength dependence of the waveplates is also related to the types of waveplates. Some types of waveplates are less sensitive to wavelength shifts. For example, the retardation performance of a Zero Order Quarter Waveplate is less wavelength-dependent than a multiple order waveplate. There are also achromatic waveplates, which are a type of waveplates composed of two waveplates made of two different materials with complementary birefringent properties. An achromatic quarter wave plate is designed to provide a flat retardation response of lambda/4 over a broad wavelength range (typically hundreds of nanometers for standard achromatic waveplates).


2. Temperature

Temperature variations can affect the optical properties of the materials used in the waveplates. The refractive index, thickness, and other material properties can change with temperature, leading to deviations in the phase shift. If the environment where the quarter wave plate or the half wave plate is to be applied is the temperature fluctuation is violent, one should choose a zero-order quarter wave plate and half wave plate or a true zero order quarter wave plate and half wave plate instead of multiple order quarter/half waveplates.


3. Polarization Requirements

The polarization state you need to generate or manipulate will guide the selection of the waveplates.

Circular Polarization: A quarter wave plate can convert linear polarization to circular polarization when aligned at 45° to the input polarization direction.

Elliptical Polarization: A quarter wave plate can also generate elliptical polarization if the input polarization is not aligned with the fast axis.

Rotation of Polarization: A half wave plate is used to rotate the polarization direction of the light by an adjustable amount. Its action depends on the orientation of the fast axis.


4. Angle of Incidence

The angle at which light strikes the waveplate affects the phase shift and polarization transformation. Most waveplates are designed for normal incidence (light perpendicular to the surface). However, if the light is incident at an angle, the waveplate's optical properties can change, causing deviations in the expected phase shift.

At large angles, optical aberrations can arise, particularly for large beams, affecting the polarization transformation. It’s important to ensure the waveplate has a sufficient tolerance to handle the expected range of incident angles without significant performance degradation.

Some particular types of waveplates, for example, polymer waveplates, offer better tolerance for deviations than crystalline waveplates.


5. Mounting

Proper mounting is crucial to ensure the precise alignment of the fast axis of the waveplate with the light's polarization. Misalignment of the waveplate can lead to incorrect phase shifts or polarization rotations. The mounting should maintain stable positioning over time and temperature variations to avoid shifts in the polarization state. If fine-tuning of the polarization is required, some waveplates come with adjustable mounts that allow for precise rotation of the plate to align its fast axis.


The Applications of Quarter Waveplates and Half Waveplates:

The Applications of Quarter Wave Plates:

1. Circular Polarizer:

The most typical usage of a quarter waveplate is to use it as a circular polarizer, transforming linear polarization into circular polarization. This transformation can be achieved by using a quarter wave plate and a polarizer. 

As shown in the picture below, after the laser beam is incident at a normal angle and passes through the linear polarizer, it is transformed into linear polarized light; the modulated linear polarized light is then incident at a normal angle on the 1/4 wave plate, the output light intensity remains unchanged, and the output light becomes circularly or elliptically polarized light, where the major axis direction and rotation direction are determined by the angle between the polarization direction of the polarizer and the fast axis direction of the 1/4 wave plate. When the angle between the fast axis of the wave plate and the polarization direction of the polarizer is +45°, the output beam is left-handed circular polarization; when the angle between the fast axis of the wave plate and the polarization direction of the polarizer is -45°, the output beam is right-handed circular polarization.

quarter wave plate use
Figure 3. Using a quarter waveplate as a circular polarizer

2. Optical Isolator
A quarter wave plate is an essential element in many optical isolator designs, typically used alongside a linear polarizer or a polarizing beamsplitter cube. An optical isolator is a device designed to eliminate or minimize undesired back reflections in optical systems, particularly in laser setups. These reflections can destabilize the laser source or interfere with the operation of optical components. In an optical isolator, the quarter wave plate serves a crucial role by converting and re-converting the polarization state of light. Together with a polarizer or a polarizing beamsplitter cube, the quarter waveplate ensures that unwanted back reflections are blocked or redirected.


quarter wave plate use

Figure 4. Using a quarter waveplate as an optical isolator


3. Optical Attenuator:

A quarter waveplate can be utilized as an optical attenuator when used in combination with two polarizers. First the laser beam is incident normally into the first polarizer, and the emerging light is converted into linear polarized light. Second, the modulated linear polarized light is incident into a quarter waveplate, and the resultant light after the processing of the quarter wavelength plate becomes elliptically polarized light. When eventually being filtered by the last polarizer, the light intensity changes. The ratio at which the light intensity is attenuated is related to the polarization state of the incident light.

quarter wave plate use

Figure 5. Using a quarter waveplate as an optical attenuator


The Applications of Half Wave Plates:

1. Continuous Polarization Rotator: 

A half wave plate can be used as a continuous polarization rotator when mounted on rotators (the rotator can be either manual or motorized). And as the amount of rotation is twice the angle between the incoming polarization direction and the fast axis of the half-wave plate, one can achieve continuous adjustments to the rotation angles by tuning the angles between the incident polarization and the fast axis of the half wave plates.


2. Variable Ratio Beamsplitter

A half-wave plate, when used in conjunction with a polarizing beam splitter (PBS), can function as a variable ratio beamsplitter, offering precise and continuous control over the splitting ratio of light. 

The PBS separates light into its polarization components: P-polarized light is transmitted, while S-polarized light is reflected. The half wave plate rotates the polarization angle of the incoming beam. By adjusting the orientation of the half wave plate’s fast axis, the polarization state of the input light can be divided into arbitrary proportions of P- and S-polarized light. Specifically, the fast axis of the HWP, aligned at an angle θ relative to the input polarization, rotates the polarization by 2θ, allowing the relative power ratio of the transmitted and reflected beams to be varied continuously from 0:100 to 100:0. This method enables fine control over the splitting ratio without requiring mechanical adjustments to the PBS, making it an efficient solution with minimal optical losses. 


Combined Usage of A Quarter Wave Plate and A Half Wave Plate as Polarization State Generator:

Quarter wave plate and half waveplate can be used together with a polarizer to form a polarization state generator. The polarizer first defines the linear polarization direction of the incoming light. The quarter waveplate then transforms the linear polarization into either circular polarization (if the polarization is at 45° to the fast axis of the quarter waveplate) or elliptical polarization (if it is at another angle).The half waveplate can then be used to modify the polarization state further, such as rotating the polarization direction or changing the handedness of the circular polarization, depending on its orientation.


quarter wave plate use

Figure 6. Using a quarter waveplate in conjunction with a half waveplate as an polarization state generator


Conclusion

Wave plates, including quarter wave plates and half wave plates, are indispensable tools in modern optics. Whether it's achieving quarter wave plate circular polarization or fine-tuning linear polarization with a half wave retarder, these devices ensure precise and convenient polarization control in critical applications. With a profound understanding the nature of quarter wave plates and half wave plates, researchers and engineers can optimize optical designs and achieve desired polarization effects with ease.

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